Vera effigies THOMAE WILLSFORD: Aetatis suae 46.  Omnia videntur formata ratione Numerorum: Boetius. 
Man's shadow oft does first appear.  
And so does his Effigies here:  
Look in his Books and there you'll find.  
Him in the Mirror of his Mind.  M: Boteler. Re. vaughan sculp:  portrait of Thomas Willsford   

Willsfords' ARITHMETIC, NATURAL, AND ARTIFICIAL: OR, decimals.  
Containing The SCIENCE of NUMBERS, Digested in Three BOOKS.  
Made compendious and facile for all ingenious capacities, viz: Merchants, Citizens, Seamen, Accomptants, etc.  
Together with The Theory and Practice united in a sympathetical proportion betwixt Lines and Numbers, in their Quantities and Qualities, as in respect of form, figure, magnitude and affection: demonstrated by Geometry, illustrated by Calculations, and confirmed with variety of examples in every species.  
By THOMAS WILLSFORD, Gent.  
LONDON, Printed by J. G. for Nath: Brook at the Angel in Cornhill, 1656.   

A PREFACE to all benevolent & pragmatical Artists who are foes to Faction, friends to Truth, and ingeniously studious of Sciences Mathematical.  

MUst daily Pamphlets still invade the Press,  
To vent Romances, almost numberless?  
And Printing to usurp, in tumults strive,  
To make't by Conquest their Prerogative?  
And must I hold my hands, when Truth I see  
Oppressed, and Science with her Monarchy  
Of mystique Arts? whose secrets some contrives  
In vain to make as common, as their Wives:  
These think there is a parity of Wit,  
And if there be not, they will Level it:  
But let them do their worst, and still relate  
Diurnal Acts, in seven Nights out of date,  
Whence they inform their Saints with things untrue  
And telling News, until there's nothing new  
But Zealous Errors under Truth's disguise;  
Like Heathens that offered Pluto Sacrifice:  
These Vice call Virtue, and by this are known,  
They think there's nothing good but what's their own.  
I do expect my Works they'll disallow:  
Or future times, because I writ it now,  
And as a general doom, 'tis understood,  
As if there's nothing here that's true or good:  
In love to Learning, I will do my best;  
And may some Artist vindicate the rest.  
Before 'tis read (lest any censures it)  
I show the subject, and the cause 'twas writ,   

To help young Students (of th' ingenious sort)  
I prove each Thesis, yet the Volume short,  
Which treats of Numbers, and what Art doth do  
Both in the Theory, and the Practice too:  
Above a reason, if you love your ease  
Reject the Demonstrations if you please  
My Rules alone may hidden truth descry,  
But best when as your Judgement dictates why:  
From bare effects, why would you gain applause  
When Reason prompts you for to seek the cause?  
For want of Art does many errors rise  
To dazzle easily dull, and vulgar eyes:  
A Horse, will by the switch, and Tutor's eye,  
Both Add, Subtract, Divide and Multiply.  
A learned beast, outdoes unskilful men  
When only guided by another's pen:  
Which was the motive, I did this impart  
Whereby to show the Principles of Art,  
From whence that you may raise such structures then  
As will prove facile, for ingenious men:  
The ground by Lines and Numbers is surveyed,  
The basis too by Demonstration laid,  
With ease and pleasure thus you may proceed  
And plainly understand what 'tis you read.  
By Line and Level here, I Numbers try;  
Arithmetic thus proves Geometry. (was best which Tract I thought  
Through all my Books  
Now at your service ready to be pressed,  
Finished long since, and do but only stand  
For a Commission, licenced by your hand;  
Or approbation from a public vote;  
Or by a major part at least, of note,  
And that they may the better judge what's fit,  
This renders here a Breviate what I writ,  
All Heights, or Distances, you can descry  
Are found and proved by sage Geometry:  
I writ of War, till I was bid to cease,  
And by the Muses, who are friends to Peace:  
Of Spheres and Stars, and Motions too of these;  
The Earth I measure, and the ample Seas:  
Forerunning Signs of Meteors I declare,  
Their strange effest, and whence their natures were:  
Yet all these Books, but my collections be,  
The form, and method, only is from me.   

All Arts from the Creation did begin,  
And all corrupted, through the stain of sin:  
For when that Adam fell, than faults in all  
As a disease, grew Epidemical:  
And so from thence ('tis to be understood)  
Man's knowledge is commixed with bad and good,  
Which to select, and orderly to place,  
Does all men's actions wonderfully grace:  
Like to th'industrious Bee, that Honey makes,  
And from that Bloom the Spider Poison takes:  
So 'tis in Arts: these times too many sees  
Star-juglers, that do search for Heavens decrees  
Writ in those Orbs, and from their forged skill  
Like Gypsies, or like Witches, which you will,  
With canting words deceive, and seem to know  
As much (I think) as any Star can show:  
They do not prove, nor seem to make appear  
As what Contingens, they with us have here;  
A fatal force of Planets they impart,  
Against Free will, an Heresy of Art,  
Born an abortive Register of Fate,  
A Child of Science, not Legitimate.   

Except my Faults, though here be nothing new,  
Acceptance (Reader) I do crave of you:  
If well, to God the glory I resign,  
Of all my works, for as his gift, they're mine,  
Who does command our Talents to employ  
As for the common good, and so do I:  
But as for all the Errors that they have,  
They're only mine; for which I pardon crave,  
And wish you would correct, what faults you see  
If not more numerous, than my Numbers be:  
Or cancel all, if all shall prove amiss,  
And please to write a better Book than this;  
It will content me (if this cannot do it)  
That it hath moved a better Wit unto it:  
I writ not these, to have a Wreath of Bays,  
But for your profit, and th' Almighty's praise:  
To both these Marks, does my endeavours tend,  
And hope in neither for to miss my end.   

Yet I could wish (kind Reader) you would look  
First on yourself, and then upon my Book;  
The Optic Science, does directly show,  
'Tis by reflection, we ourselves do know;  
And if your errors there you cannot find,  
This Distich always place before your mind:  
All men have faults, from whence, what e'er we do  
Our works are humane, thence erroneous too:  
My oversights, if you correct them true,  
It will be thought humanity in you:  
I will with those most cordially dispense,  
That think themselves are clear from all offence,  
Although they question this; since none we see  
But Fools & Children that from crimes are free:  
As for the Wise, whose judgement's not exact,  
But censures most intentions by the Fact;  
I there may suffer much, but know not well  
As yet the faults, nor here shall never tell  
How to be free; therein I may be crossed,  
To all ungrateful men my labour's lost,  
If by a rigid judge it must not live,  
Before a jury can their Verdict give.   

I'll write no more, lest you my Verses read  
Should be by them deterred for to proceed,  
Or having read them over, think the best  
To be at first, and nothing in the rest;  
My pains of writing 'twill again renew,  
That I have lost my time, and troubled you:  
If so, then let the Press be quite exiled,  
Lest that more virgin paper be defiled;  
And where I have th' unhappy Author been,  
May these be winding sheets to bury'm in,  
And bound to rest, from Sects of silly men,  
As from assaults of a malicious pen;  
May dark Oblivion undertake the trust  
To Shrine them up, with other Deeds in dust,  
And never suffer more to come to light,  
To stand the censure of a Proselyte;  
Yet when these Sects of Zoilists shall be dead,  
May this revive, and be by Artists read:  
Who will my faults with mercy overlook  
And find it was no R writ the Book:  
Whereas all hackney Writers, future times  
Will know their Works are volumes full of crimes:  
And so (perhaps) with scorn will let them lie,  
Or having read them, laugh and throw them by.   

I'll here conclude, and plead no more for these  
But prostrate them to censure as you please;  
And lest that any other bears the blame  
To show whose 'tis I here subscribe my name   
A friend of thine As thou art mine, Thomas Willsford.   

To his honoured Uncle Mr THOMAS WILLSFORD, On his Tract of Arithmetic.  
WHat Numbers may your numerous Art express?  
Which as your Ingeny, is Limitless:  
Whose Corruscations meet the rapt desire,  
And Scorch the Eye that Looks but to admire.  
Arithmetic a double Basis rears,  
For this side Verity; that Reason bears:  
But you the middle Column do remain,  
By Demonstration you the Arch sustain.  
Envy, and Folly, shall as Ciphers stand;  
And while they give your Figures the left hand,  
Augment th' it value; the success of those  
Who by Invectives, do advance their Foes;  
I must suspend my sense, since every Line,  
Commands our gratitude, and speaks a Mine  
Of public good: yet I have hopes I may  
Discharge one thought at least, and freely say,  
While men th' advantage of your Works Divide,  
Our profit and your praise is Multiplied,  
Edward Boteler.   


An INDEX or CONTENTS of the Principals of the first Book in each Section and Paragraph of Natural Arithmetic.  

Sect. 1. Parag. 1.  
THe Definition, Elements, and Species of Arithmetic, page 1  
Addition, the first species of Arithmetic, parag. 2. pag. 6  
Subtraction, the second species, parag. 3. p. 15  
Multiplication the third, par. 4. p. 26  
Division the fourth, par. 5. p. 40   

The second Section  
Shows the definitions, terms, and values of fractions, with their Reductions from one denomination to another, parag. 1. pag. 63  
Addition of fractions, proper, improper, or compounded, parag. 2. p. 86  
Subtraction of fractions either proper, improper or mixed, par. 3. p. 93.  
Multiplication of Fractions, proper, improper or compounded, par. 4. p. 103  
Division of Fractions or broken numbers, either proper, improper or mixed, par. 5. p. 109    

An INDEX showing the principal CONTENTS of each Paragraph in the second Book.  

DEfinitions of quadrat Roots, page 121  
The extracting of the square Root in whole numbers and fractions, parag. 1. pag. 122  
To extract the quadrat Root from an irrational number in any proportion assigned, pag. 128  
The definition of a Cube, p. 136  
The extraction of the Cube Root, both in whole and broken numbers, parag. 2. pag. 137  
General Rules for extracting of Cubique roots, pag. 142  
The extracting of Cubique roots from irrational numbers in any given proportion, pag. 150  
The reason and cause of the operation in extracting of Cubique roots, pag. 159  
How to extract all other Roots in whole numbers or fractions, as Biquadrats, squared Cubes, etc. parag. 3. pag. 162  
How to extract all Roots in any proportion that shall be assigned, pag. 165  
A reference of numbers as in relation to their quantities and qualities, parag. 4. p. 166  
Natural progressions and Arithmetical proportions, with their additions, par. 5. p. 172  
Geometrical progressions and proportions with the addition of their numbers, par. 6. p. 179  
Universal Axioms in Arithmetic, accommodated to the Rules of Practice and proportion, parag. 7. p. 190  
Canons in Arithmetic, with definitions and divers rules of proportion, single and double, direct and reverse, par. 8. p. 201  
The rules of practice direct and reverse, both in whole numbers & fractions, par. 9 p. 212  
The double Rule of Three direct and reversed, performed at two operations, or at one, in whole numbers or fractions, par. 10. p. 220  
The rules of society or companies, both single and double, by whole or broken numbers either in gain or loss, par. 11. p. 226  
The rules of Alligation, or mixture of divers simples according to a common price, or in proportion to any quantity, par. 12. p. 233  
The rules of Alligation, in compounding of Physical simples, according to their qualities in any of their degrees, par. 13. p. 250  
The rules of single false positions, or by false supposed numbers to discover the truth, par. 14. pag. 262  
The double rule of false, wherein by False positions and errors the truth is discovered, par. 15. pag. 269    

An INDEX or CONTENTS of all the principals in each Section or Chapter of this the third Book of Artificial Arithmetic.  

Sect. 1. Chap. 1.  
THe definition of decimals, and the reduction of vulgar fractions to artificial numbers, pag. 287  
Annotations of decimals, p. 291  
Numeration of decimals, p. 293  
Addition of decimals, with whole numbers and fractions commixed together, ch. 2. p. 298  
Subtraction of artificial numbers compounded with integers and fractions, ch. 3. p. 301  
Multiplication of decimals, with fractions, or mixed with integers, ch. 4. p. 303  
Division of decimals with integers commixed, and how to find their quantities, ch. 5. p. 306  
Tables of the Coins, Weights and Measures commonly allowed of in England, ch. 6. p. 314  
decimal Tables of the English Coins, Weights and Measures, and also of Minutes and Seconds calculated to 7 places, ch. 7. p. 321     

A PROEM, or PROLOGUE, to satisfy only the ingenious Students of these liberal Arts, proving the antiquity, excellency, and use of Arithmetic as an element and introduction to other noble Sciences.  
IN the beginning of time, the omnipotent Creator, and Artificer of the World, laid the structure of it in Number, Weight and Measure; and although man's imbecility cannot comprehend this exquisite order, and as the Sacred Records do testify, Man shall never find it, from the Creation unto the Consummation thereof, yet delivered this as a Pattern for us to follow; so here I should not need to make a search, or inquire after the authorities of men, but that some will rather expect Humane reasons, than what is Divine, which is above the reach and capacity of mortals; whereas this is but a dark shadow of Knowledge, made comprehensible to our weak senses, and by methodical Rules accommodated to humane use, which is the tract, or common way that I must move in, and endeavour for to direct all in the nearest path that leads every one to his desired journeys end, in any of these useful Arts, which I undertake to render unto the public view, and place Number first as an introduction to the rest.  
Yet candid Reader, expect nothing here of me but a volume compiled out of others labours, and they formerly from the light of Nature, the dictates of our first Parent, and he from the inspiration of the Creator who is omnipotent and eternal, 
Eccles. 17.5.  whose endowments (conferred on Adam) had continued down as hereditary from one generation to another, had he not transgressed the just Precepts of his Maker, for which ingratitude he was expulsed the Garden of Eden and the sight of God, who withdrew his favours, and then innumerable errors obtenebrated the understanding of his succeeding Race, by sin made servile, Experience the Mistress of humane Sciences, and vigilant Industry her Usher; and by these means, declining ages, illustrated and facilitated Arts to common capacities; yet so, as that the most learned do find a Plus ultra, which they cannot apprehend, but as derived from the fountain of all Knowledge, who is incomprehensible in all his works, and that under the Sun, there is nothing new. Yet in every age, God hath pleased to illuminate some above the reach of other mortals, and that their inventions seems to Man as new, and those perhaps in the following age decline again, until enshrined by Oblivion, or suppressed, while ignorance gets the upper hand, by the assistance of the rude Vulgar, professed enemies to Art. This fantastic age is so inquisitive after novelties, that there's few that truth affects, she being old, and still the same, and if I should please none, than my intentions are frustrated, and part of my labours lost, for I would not willingly displease any, except the Proselytes of the times, who I know will bring in Apuleus upon his Golden Ass against me; who says Miserimi est ingenii semper uti inventis & non inveniendis; and therefore I will endeavour to add something to the method of these copious Sciences, although it be but as the Wren when she pissed in the Sea, encouraged thereunto by the Adagy which affirms, Facile est inventis addere, yet not expecting to satisfy Novelists any more, then to see the stability of a weather Cock; but as an ornament to my work, for to have something that may seem new to men, and to delight the Reader; and being there is a natural sympathy betwixt Number and Magnitude, I will delineat my grounds of Arithmetic, with Geometrical demonstrations, whereby to fortify myself against the enemies of this noble Art, yet build no Labyrinths, nor make the approaches intricate, unto the understandings of ingenious men, nor yet a burden to the memory of the Reader, nor with figurative expressions, or doubtful senses, like the interpretation of Oracles, but as clear and short as I can; and in pursuance of this form I will cast by, and obliterate the errors of many writers, in their tumults of superfluous words, terms, definitions, prolix, and ambiguous discourses, as Metaphors, Allegories, and many other implicit speeches, to show their learning, and not the ready ways to Sciences, and in fine prove nothing with their Dilemmas, but their own follies, and the Readers loss of time; whereas (I hope) to find a friendly acceptance, an ample recompense of my labours, and thanks from all young students of these Arts, who have or might have been perplexed and deterred from proceeding in rugged paths, where only empty words have been their guides, until they have lost themselves in obscure Maeanders, almost unextricable with their Conductors, or unexplicable to the apprehension of many Tutors: whereas Arts should be delivered plain, pleasant, and perspicuous in themselves, which if I have with a happy Genius performed, and made more facile than other Authors have delivered it, render the glory to God, who hath given me what I have, and I do account it a greater honour to have received it from the Immense Deity, than if it could have been of my own conferring or election, for he is sole Lord and the Artificer of all, and so St. Augustine calls Him, and says, Quicquid te in arte delectaverit Artificem commendat: And thus the Regal Psalmist 99 ver. 3. Scitote, quoniam ipse est Dominus Deus noster, ipse fecit nos, & non ipsi nos.  
As for the antiquity of natural Arithmetic in the practice, it is generally conceived co-Aged with the World, or the infancy thereof, a thing which little Children do naturally covet to learn, as they do discourse, a property annexed unto every rational Creature; the first practitioners of this Art are recorded to be Seth, and his succeeding generation, in imitation of their parent, whom they esteemed as a God, for his virtue and learning, Josephus lib. 1. Antiq. cap. 3. affirms, that his race were industrious, and of ingenious dispositions, and the Inventors of Astronomy, whereof Arithmetic is the ground: thus this knowledge descended unto Noah, and with the reparation of Mankind revived again by the industry of the Syrians, who were expert in all the Mathematical Sciences: the Phoenicians are recorded by Strabo, lib. 16. to be famous in Arithmetic, Navigation, 
Josephus lib. 1. c. 16.  and in all Warlike Arts: the Chaldaeans instructed Abraham, and he the Egyptians in Arithmetic and the Motions of the Heavens: from hence these Sciences did arrive in Greece, and from learned Athens transported to Rome; and from thence dispersed over Europe unto the Britain's, where all learning flourished, until suppressed with heresies and heathenish impieties; in the time of the English Saxons, Religion returned again, with the Liberal Sciences, and all kinds of Learning attending upon her train, England twice acknowledged the learned Tutor unto France; and since that, all Arts have flourished here, yet less in the theory, than in the practic part, every one not born to be an Artist according to the Adagy, Non cuivis homini contingat adire Corinthum.  
The use and excellency of Arithmetic is manifested by many ancient Writers, and grave Philosophers, who placeth the Art of Numbers as the Primum mobile to all Mathematical Sciences, it clears all difficulties in Quantities, it proves all Angles, Lines, and Superficies, it measures the magnitudes of all Bodies, the gravity and proportion of Weights, it discovers the harmony of sounds in Music, it solves all Enigmatical questions and Problems in Geometry, cosmography, Geographie, Astronomy, Navigation, Fortification, Architecture, and the Optic Sciences, and in multitude of Military propositions; and that you may not from empty words only receive some satisfaction herein of many examples, I will instance a few. Archytas Tarentinus, one of Plato's Disciples in Geometry, and so famous in Arithmetic, that he was the wonder of the times, and the glory of his Country, both in public and private affairs, the People happy under him their victorious General, renowned for his learning, and recorded to posterity for his knowledge in this Art; and thus writeth of him the most ingenious Poet: Horace, lib. 1. Ode 28. in Archytam, Te maris, & terrae, numeróque; carentis arenae Mensorem cohibent, Archyta.  
Arithmetic is very useful to Merchants, and to all in general that drive a trade either at home, or in foreign parts; by this Art, knowing the rates in exchanging of Coins, Weights, and Measures, and converting one Species into another; in keeping their accounts of gain or loss, of Debtor and Creditor; which moveth Caelius Rhodiginus, lib. 18. Lectionum Antiq. for to suppose the Phoenicians exquisite Arithmeticians, by reason of their commerce and trading; and the Egyptians famous in that, and the knowledge of Geometry, because of the annual inundations of Nilus, thereby draining their grounds, measuring of their lands, and placing of their bounds; thus making Necessity the Mistress of Arts: but howsoever, useful it is, in regulating of most humane actions or employments, in times of War or Peace; by Arithmetic there may be found any true proportion in the mixture or composition of Metals, as you may see by Archimedes in examining of the Crown made for the King of Sicilia, in which this famous Artist did discover how much true Gold there was in it, and how much adulterate mettle; and now by numbers 'tis commonly to every Artificer in that kind: of excellent use it is in composition of Simples, and making of Medicines, according to any quantity or quality propounded, or in respect of the temperature in any of the 4 degrees, viZ. Hot, Cold, Dry, Moist, but this is out of my Element; yet lest that any should say I talk like an Apothecary, Hypocrates the father of Physicians, commanded his Son that he should studiously labour in Arithmetic and Geometry, not only as for the splendour of his life, but also for the excellent use they had in composition of Simples, and in knowing the order and parts of the body, an epitome of the World; and for this also is Galen recorded another father of the Herbalists.  
Divine Plato in his Commonwealth commends the study of Arithmetic, and attributes so much to the praise and glory of it, that he thinks men mad or foolish that are quite ignorant of the Art: Aristotle in libro de Aud. conceiveth the Organs of hearing to have great force in figures, in the difference and distinction of Sounds. Ptolemaeus Alexandrinus, in libro 1 cap. 2. de Musica, affirms as much in honour of it, and in other of his books says, that (in all humane knowledge) Arithmetic and Geometry obtains the first degree of certainty; and as for Natural Philosophy (although studiously to be laboured in by all that would pretend to any knowledge, worthy of a Rational Creature, or refined from the dregs of the common people) do but behold (says he) the manner of their demonstrations, they may be rather called conjectures then a Science, for their many, and diversity of opinions, whereas these run constant in one channel; and as for Arithmetic and Geometry, they do concur together in their demonstrations; the first represents it to the Imagination, the other unto the Sight; the one makes it perspicuous to the internal Sense, the other visibly demonstrates to the external; thus one proves the other, and Number exactly discovers all the parts certainly the same in every operation, and true to 1/10, 1/100, 1/1000, or in any other proportion greater or less, as shall be required; decimal Arithmetic extracts its original from producing the nearest Square and Cubike Roots, etc. out of irrational numbers, but whose invention at first I find not any where recorded; of late years it was put into method, that it stands now like an Art of itself, and hath its Axioms and rules in Addition, Subtraction, Multiplication, and Division, with whole numbers and fractions together: to search for men's names enshrined long since in dust, would prove in vain, or for those, who have been famous in the Art of Numbers, were like Archytas numbering the sands of the Sea; for this Science hath been facilitated by many since Theophrastus writ his 2 books of Numbers, in the days of Alexander the Great; or Pythagoras, one of the 7 wise men of Athens, from whose Tables the Logarithmes do derive their conception; but in respect of the fo●m, method, and the excellent use of them in the practic part of all Mathematical Sciences, they are justly called the invention of John Neper, Baron Marchiston in Scotland, about the year of Grace 1610. and in the year 1614 the Author published a book of them, entitled, [Mirifici Logarithmorum Canonis descriptio:] being thus exposed to the public view, it was illustrated by many, but first in England by Mr. Briggs Professor of Geometry in London, yet before his masterpiece came from the Press, there arrived one in Folio from the University of Louvain: Of these two kinds of Arithmetic I will treat more at large hereafter, and now proceed no farther in showing the antiquity, excellency and use of Numbers, but conclude with Socrates: Nunquam animadvertisti, qui natura Arithmetici sunt, eos ad omnes Artes, percipiendum perspicaces & acutos esse?   

A DIRECTORY FOR All young Arithmetitians, and Students in the practice of Numbers; wishing their increase of knowledge, to the wisdom of our forefathers, and not to deride their Dictates, as new Courtiers their old fashions, whose sole ambition is; Os Populi meruisse: and theirs was, Ad Laudem & Gloriam Dei.  
THe secret force of Numbers I impart,  
And doubtful ways, but for the rules of Art;  
Like men at sea (who cannot find a shore)  
Without a Compass, Helm, or yet an Oar:  
The course in Numbers, dubious is as these,  
Ample as Earth, and spacious as the Seas:  
Be careful then to keep my Rules in mind,  
The hasty Bitch brings forth her Puppies blind,  
And so will those, who labour to proceed  
Before they understand what 'tis they read.  
Let not your Thoughts without your judgement run  
Like jades that tire before their journey's done:  
And oft unskilful men, who ride too fast,  
Do lose their ways, or wearied prove the last:  
Besides with haste, they will neglect to see  
The wind, and the turn that there be.  
Who Goods have lost, or hidden Treasure looks,  
Must search byways, and all their crooked nooks,  
Rocks, Hills, and Dales, and places under ground,  
When in the common Road they are not found,  
Which way to take, your judgement must direct,  
And what 'tis common Numbers can effect:  
If they prove able, than what Rules to choose,  
And whether False-positions you must use:  
If you in vain have vulgar Figures tried,  
And Rules of False proves an erroneous guide,  
The Cossick numbers will direct you right  
Through all Meanders, though involved in Night;  
They'll search the land, the seas, and every creek,  
And will conduct you, to the thing you seek;  
For by their strict pursuit I make no doubt,  
But what's in Number hid, they'll find it out  
If known to Art, and that there be a way  
No Bloodhound shall hunt drie-foot so as they.  
Too fare let not your inquisition go,  
Examine often whether right or no;  
With too much haste, you may over run the mark  
Which oft does lie concealed within the dark;  
And false conceptions of the sight there be,  
For all that glisters is not Gold we see:  
This, Error will unmask in Truth's disguise  
With all her painted train of Fallacies  
Which various are, so leave no Rule behind,  
For you may be in darkness till you find;  
Or soon to light, you'll this Dilemma bring  
From Art, and Nature, that there's no such thing:  
Aequation will the thing that's sought produce,  
Or show the Quaerie's false, or not of use:  
Rules I will give as many, as you'll need.  
Farewell: and in the name of GOD proceed.  Festina lente. Serius absolvit, qui nimium properat.  

Natural ARITHMETIC, Compendiously discussing and explicating the theory and practice of the Art of Numbers, divided into Sections, and those again into Paragraphs.  

Definitions: Sect. 1  

Parag. 1.  
ARITHMETIC is the Art, or Doctrine of Numbering; derived from the Greek, and signifies Number; the subject of Arithmetic, as Magnitude is of Geometry.  
Of this Art there are two kinds, viz: Natural and Artificial: the first of these is so denominated from having proper Figures, and Numbers significant of themselves; whereas the other is Figurative, or founded upon art: of this Arithmetic there be several kinds, of which (God willing) I will treat of in my third Book, and now proceed.  
Number is defined to be a discrete quantity, being divisible in its parts, consisting of Unites, by which every thing is numbered; and in this hath proper and peculiar characters.  

Annotation.  
This describes the Characters, as the Elements to this Art; by which Number is expressed, and are in all but ten, which in order are these, viz: 1 2 3 4 5 6 7 8 9 0 one two three four five six seven eight nine a cipher Nine of these are called significant Figures of themselves; the little circle or cipher represented by the letter O, signifies nothing of itself, yet increases the value of number according to its place or position.  
The first of these figures is called 
1 A.  a Unite, which simply of itself is no number, but the beginning of all number or quantity; as a point in Geometry is the first of magnitude represented by the letter A.  
All the other eight significant figures 
A— B  are composed of Unites, as a Line in Geometry is the continuation of Points, represented by A. B so that 2 is composed of two Unites; 3 made of three Unites, and 4 contains four Unites, and so of all the rest.   

NUMERATION.  
THis shows the value or quantity of any number given; or how to express in proper figures any number assigned or propounded, and to declare the quantity of each figure, according to its place or position, their contents increasing by degrees ascending from the right hand towards the left, as do the letters and characters of the Hebrews, and all the Eastern people, from whence this Art of Number extracts its original.  
In this amplification of Numbers you must observe the figure on the right hand denotes simply itself, as 2 two; 4 four; 6 six; 9 nine; 0 nothing &c. the next degree, or second figure towards the left hand is ten times its own quantity; the third place a hundred times; the fourth degree a thousand times; the fift place ten thousand times; the sixth figure in order is a hundred thousand times its own number; the seventh place a million; the eighth ten millions; the ninth place a hundred millions; and so proceeding without end, increasing every degree in this manner following, reiterating Unites, Ten, and Hundreds.  

A Table of Numeration, continued.  Degrees. 1 1 One 1 1 Degrees. 2 10 Tennes 21 2 3 100 Hundreds 321 3 4 1000 Thousands 4321 4 5 10000 Ten thousands 54321 5 6 100000 Hundred thousands 654321 6 7 1000000 Million 7654321 7 8 10000000 Ten millions 87654321 8 9 100000000 Hundred millions 987654321 9  
By this Table of Numeration you may observe how the figures by degrees do increase by ten, proceeding from the right hand towards the left, in value according to their places; as for example: the sinister Table gins with a Unite, unto which annexing a cipher, and then the second degree will be 10, that is in value ten times the first: unto that annex another cipher, and then in the third place it is 100 a hundred, that is ten times greater than the second degree, the fourth place 1000 a thousand, that is ten times 100, etc. The second Table I also here begin with a Unite, before which towards the left hand 1 place 2, and then in the second degree it makes twenty, so both the figures 21, and 2 in that place 10 times its own number: then put 3 before them, and it will be the hundreth place; and the character being 3, denotes three hundred; the three figures making in all 321 three hundred twenty and one: the fourth degree obtains the thousand place, being ten times greater than the third degree; and having the first figure before it 4, the value of the four figures will be 4321, that is, four thousand three hundred twenty and one: the next degree is ten thousand, and the character being 5, it must be fifty thousand, and the whole value of those five figures 54321, and so all others proceeding in order by ten.  
Yet the better for to effect the computation of great numbers, when their degrees cannot easily be contained in memory; than it is necessary for to put periods unto every century, or three figures, and so numbering them in unites, ten, hundreds; and then in expressing of their values, observe their places, and the quantity of each figure as it is simply of itself, and note that after the first period, their places are Thousands, and after the second period Millions; and after the next period Millions of millions: and so proceeding with reiteration of unites, ten, and hundreds, until their periods are put unto them; by this means avoiding errors, arising in numbering of great quantities, and mistaking their degrees, as in this example shall be illustrated, and suppose this number given, whose quantity is required, viz: 1, 234, 567, 890 which having pointed, read thus: one Million of millions two hundred thirty four millions, five hundred sixty seven thousand, eight hundred and ninety. If this had been the number given 1 0, 0 0 0, 0 0 0, 0 0 0 it would have been thus expressed, ten Millions of millions: or thus, ten th●usand Millions. Observe this order well, and then the computation of any Numbers will prove as easy as they be necessary for to be known or expressed, according to their places in degrees.  
Number is also for to be considered, whether it be of one, or several den●minations; those are said of one denomination, when they do consist of one number, weight, or measure, whether they be all of one entire number, or in the same parts.  
All numbers of several denominations are said to be mixed, when they do consist both of Integers and Fractions collected in a sum, viz: Pounds sterling, Marks, Nobles, Shillings, Pence; or in respect of weight in Tons, Hundreds, Stones, Pounds, Ounces, Drams, and Grains; or in long measures, viz: Leagues, Miles, Furlongs, Perches, Yards, Feet, Inches; or any other parts or fractions.  
Every number is subdivided into three parts, and is either Simple, decimal, or Compound, and are thus distinguished.  
1. Simple, are only those, who do consist of themselves, as do the 9 significant figures, viz: 1, 2, 3, 4, 5, 6, 7, 8, 9  
2. decimals or Tennes, are those who do consist of any significant figures, with cyphers annexed unto them, as 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, etc.  
3. Compound numbers are all those which are made by Simple and decimal numbers commixed together, viz: 12, 23, 34, 45, 56, 67, 78, 89, 91, and so are all others which are composed of significant figures.  
From Numeration there proceeds four species, viz: Addition, Substraction, Multiplication, and Division, which in order are treated of in the follow- Paragraphs or Chapters.    

ADDITION. Parag: 2. The first species.  
A Species is here defined to be a certain form of working by numbers, in which four this Art depends, the Rules are framed, and all questions in Arithmetic performed.  
Addition teacheth the contracting of many several quantities or numbers into one total sum.  
To effect this; place the greater number first, and right under that the next, and so in order place the several sums, this is convenient in the respect of form, but of no necessity; but this is, that you always observe in Addition for to put unites under unites, ten under ten, hundreds under hundreds, etc. that is, every figure in order underneath one another according to their degrees or places, from the right hand; this done, underneath them all you must draw a line, as by the following Paradigma.  

Example 1.  
Let the three numbers given be A 320 B 241 & C 231, of any one denomination, but here suppose them grains of Barley, the least denomination of English measure, four of them making an Inch, and all these a statute Perch, whose sum is required; and having placed them under one another, as by the letters A, B, C: begin with the figures first on the right hand, and say 1 and 1 makes 2, which writ down beneath the line under unites, the cipher of itself being nothing: then go to the next place, and say, 3 & 4 makes 7, and 2 is 9; which place in the second rank from the right hand: and proceed to the next place, where say 2 & 2 makes 4 and 3 will be 7, which place likewise beneath the line under hundreds: and so continue if there be any more numbers, the total sum of these is 792, as in the example: and so many grains of Barley or quarters of inches there be in 16 feet and a half.  
An illustration Arithmetical.  
Number consists in the addition of unites, as a Line does in Geometry by the continuation of points, and is called the first quantity, and so is number discreet, whose sum is nothing else, but the putting of several numbers together, as is the addition of Lines: as in the last example, admit three lines given for to be added together, each line consisting of so many equal parts respectively, as were the numbers propounded, viz: the line A divided into 320 parts or quarters of inches, each equal to a grain of Barley: B containing 241, & C 231 of those equal parts, all which added together will make the line A B C D, the true sum of the three lines given, being 792, as in the last Scheme appears, the quarters of inches contained in a Statute Pole: in the same manner if 12340 were to be added unto 87659, the total sum would be 99999.  
But it will most commonly happen, when great numbers are to be added, that the sum in most ranks will exceed the quantity of any one significant figure; if it be a Decimal, place a cipher under that rank, if a compound number write the simple significant figure, and add in the decimal to the next place as an unite; if two ten as 2, three ten as 3 etc. according to the order and degrees of Numeration, for every figure is no more than its own quantity, but in respect of the degree and place, as by the following examples will be made evident.   

Example 2.  
There was an old man, whose  44 Years 12  44 The total 100 age was required, to which he replied: I have 7 sons, and there was 2 years between every one of them; in my 44 year, my eldest son was born, which is just now the age of my youngest. First set down the Father's age; after that he had 6 sons which made 12 years more, writ that down, and the age of his youngest, which was equal to his Father's age at the birth of his first son, all which together makes the old man's age: so begin with the first rank, and say 4 & 2 makes 6 and 4 more is 10, and being there are more numbers to come, set down a cipher in the unite place, the decimal being but 1; in the next add it to 4 which makes 5 and 1 will be 6 and 4 makes 10; under the second rank place another cipher, and being there are no more figures, place the decimal, or unite in the third place: the total sum is 100, the old man's age required, as in the Paradigma appears.   

Example 3.  
There was a Traveller at the 
From Barwick to London.  Miles— 60 48 55 47 48 The total is 258 Town of Barwick inquired how fare it was to the City of London by computation, and was thus answered: from hence to Durham it is 60 miles, from thence unto the City of York 48 miles, then to Newarke 55 miles, from thence to the Town of Huntingdon 47 miles, and from thence to the City of London 48 miles: having placed these numbers in order, their sum will contain the whole distance, therefore say 8, 7, 5 & 8 makes 28, and according to the former rules, set down 8 the simple figure of this compound number, under the rank of unites, and add the two decimals into the next place, saying, 2 & 4 makes 6 and 4 is 10 & 5 will be 15 & 4, 19 & 6 will be 25; writ under that rank the simple number 5, and go the two decimals, but being there are no more rooms or places, writ down the decimal in the next degree or place to the last figure, and so you will find the total sum 258 miles, the true distance by computation from Barwick to London, as in the example.  
Yet note when numbers given are of divers denominations, in adding of those sums together, you must always observe for to begin with the lest first, on the right hand as before, and when their sums do amount unto any compound number consisting of integers and parts, subscribe the fraction, or part, underneath that column, whereon the head is writ the denomination, which is various both in number, weight, and measure; the integer or integers must be added as unites, to the next place towards the left hand: of this I will show you some examples; but to know the parts, I must refer you to practice, they being over-numerous to be subscribed, as in the numeration of mixed numbers was already intimated.   

Example 4. Addition of mixed numbers in Coin.  
This Table is divided into Li. So. De. 14567 8 9 3289 16 3 987 13 4 81 6 8 9 14 11 18935 19 11 3 columns, for so many several denominations: upon the head of the column to the left hand there is writ Li. denoting Libra signifying a pound, not here in respect of common weight but money, and for definition is called Pound sterling; and is an integer, according to the English account in Coins: the next column is noted with So. for Solidum, a coin of brass used by the Romans, but with us of silver, and signifies a Shilling, 20 of those pieces making 1 Pound sterling: the third column, or first on the right hand is noted with De. for Denarius, which signifies 10 for so many pieces it contained of the Romans lest coin; it hath had a various estimate in our English coins, as it signifies a Penny, the twelfth part of a shilling: for until the Reign of Henry the sixth a Penny was the twentieth part of an ounce, and in his Reign made the thirtieth: by Edward the fourth 40 pence in an ounce: by Henry the eighth there were allowed 45 pence to an ounce: and by Queen Elizabeth an ounce of silver was divided into sixty parts, called pence, as at this day; so much for the Coins, and now for the adding of them together, begin with the least denomination first, where you will find ●5 pence or 2 shillings and a 11 pence, which writ down, and add the 2 shillings to the next column, saying, 2 & 4 is 6 & 6 is 12 & 3 is 15, and 6 is 21 & 8 makes 29, and then reckon down the ten, saying, 29 & 10 is 39 & 10 will be 49 and 10 more makes 59, that is 2 pound and 19 shillings which subscribe in the proper column, and add 2 unto the pounds, saying, 2, 9, 1, 7, 9, 7, makes 35, that is 5 and goe 3, then 3, 8, 8, 8, 6, makes 33, that is 3 and go 3, which add to the next rank, as 3, 9, 2, 5, that is 19, subscribe the simple number, and add the decimal to the next place, as 1, 3, 4 maketh 8, which as a simple number set down, and lastly, 0 & 1 is 1, which prescribe according to its place, and the whole sum is 18935 L.— 19 S.— 11 D. as by the example in the Table.   

Example 5. Addition of mixed numbers in Weight.  
Of Weights there be two kinds Lb. Ou. Dr. 99 15 7 110 14 6 87 12 5 108 13 4 56 9 3 84 6 1 548 8 2 chief used in England, viz: Averdupois, and Troy weight; the first of these, which is also called the Civil or Merchant's weight, is divided into two kinds, the greater, and the less: the integer of the greater, is 112 pound to the hundred weight, and those subdivided into Quarters, Stones, and Pounds etc. by these are weighed the most gross commodities; or used by Merchants and Wholesale men. The lesser weight by those who sell by Retail, whose integer is a pound, and usually marked as is the head of the first column, or thus lb this integer is subdivided into ounces and signed as the second column, and often thus ℥ 16 making a pound, every Ounce is subdivided into 8 Drams or Dragmes, noted as in the table, and often thus ʒ, every dram into 3 Scruples, which usually hath this mark ℈, etc. But leaving these parts of intigers for some other time, and sum up these in the Table according to our prescribed method, and begin first with the least denomination of this mixed number, which here are Drams, which column added together makes 26, that is 2 Drams, and 3 integers for to be added unto the next column, which according to the former order of Addition, will make the next column 72 Ounces, that is 8 ounces, and 4 integers in the next column being pounds, which being added into the pounds according to the rules of Addition, will make the sum of the next column 548, and the total 548 lb- 8 Ounces- 2 Drams; which according to the greater or grosser weight may be thus expressed 4 C. 7 St. 2 lb 8 ℥. 2 ʒ.   

Example 6. Addition of mixed numbers in long or radical Measures.  
The Measures in England are Pe. Ya. Fe. 318 5 1 299 4 2 48 5 1 167 0 2 98 4 1 319 5 1 1253 3 2 more various than either of the former two, in this example I will only use Perches, Yards, and Feet, and the least of these admitting of many subdivisions, but in finding or measuring of great distances quite unnecessary; as a League containing 3 Miles: one Mile 8 Furlongs: one Furlong 40 Pole or Perches: one Perch 5 Yards and a half: and 3 Feet a Yard: in this example a statute Perch is the integer, and 'tis supposed that the distance measured between 6 stations are here set down, whose sum is required, which to find, begin first with the least denomination, where I find 8 Feet, that is 2 Yards or integers in the next column; so setting down 2 feet in its proper column, I add the other 6 feet as 2 yards or integers to the next column, which will make the sum there 25, that is 3 Yards and 4 Perches: where I subscribe the parts, and add the 22 yards as 4 Perches to the next column, which will make the first rank of figures there 53, and these being in this column, all of one denomination, I subscribe 3, and go 5 decimals in the next rank, as by the former method, and so continue on, until they be all added together, whose total sum is 1253 Porches, 3 Yards, 2 Feet, for the distance of the 6 stations; and being 320 Porches make one Mile, containing 8 Furlongs, and every one of those 40 Pole, you may write the total of the distance thus, 3 Miles, 7 Furlongs, 13 Poles, 3 Yards, 2 Feet: here you must observe in subscribing the parts to note when you have done, whether the fractions or parts of several denominations will not make an integer, as sometimes 'twill happen, when a part of a denomination must be had to make an integer; as in this case where 5 yards, 1 foot, and 6 inches, makes a Porch, or 16 feet and a half, and so it would have happened here, if the sum remaining in the column of yards had been 5; and so in divers other cases might be instanced.    

SUBTRACTION. Parag. 3. The second species.  
THe property of this Species, is to find the difference between any two numbers given; which is found by drawing, or taking one number from another, the residue or remainder is the difference between them, which consequently being added to the lesser, will make a number equal to the greater, the difference being the excess between them: as if 1 were to be taken from 9, the remainder or difference will be 8, which excess added to 1, will make it 9, equal to the greater number given: so if 6 were substracted from 8, the remainder will be 2, which difference added to 6, maketh 8 the greater number, or equal to it: for nothing that is compounded can be the same: Yet 2 Pints will make 1 Quart, and 4 Quarts 1 Gallon, but leaving this for Logicians to discuss, I will proceed.  
When any subtraction is to be made, the lesser must be deducted from the greater, or equal things from other quantities that are equal, but a greater from a less cannot be deducted in the whole, yet in particulars they may, when the praecedent numbers towards the left hand shall be greater, from whence the subtraction is to be made, as shall be illustrated by examples, and first observe to write in figures the greater number, and under that the lesser, and in such order as was prescribed in Addition, according to their ranks and places, whether they be Ciphers or significant figures as unites under unites, ten under ten, etc. Yet notwithstanding when the lesser number is under the greater according to their places, a significant figure may be under a cipher, or a greater under the less; in such cases, take an unite from the place before it, towards the left hand, it will be 10 in that rank, which add to it, and then subtract the former number, and keep the unite which was borrowed, in your mind, and add that to the lower number in the next place; for the upper number was to have been an unite less, therefore if the lower precedent number be made a unite greater it is all one: as if 9 were to be subtracted from 10; and placed according to my direction, it cannot be taken from a cipher alone, but as a decimal it may, and 1 will remain. Again, admit 19 to be subtracted from 21, 9 cannot be taken from 1, therefore suppose an unite borrowed of 2 & then the former 1 will be made 11, from whence take 9, and there will remain 2, now 'tis all one whether I make 2 in the 21 but 1, or the decimal 1 I borrowed be added to 1 in the 19, and so make that 2, which is the best way, and then take 2 from 2 and nothing remains but 2, the first subtraction; for the difference betwixt 21 and 19 If several numbers were to be subtracted, from divers other sums, collect them all into one by Addition, and likewise those which are to be subtracted, and then find the difference between them; for this rule or Species admits but of two sums; these principles observed, the rest will be cleared by examples following.  

Example 1. Arithmetical and Geometrical demonstrations.  
Number in Arithmetic, and a Line in Geometry have an undoubted Symmetry, or proportion in all their parts, being both of the first quantity, as was said and proved already: and here in this example, if the line C were to be taken from the line A or laid upon it, the line C would extend itself to B; from whence 'tis evident, A, B, would be the difference betwixt them, which remainder is equal to B, Again, suppose the line A were a Statute perch or Pole, and divided into inches, it would contain 198 of those equal parts, and admit the line C were 12 foot or 144 inches, to be taken from A, having placed them right under one an other, begin with the unites, and say; take 4 from 8 and there will remain 4. then go to the rank of decimals and say, take 4 out of 9 and there will remain 5, which place under the line in the second rank, then proceed to the place of hundreds, where you will find 1 taken from 1, and nothing will remain; so 54 inches is the residue, or difference between the two lines which was required.   

Example 2.  
Henry the 8, in the year Anno Domini salvatoris 1653 Anno Domini salvatoris 1536 The difference of years 117 of our Lord & Saviour 1536, made himself the Head of the Protestant Church in his own Dominions, and Thomas Lord Cromwell his Vicar General under him: and in the year of Christ 1653, Oliver Cromwell Lord General for the new State, was made Protector over all the three Kingdoms; and here it is required the annual revolutions, or the years elapsed between them. First, set down the year present, which is the greater number, and under that the less, each figure according to his degree or place, then say, take 6 out of 3, which cannot be done, without borrowing one of 5, that is 10 in that place, which added to 3 will make 13, so take 6 from 13 and there will remain 7, which place beneath the line, drawn to part the remainder, from the numbers to be subtracted, and under 6, then say 1 and 3 is 4, which taken from 5, and there remains 1, which writ under 3, then take 5 out of 6 and there will remain 1, which set down in its place, then take 1 from 1 and nothing will remain, so the difference between these years is 117, and so many annual revolutions of the Sun are elapsed.   

Example 3.  
There was a Merchant  lb. Money borrowed 5000 Money repaid 4975 Remaining due 025 who had borrowed at one time 3200 pound, and yet to go forward with his designs, was constrained to take up more as 1500, and presently after that 300 l, upon the return of his adventure, the Merchant sold his goods so soon as he could, to pay his Creditor, whom he brought at one time 4975 li. & it was required to know what the difference was between the money borrowed, and the sum paid. First, collect together all the several sums that were lent, which will make 5000 lb and set it down, and being 'tis evident, the money repaid was less, subscribe that under the greater, then draw a line and subtract them, saying, take 5 from 10 and there will remain 5, and 8 out of 10 there will remain 2, which writ in the second place: and proceed, saying, 1 which was borrowed, and 9 will make 10, take 10 from 10 and nothing remains; then again say 1 and 4 makes 5, take 5 out 5 and nothing will remain: the last cipher was set down to keep the place, being there were more to be subtracted, and something might have remained, which is not at the first seen to young practitioners, and no prejudice to any: the difference is 25 li. and so much money is remaining due to the Creditor, besides Interest, which according to the time is to be considered.   

Example 4.  
There was a Merchant  lb. The Bank 1900 The sum known 892 The difference 1008 who upon a foreign employment made two of his servants Cash-keepers in his absence, and commanded them to put all the money they should receive into a Counter or common Bank, whereof he took the key himself, and left in his treasury 95 li. before his return, one of the Cash-keepers died, and calling the other to an account of what money he had received, he made it appear by Bonds and other cancelled Deeds, that there was 797 li. put into the common Treasure by him; but what his fellow servant had done he knew not: the Merchant opening his Counter found in it 1900 li. and it is required to know how much money the servant which was dead had received: first add the stock which was left, viz: 95 li. to the money received 797 li. the sum is 892, which as in the example, set underneath the total sum found in the Bank, viz: 1900 li. the difference will show 1008 li. the true sum, which the other servant had received.   

Example 5. Subtraction of mixed numbers in Coin.  
There was a Farmer, who had L S D Q 1000 0 0 0 999 19 11 3 0000 00 00 1 borrowed 1000 li. of an Usurer; or at least wracked up, by extortion, with Interest upon interest until extended to that sum as that the Farmer knew not what to do; the Usurer having more Law of his side than equity, less mercy than a she Wolf upon her prey, and now an opportunity by the extremity of Law, and virtue of his Bonds, the day of forfeiture near at hand; not commiserating the poor man's case, or complaints, or admitting of delays, in prolonging time, any more than death: the Farmer in this exigent made recourse to his friends, engaging them, pawning his goods and credit, so at last with the help of his Wife's butter-money, he raised the sum subscribed in the margin: the Usurer seeing 999 li. refused to receive it, gaping still after the forfeiture, but in vain, for with the broken money 'twas right to a farthing, as by Subtraction will appear in the example; take 3 farthings out of nothing you cannot, then borrow an intiger in the next place which is 1 D, or 4 Q, then say, take 3 out of 4 and 1 Q will remain; again 1 D borrowed and 11 D is 1 shilling, which take from nothing you cannot, but from an intiger borrowed in the next column you may, and nothing will remain: then proceed and say, 1 S borrowed added to 19 S makes 1 Li. or 20 S. and being there is a cipher over it, borrow an integer in the next place which are L. 1 being 20 S. take 20 out of 20 and nothing remains, therefore subscribe a cipher as before: then in the column of pounds sterling, say, 1 borrowed and 9 is 10, take 10 out of 10 and nothing remains; set down a cipher, and say again 1 borrowed and 9 is 10, take 10 out of nothing I cannot, borrow 1 of the next place, as before, (these being all of one denomination) then take 10 from 10 and nothing remains, and so proceed till all be subtracted and nothing remains, so in conclusion there is only remaining 1 Q. which stands for Quadrants; in our Coin it is taken for the fourth part of a Penny, that is, a Farthing, and he who pays to that, the Proverb declares an honest man.   

Example 6. Subtraction of mixed numbers in Weight.  
The letters in the head of this lb O P G 670 3 18 20 369 8 19 4 300 6 19 16 Table denotes in each column the pounds and parts according to Troy weight, lb pounds; O ounces, whereof 12 makes one pound; P for peny-weight, twenty of those makes an ounce; G grains, 24 of them do make a peny-weight. There was a Merchant that brought over two quantities of unrefined silver, one Mass weighed 670 lb. 3 O. 18 P. 20 G. The lesser Mass 369 lb. 8 O. 19 P. 4 G. when they were refined, and in Lingots; the lesser quantity at 5 shillings the ounce, did come but to so much as was their difference in weight between either Mass, which is here required; having placed them according to their pounds, and parts, as in the Table; subtract one from the other, beginning with the lest first, as in the other examples, and say, take 4 G. from 20, and there will remain 16; then to the next column, where 19 Penny weight is to be taken from 18, but cannot, therefore take 19 from 38, and there will remain 19 P. 1 ounce which was borrowed and 8 will be 9, take 9 from 3 is impossible, take 9 from 15 and there will remain 6 ounces, 1 lib. borrowed and 9 is 10, take 10 from 10 and nothing remains, 1 & 6 will be 7, take 7 out of 7 and nothing will remain, subscribe a cipher as before; and lastly, take 3 from 6 and 3 will remain; so the difference is 300 lib. 6 O. 19 P. 16 G. which at 5 shillings the ounce, that is 3 li. sterling the lib. comes unto into money 901 L. 14 S. 11 D.   

Example 7. Subtraction of mixed numbers in Measure.  
This example is of Land measure, Acres R P 9012 1 8 912 1 3● 8099 3 18 wherein observe generally throughout England, that although there be Customary & Statute-measure, whereof the first is very numerous, yet in all and every one, there are 40 square Pole in a Rood, and 4 Rood in an Acre, and the question is here of a Man who had two sons, and in his last Will bequeathed to his eldest son 9 Lordships but no Manors, which were in content upon the survey 9012 Acres, 1 Rood and 8 Perches; and to his youngest son he gave Nonsuch and Drownland in a County of the Moon, containing by supposition 912 Acres, 1 Rood, and 30 Pole, surveyed by Telescopes, and 'tis required to know how much one brother had in content more than the other; having placed them, as in the Table, take 30 P. from 48 and there will remain 18, 1 I borrowed and 1 makes 2, take 2 from 5 and there will remain 3 R. then say again 1 & 2 is 3, which taken from 12 and there will remain 9, then 1 & 1 makes 2, that taken from 11, and 9 remains, which subscribe as before; then say, 1 & 9 makes 10, take 10 out of 10 and nothing will remain, there subscribe a cipher, and being there are no more figures in the lesser number, subtract the last 1 which was borrowed out of 9 and the remainder will be 8, so the residue or difference between them is 8099 A. 3 R. 18 P. as in the Table you may see.  
An Examine of Addition and Subtraction,  
Or a trial whether you have done right or no, there be divers rules, but the best and general way, is to make one of these Species prove the other; as when divers numbers are added together, from the sum subtract the parts, and you will produce the former numbers; and where a subtraction is made, to try that; add the difference or remainder, to the numbers subtracted which is least, and the sum will give the greater number; or from any number subtracted, take the difference from the greater given number, and the remainder will be the lesser number that was subtracted: for always, where there are 2 numbers subtracted, as the remainder is the difference betwixt those numbers given, so the lesser given number is the difference between the remainder, and the greater given number; for every remainder must be less than the greater number, by the quantity of the lesser, therefore 'tis the difference between them; yet though 'tis evident I will illustrate it with examples in both Species, how they mutually do try one another.  
In the first example of Addition, if from the sum of 792 you take C, 231, the remainder will be 561, the sum of A and B, from whence subtract either of their parts, and the other will remain, as take away A 320, and there will remain 241 for B, or if 241 had been subtracted from 561, there had remained 320 for A, and thus you may proceed in any other, either of one denomination or of many.  
To try whether Subtraction made, be true, take the last example of mixed numbers, and say 18 and 30 is 48, that is 8 Pole and 1 intiger in the next column, so proceed, 1 and 3 makes 4 and 1 will be 5, that is 1 Rood and go 11 Acres; then 1 and 9 is 10 and 2 will be 12 that is 2 and go 1 decimal, which with 9 and 2 will be 11, subscribe 1 and go 1, which added to 9 makes 10, a cipher and go 1, lastly 1 and 8 makes 9, so the sum is 9012 A, 1 R, 8 P, as before: or to try this by subtraction, take the remainder from the greater number, their difference will be the lesser number which was subtracted, if it were rightly performed, as take 18 from 48 and there will remain 30 P, take 4 Rood out of 5 and there will remain 1 R, take 10 Acres out of 12 and there will remain 2, take 10 out of 11 and there will remain 1, then take 1 out of 10 and the remainder will be 9, lastly 1 and 8 is 9, take 9 from 9, and nothing remains, and the other figures subscribed will make 912 A. 1 R. 30 P. the lesser given number: and thus by finding either of them by the remainder, it proves the work undoubtedly true.    

MULTIPLICATION. The third Species, and fourth Paragraph.  
MVltiplication is the increasing of any number by another, so often as there be unites in one of the numbers; or from any two numbers given to find a third, which shall contain one of the numbers as many times as there be unites in the other.  
To Multiplication there appertains three principle members, viz: the Multiplicand, the Multiplier, and the Product.  
The Multiplicand, is the number given for to be multiplied, and is properly the greater number in respect of order, but of no necessity; for 'tis all one whether 4 be multiplied by 3, or 3 increased by 4, 'twill be either way 12.  
The Multiplier is usually the least, and always the lower number: it is to increase the Multiplicand as many times as there be unites contained in the Multiplier; as if 6 were to be multiplied by 4, they would produce 24, and so would 6 times 4; that is 4 added 6 times together, for the number of unites contained in the Multiplier.  
The Product is the number produced by the mutual increasing one number by another, as the Multiplier by the Multiplicand; as if 9 were multiplied by 2, the Product will be 18, or 8 times 9 is 72.  
All numbers increasing but according to the unites contained in them, you may divide the Multiplier, the Multiplicand, or both of them into what parts you please, then multiply those parts, and add them all together, they will produce the same number; for the whole is in the same proportion to the whole as all the parts are, as in the last example; divide 9 the Multiplicand into what parts you please, as 2, 3, 4, these making 9 in all, then increase all those parts by 8 the Multiplier, and they will produce 16, 24, 32, whose sum is 72, and so is 9 multiplied by 8, which Multiplier divide into what parts you will, as admit in two, viz: 3 & 5, and then multiply the former parts of the Multiplicand, which were 2, 3, 4, by 3, and then by 5; the Products will be 6, 9, 12, & 10, 15, 20, all which Products added together make 72, that is 8 times 9 as before, and the like of any other; for as an unite or 1 is to the Multiplier, so will the Multiplicand be in proportion to the Product, and the contrary, as by examples shall be illustrated.  

A Demonstration.  

A Table of Multiplication.  A    F    B 1 2 3 4 5 6 7 8 9 2 4 6 8 10 12 14 16 18 3 6 9 12 15 18 21 24 27 4 8 12 16 20 24 28 32 36 5 10 15 20 25 30 35 40 45 6 12 18 24 30 36 42 48 54 7 14 21 28 35 42 49 56 63 8 16 24 32 40 48 56 64 72 9 18 27 36 45 54 63 72 81 D    E    C  
This Table consists of the mutual multiplication of all the nine significant figures, proceeding from an unite to 9 either way; that is, from A to B, and from A to D, according to the orderly succession of these numeral characters, every one having their several Products, or number of unites contained in their multiplications, which ought for to be learned without book, and imprinted in your memory; yet to facilitate the labour of young beginners, I have here subscribed this Table, easy to be understood, and a readily to be remembered, divided into 91 little squares; every one containing the Product, made by the simple figure in the head of the Table over it, and that in the side right against it, on the left hand; every one increasing, or decreasing in his column, according to the unites in the Multiplier, or Multiplicand; as for example, admit 5 & 6 were to be multiplied, I look the one number in the head of the Table, and the other in the side, in their common square or angle you will find 30, their true Product; and observe how that column under 6 increases and decreases by 6, and that of 5 by 5: thus, 9 times 9 will be found 81, and their columns increased by 9, and so the like of any others.  
And farther you may here observe how every simple figure multiplied in its self, is terminated, until it returns to its own character again; as for example, 1 multiplies nothing, but still remains an unite, being no number of itself; 2 multiplied by 2 ends in 4, 8, 6, and fourthly is terminated in 2 its root; 3 by 3 ends in 9, 7, 1, and fourthly in 3 again; 4 by 4 ends in 6, and secondly in 4, next 5 & 6 multiplied in themselves, are terminated in their own figures, and from thence are called Circular numbers; 7 multiplied by 7 ends in 9, 3, 1, and fourthly, in its self; 8 by 8 ends in 4, 2, 6, and fourthly, in its self again; 9 multiplied by 9 will end in 1, and in every second multiplication is terminated in 9, it's own root, or number again produced.  
Multiplication is a quadrature and hath this Analogy or proportion with a superficial square called in Geometry the second quantity, a figure composed of lines, whose sides are divided into parts, and intersected with paralle's, or equidistant lines, as is the last figure A, B, C, D, making thus both Squares, and right angled Parallellograms, equal to the numbers multiplied in themselves together; as for example, the Table or square A, C, hath every side divided into 9 equal parts, as admit Poles, Yards, Feet, Inches or what you please; suppose these sides as A, B, and A, D, in Feet, and intersected with lines, the whole Square will contain 81 square Feet, for the true superficial content of it, so if 9 were multiplied by 9, the Product will be the same, or if the long Square A, F, E, D, were given, the superficial content of this Geometrical figure, called a Parallelogram, would be 45 square Feet; and so is the product of 9 multiplied by 5: If it were required to know how many feet there were in a Yard square, 3 Feet makes a yard in length, therefore if every side of such a square were divided into 3 equal parts, and intersected with right lines, there will be found 9 square feet, as in the Table will appear; and so 3 multiplied by 3 will produce 9, and in these, the proportion is continued; as an unite is unto the number, or side of the Square given, so will the side be to the whole Square; or as 1 to the Multiplier, so will the Multiplicand be unto the Product, as in the former examples; as 1 to A, B, 9, so will A, D, 9 be to 81 the whole Square or Product, or as 1 to A, F, 5, so A, D. 9 in proportion to 45 the long Square A, F, E, D, and so the like of any other in this kind whether greater or less, and so much for the Sympathy betwixt Multiplication, and Geometrical Squares of the second quantity.   

The way and form of Multiplication when there is more than one significant figure in the Multiplicand, Multiplier, or in both of them.  
When there be several figures to be multiplied both in the Multiplier and Multiplicand, set down first the greater number, and under that the lesser, according to their degrees or places, as unites under unites, ten under ten, hundreds underneath hundreds, etc. as hath been directed in divers other examples already; this done, draw a line underneath them both whereby to separate them from the numbers increasing by their mutual multiplications, proceeding from the right hand in order towards the left; every figure of the Multiplier, must be increased or multiplied, into every particular figure of the Multiplicand, from whence there will arise so many Products, as there be figures in the Multiplier; the first figure in every Product ought to be placed exactly under that figure which multiplies, and so in order with a convenient distance towards the left hand, for all the several Products must be added together, whereby to find the result or total of them: and in every figure as you multiply, set down the Product if under 10; but if a decimal or decimals, subscribe a cipher; if a compound number, writ down the significant figure, and keep the decimal or decimals in your mind, and as you multiply the next figure, add them in as unites; and so proceed until every figure of the Multiplier be increased, by all the figures of the Multiplicand in order as was said already, and shall be now illustrated with several Paradigmas following.   

To find how many days there be contained in a common Year consisting of 52 Weeks and one day. Example 1.  
Having set down the 52 Multiplicand 52 The Multiplier 7 The Product 364 The odd day 1 The total of days in a year 365 Weeks, subscribe underneath it, the days in one Week, which are 7, and being a simple number, I set it under 2, in the unite place; and since there are so many days in one week, there must be 52 times so many in a year, besides the odd day: then say 7 times 2 makes 14, for which subscribe 4 below the line, and right under 7, and keep the decimal in your mind, they say 7 times 5 (as by the Table of Multiplication) will make 35, and the 1 decimal in mind makes 36, therefore subscribe 6 right underneath the 5, and keep the 3 decimals in mind; but since there is no more, writ it down, and the product is 364, the number of days in 52 weeks; and being there is one day more in a year, add that in, and then the total will be 365, the true number of days in the Sun's annual revolution, the thing required.   

The days and parts of a year being known, to find the number of hours contained in it. Example 2.  
First set down the greater Multiplicand 365 Multiplier 24 The Products 1460  730 The total 8760 number, and then the Multiplier, according to the places of the figures, in this 24 being the hours contained in a natural day, and then say 4 times 5 is 20, for which subscribe a cipher, and keep 2 decimals in mind; then 4 times 6 is 24 & 2 is 26, that is 6 and go 2, then say 4 times 3 is 12, and 2 in mind is 14, and being there are no more figures in the Multiplicand, writ them down, and then the first Product will be 1460; then go to the second figure of the Multiplier, which here is 2; and say, 2 times 5 is 10, subscribe a cipher under 2 the Multiplier, and go 1 decimal, than 2 times 6 is 12 and 1 makes 13, that is 3 and go 1, than 2 times 3 is 6, and 1 in mind makes 7, which writ down under the 1, and then add the Products together, the total will be 8760, the number of hours in 365 days, and since the Julian account, makes the magnitude of the year for to consist of 6 hours more, add 6 unto the Product, and the total will be 8766 hours contained in a common year.   

To find how many minutes are contained in a year. Example 3.  
Here you are to know the Multiplicand 8766 Multiplier 60  0000  52596 Product 525960 parts or minutes of an hour, which are 60, those must be the Multiplier, and 8766, the hours contained in a year the multiplicand, then say no times 6, or 6 times nothing, is nothing, therefore subscribe so many cyphers, as there be figures in the Multiplicand, and then begin with 6, and say 6 times 6 is 36, there set down 6 beneath the line right underneath the Multiplier 6; then say 6 times 6 makes 36, and 3 decimals in mind will be 39, there subscribe 9 and go 3, then 6 times 7 will be 42, and 3 in mind is 45, that is 5 and goe 4, then 6 times 8 is 48, and 4 decimals in mind will make 52, which subscribe, there being no more figures to be Multiplied; the product of these is 525960, and so many minutes there are in a vulgar year, but this and all such questions are performed a readier way, as you shall see by the examples following.   

Breviates, or compendious Abridgements in Multiplication exemplified. Example 4.  
If any number be given Multiplicand 1000 Multiplier 25 The Product 25000 for to be multiplied, which hath only cyphers, with an unite upon the left hand; being 1 is no number, and consequently multiplies not, annex all the cyphers to the significant figures towards the right hand, and the work is ended, as suppose 1000 Soldiers, and every, one received for his pay 25 Crowns, the product will be 25000 Crowns for the whole sum paid; and so for any other, as 10, 100, 1000, 10000, etc. but if there be any other number given to be multiplied, with cyphers following any significant figure or figures, in all such cases neglect the cyphers, and making the lesser number than Multiplier, and proceed as before, and to the product annex all the cyphers that were towards the right hand either in the Multiplier, Multiplicand, or in both of them, as in the former example where 60 was the Multiplier, if 8766 had been multiplied by 6, the Product would have been 52596, to which if a cipher had been annexed, the sum would have been 525960, as before.   

When cyphers in any Multiplier are included between significant figures, the multiplication may be easily abreviated thus. Example 5.  
Admit 9080 were a number Multiplicand 908 Multiplier 309  8172  2724 Product 280572 given for to be multiplied by 3090, here according unto the last Paradigma, I do neglect the cyphers in both numbers, and so set down 908, and under that 309; then draw a line underneath them as in the Table, and beginning with 9 the first Multiplier, say. 9 times 8 is 72, both which figures subscribe, because a cipher follows in the Multiplicand, then say 9 times 9 is 81, which also set down, there being no more to come the cipher is next, which makes nothing, but only assumes a place; therefore go unto the next significant figure which is 3, and by reason of the cipher in the Multiplicand you may say, 3 times 8 is 24, and 3 times 9 is 27, which subscribe in order observing punctually to place the first figure of that Product under the figure of the Multiplier, as in this 4 under 3, and so in order proceed towards the left hand, by this means the place of the cipher is gained, and being between the first and last Product can be nothing, nor have a place in adding them together: the Product of these figures is 280572, and being the Multiplier and Multiplicand had either of them a cipher in the first place on the right hand, annex the two cyphers to this sum, and then the total will be 28,057,200, the solution required.   

When any number is given for to be multiplied by one figure, or by two, if one be an unite, it may be performed compendiously thus. Example 6.  
It is required to know  112 Multiplicand A 896 The Product B 5376 The total A, B, 14336 how many ounces there be in 800 li. of the Merchant's weight; first set down their hundred, which is 112 li. then draw a line underneath it, and 8 being the Multiplier, you may keep it in mind, and say 8 times 2 is 16, that is 6 and go 1: then 8 times 1 is 8 & 1 will be 9, which subscribe, then say again 8 times 1 is but 8, which having set down, the Product of 8 hundred weight is 896 lib. as in the Table, which is to be reduced into ounces, 16 of them making 1 li. an unite being here in the second place towards the left hand multiplies not, but will produce the Multiplicand, and remove it one degree farther (according to my former prescribed rules of Multiplication) therefore in all such cases as these, you may neglect the unite, and keep the other figure in mind for your Multiplier, and place your Product a degree backward, as in this example, A is the Multiplicand for to be increased by 16, or 6 kept in mind, then say 6 times 6 is 36, that is 6 and go 3 decimals, than 6 times 9 is 54 and 3 will be 57; which is 7 and reserve 5; then say 6 times 8 will make 48 and 5 is 53, which figures subscribe, there being no more to be multiplied, so this Product is at B 5376, which is placed one degree towards the right hand, whereby the unite place in A the Multiplicand, may stand in the decimal place of the Product, and so it would have done according to the common way of Multiplication, but then beneath it, as 'tis now above it: these two numbers must be added together, whose total is A, B, 14,336, the true number of ounces in 8 C lib. gross weight; the line drawn between A & B is here for directions only, and in your work is better omitted.   

How to multiply the shortest way with two figures, when the first figure on the right hand is an unite. Example 7.  
It is required to find Multiplicand 6336 Product 25344 The total 259776 how many inches there be in 41 Miles, one containing 320 Perches, every Perch in length 16 feet and a half, and 12 inches the length of one Foot; now in a Pole there is 33 half feet, which multiplied by 6 inches the Product will be 198, the number of inches in a Perch or Pole, that multiplied by 320, the Perches in a Mile, or by 32, the Product will be 6336, to which add a cipher and the Product is then 63,360, the number of inches in a Mile, and 41 is the distance given: now omitting the cipher, set down 6336 which is the Multiplicand, and 41 is to be the Multiplier, an unite can only produce 6336, and being the 1 is to the right hand, set the Product of the other figure one degree towards the left hand, keeping that Multiplier in your mind, and say, 4 times 6 is 24, that is 4 and go 2, then 4 times 3 is 12, and 2 maketh 14, that is 4 and go 1, then 4 times 3 is 12 and 1 decimal in mind is 13, that is 3 and go 1, then 4 times 6 is 24 and 1 is 25; the total is 259776, to which annex the cipher omitted on the right hand, the total is 2,597,760, the number of inches contained in 41 Miles, the thing sought.   

Example 8.  
If you are to multiply by 11, 28800 28800 28800 28800 316800 316800 it is all one whether you place the number given as the Product, a degree forward or backward, for it comprehends both the last examples in one, as in this Paradigma will appear, there are 11 half Yards or Cubes, in the length of a Perch, 320 makes a Mile, 3 Miles a League, and the distance from London to Norwich is 30 Leagues, and how many Cubes or half Yards are there contained between those Cities, in one League there are 960 Perches, in 30 Leagues 28,800 Perches, and this Multiplied by 11 is the question required, in all such cases you have no more to do, but subscribe the same number again, which was to be Multiplied, observing to remove all the figures one place or degree either backwards or forwards, the sum of those two numbers will give the product, as you may see in the example; all these ways are as easy as shorr, if well observed: but otherwise much more subject to error then the common way.   

Two several explications of Multiplication continued. Example 9  
These which some calls continual Multiplication, is nothing but the Multiplying of several numbers into one product; that is, the third number multiplied into the product of the first and second, and that into a fourth, etc. As for example. 4, 7, 10, 13, admit the numbers given, 'tis all one where you begin but let it be at 4, and 7 multiplied makes 28, that by 10 will be 280, and this product multiplied by 13 makes 3640, the first question solved; secondly when the Multiplicand consists of several denominations, as 5 L. 6 S. 8 D. to be increased by 6 produceth 32 L. viz. 6 by 8 is 48, a cipher and go 4 S. then 6 times 6 is 36, and 4 in mind is 40 S. a cipher and go 2 L. then 6 times 5 is 30, and 2 is 32 L.    

DIVISION. Parag: 5. The fourth species.  
DIvision is to separate any quantity given into any parts assigned, or to find how often one number is contained in any other; or from any two numbers given to find a third, that shall consist of so many unites as the one number is comprehended in the other.  
To Division there belongs three principle members or parts, viz. the Dividend, the Divisor, and the Quotient.  
The Dividend is the number given for to be divided, and is the greater number.  
The Divisor is the number by which you must divide, and is the lesser number.  
The Quotient is that number, which is produced by division, and contains so many unites, as the Divisor is comprehended in the Dividend, from whence this Analogy or proportion proceeds, viz. as the Dividend is to the Divisor, so will the Quotient be unto an unite, and the contrary; as the Divisor is to an unite, so will the Dividend be to the Quotient.  

Division proved by Arithmetical and Geometrical Demonstrations.  
To illustrate this Species, there was a man had 6 Daughters, to whom he bequeathed 30 fields or enclosures, all of equal value and content; and so, as af-after his de●th, every one should have an equal portion or share in the land, and if any died before they married, the rest were to have it divided amongst them equally; suppose all the fields to be enclosed with the square A, B, C, D, the Father departed, the land was thus divided: 30 is the number of fields, which is the Dividend, 6 the number of Daughters must be the Divisor, as A, B, then will A, D, 5 or B, C, be the Quotient, for 6 is contained in 30, 5 times, that is so often as there be unites in the Quotient, as by the number of little squares is evident, within a year, 5 of these coheires married, and the youngest was prevented against her will you know by untimely Death who by this means caused more divisions in the estate, for now 5 was the Divisor, and 6 the Quotient, for so many times 5 is contained in 30, which is in proportion to A, D, 5 as A, B, 6 is unto an unite, this is directly contrary to Multiplication, for A, B, multiplied into A, D, or which is all one C, B, in C, D, will make 30, for the whole square A, B, C, D, which divided by one of the sides will produce the other, and so any number whatsoever that is multiplied by an other, will be divided by the same, and Multiplication and Division, do try and prove one an other, as shall be shown hereafter. This is the foundation, and now I will show the operation of it, illustrated with divers Paradigmas.   

Example 1.  
There was a man who by his industry had gained 963 li. and falling sick (past hope of recovery) made his Will; he had three Sons, to whom he bequeathed his estate impartially, not swayed by the custom of the Country, nor overruled by his affection, not knowing which of them would deserve it best, he distributed his estate equally to them all, which after his departure was accordingly divided; and first of all in division set down the Dividend, which in this example is 963 li: the Divisor is 3, which place under the first figure on the left hand viz. 9, then see how many times 3 is contained in 9, which will be found 3 times; and having made an arch, or such a line, as in the example (upon the right hand of the Dividend) set down the Quotient, which must be multiplied into the Divisor, and the product subtracted from the Dividend as 3 times 3 is 9, take 9 out of 9, and nothing remains; then remove the Divisor one place or degree towards the right hand, as under 6; then find how many times 3 will be contained in 6, which is twice, therefore put 2 in the Quotient, which multiplied by 3 makes 6, which take from 6 the Dividend, and nothing remains, remove 3 again to the next place, as under 3, which is there contained but once, put 1 in the Quotient, which times 3 is but 3, and that taken from 3 the Dividend, nothing remains; every one's share being 321 L, and so many times 3 the Divisor is contained in 963 L, the Dividend, if any thing remains in making of subtraction place it over the figure or figures, from whence subtracted, and cancel those figures you have done with all, With a light dash of your pen, and if any thing remains at last, place it beneath your work, in this example nothing remains, in whi●h you may see the form, and precedents in those following.   

Example 2.  
There was a man beyond sea who had received of a Merchant's Factor by Bills of Exchange 1726 Crowns, and 'tis required to know how many pounds Sterling it is; 4 of those pieces making 20 shillings or 1 lb. therefore 4 must be the Divisor, having set down 1726 the Dividend, and made a place for the Quotient, 4 being not contained in an unite, therefore set 4 under 7, and see how many times it will be had in the compound number over it, viz: in 17, and you will find 4 times, then put 4 in the Quotient, which multiplied into the Divisor, the Product is 16, which taken from 17, there will remain main 1, which place over 7, and cancel 17 of the Dividend; and 4 the Divisor, place again under 2; then seek how many times it will be contained in 12 (which is over it uncancelled) and you will find 3, which put into the Quotient, and that multiplied by 4, the Product is 12, which subtracted from 12 nothing remains, for order place a cipher over 2, and cancel the 12: lastly, remove 4 the Divisor under 6, in which, 4 is but once contained, therefore put 1 in the Quotient, and subtract 4 from 6, and 2 will remain, which writ down under all the work; where you may see 1726 Crowns, being divided by 4 (the parts of a pound Sterling) the Quotient is 431 lib. and 2 Crowns or 10 shillings more, as in the operation appears.   

Example 3.  
Here is given 2000 Drams, and it is required to know how many Ounces there be contained in that weight; being 8 Drams do make an Ounce, 8 is the Divisor, which cannot be had in 2, the first figure of the Dividend, therefore set it under the second place towards the right hand, and then see how many times it is contained in both those figures, viz: in 20, 3 will be too great, because 3 times 8 is 24, therefore take 2, and set it in the Quotient, and multiply it into the Divisor, saying, 2 times 8 is 16, which Product subtract from 20 and 4 remains, which place over the cipher; then remove 8 the Divisor to the next place towards the right hand, and look how often 8 will be contained in 40, and that is found 5 times just, which put in the Quotient, and say 5 times 8 is 40, take 40 out of 40, and nothing remains, and cancel as you go all the figures you have done withal; then remove 8 to the next place, which is under the last cipher, and being there is nothing more, put a cipher in the Quotient, and the work is ended, the Quotient 250 Ounces, nothing remaining, and so many will be found in 2000 Drams, the question solved, hitherto hath been examples with but one figure in the Divisor, the next with many.   

How to divide, when the Divisor consists of more figures than one. Example 4.  
There is a distance given between two places containing in feet 15000, and it is required to know how many stadiums it is, one consisting of 625 feet, 8 of them making an Italian Mile, or a thousand paces, 3 of those Miles a League. Here in this question 15000 feet is the Dividend, or number to be divided, 625 the Divisor, the number of feet in one stadium: observe (as before) to place the foremost figure of the Divisor under the first figure upon the sinister hand of the Dividend, if it will contain it, and the other figures following; if not, remove it one place towards the right hand, and then set down the rest of the Divisor in order: this done, you are to choose such a figure for the Quotient, as that being multiplied into all the figures of the Divisor, the Product shall be equal to the figures over them, or the nearest less; which you may find by Multiplication, but by the first figure of the Divisor sooner, and as certain, if you observe the figures following, and by practice will be made easy, in this example 625 is the Divisor, the first figure 6, cannot be had in 1, the foremost of the Dividend, therefore place 6 under 5 and the rest in order, under the other figures: then look how many times 6 will be found in 15, 3 will be too great, for 3 times 6 is 18; then take 2, which put in the Quotient, and then multiply it, into the first figure on the right hand of the Divisor viz. 2 times 5 is 10, over which stands a cipher, subtract 10 from 10 and nothing remains, set a cipher and go 1, then say 2 times 2 is 4 and 1 borrowed is 5, take 5 from 10 and 5 will remain, which set over the cipher, and cancel 0, and go 1, then say 2 times 6 is 12 and 1 borrowed is 13, take 13 from 15 and 2 will remain, cancel 15; and the Divisor as you make subtraction: this done, remove the Divisor one degree farther, as in this example, set 6 below 2 of the former divisor, 2 under 5, and 5, under the last cipher here of the Dividend: then see how often will 6 be in 25 (as yet uncancelled) and 'twill be found 4 times, put 4 in the Quotient, and then say 4 times 5 is 20, over which stands a cipher, take 20 from 20 and nothing remains, set a cipher over it, and keep 2 decimals in mind, then say 4 times 2 makes 8, and 2 will be 10, which subtracted from 10 nothing remains, place a cipher over it, and cancel the figures when subtraction is made, then say 4 times 6 is 24 and 1 in mind is 25, which taken from 25 nothing remains, the reason, of placing cyphers when nothing remains is, that there may be a significant figure remaining after them to the left hand, which is in value according to its place or degree; and observe that the remainder is less than the Divisor, or the work is false in multiplying or subtracting, in this example 24 stadiums is the Quotient, that is one League, or 3 Miles, for so many times is 8 contained in 24, the thing required.   

To divide a number of divers denominations, into any parts assigned. Example 5.  
To effect this,  Li. S. D Q. A 71024 16 3 0 B 3945 16 5 2 there are 2 ways, the one is for to reduce them into one and the least denomination, and then for to divide that sum by the number given: and if there be any remainder, you may reject it, as not of common use: when this is done, reduce the Quotient into several denominations again: as admit A were the number given in pounds, shillings, and pence, for to be divided into 18 equal parts: this reduced into the least denomination of Coin (as Farthings) will be in one sum 68,183,820 Q. this divided by 18, will be 3,787,990 Q. which divided by 4, the number of Farthings in a Penny, the Quotient will be 946 997 D. 2 Q. the Pence divided by 12 will produce 78916, S, 5 D, the shillings again divided by 20 (so many making a Pound sterling) the Quotient will be 3945 L, 16 S the remainder, so the 18 part of A 71,024 L. 16 S. 3 D. will be B 3945 L. 16 S. 5 D. 2 Q. This is a common way to those who are not conversant in fractions, but this following is better, shorter and not so subject to error, and thus.  
A more compendious way.  
First set down the greatest denomination if the Divisor will be contained in it; if not, multiply it into a less denomination by the parts, and to the Product add the parts of that denomination (if there be any) and then divide it by the given Divisor; and if there proves a remainder, multiply that by the parts of the next denomination, and so proceed to the last, as in this example 71024 li. is the Dividend, 18 the Divisor: the operation will be as in the former Paradigmas, by which you will find here a Quotient of 3945 li. and the remainder 14 li.  
The last remainder was 14 li. which multiplied by 20 shillings (the parts of a pound sterling) the Product will be 280 shillings, to which add 16 S. the sum 296 for the Dividend, which divided by 18, the Quotient will be 16 shillings, and the remainder 8 shillings.  
In the last division 8 s. was the remainder, and being there are 12 pence in 1 shilling, that multiplied by 12 the Product is 96 pence, to which add in the 3 D in the question, the sum will be 99 for the Dividend, under which subscribe the Divisor 18 and divide: the Quotient is 5 D. the remainder 9 D.  
The remainder last being 9 Den. which multiplied by 4 makes 36, the Product in Farthings, which divided by 18, the Quotient is found 2 Farthings, and nothing remaining, so the 18 part of 71024 Lib. 16 S. 3 D. is found to be 3945 Lib. 16 S. 5 D. 2 Q. as before.   

A succinct and perfect way of Division, without removing the Divisor, or cancelling a figure. Example 6.  
This way of all other that I have seen is the most compendious, facile, and conformable to practice, this species of Division being contrary to Multiplication, as in the effect, so in operation, as by their trials will more evidently appear, and first of all, you are to observe to set down the Dividend, and upon the left hand of it (in the same line) the Divisor; then draw a line right down with your pen, to part the numbers, and draw an other line unto that, for to separate the Divisor from the Quotient, which is to stand under it, as in this example, where 1638 years or days are given for to be divided by 7, make a point with your pen, under that figure of the Dividend, which first contains the Divisor, reckoned from the foremost figure of the left hand, supposing the point for to be a period to that number; as here I make a point under 6, for 7 cannot be had in one, but in 16 it may, and twice, set 2 in the Quotient under the line, beneath the Divisor 7, which multiplied together is 14, that subtracted from 16 the remainder is 2 which place under 6, in this form and method you must proceed, never subscribing the products, but committing them to memory, for they serve but till subtraction can be made, therefore unnecessary, and would encumber the work, with supernumerary figures, for which cause they were omitted in all my former examples, and yet this a briefer way, as will evidently appear, make an other point even with the remainder 2, and under the next place of the Dividend as under 3, and ask how many times 7 will be found in 23, the answer will be 3, then say 3 times 7 is 21, which taken from 23, and 2 will remain, which subscribe under the last point, and make an other against it under 8; then see how often 7 will be had in 28, which will be found 4 times, which multiplied by the Divisor 7, the product will be 28, which taken from 28 nothing remains, the Quotient 234.   

To find the Cycle of the Moon, or Golden number. Example 7.  
This Prime or Cycle of the Moon is a number proceeding from an unite increasing every year 1, unto 19, and then gins again, this revolution is to find the difference betwixt the Solar and Lunary year; for in this period of time all the Lunations and Aspects of the two Luminaries do return again to the same places as they had been before; but this computation exceeds the truth in time 1 hour and a half in 19 years or very near, this number may be always thus found: Our Blessed Saviour was borne in the first year of this Number, which you will find, if you add one to the year of ou● Lord Jesus, and divide the sum by 19, the remainder is the Golden number, and when nothing remains 'tis 19, the Quotient is the revolution or period of this Cycle, since the Sacred Virgin was a Mother, and for the more illustration of it, take this Paradigma; in the year of Grace, 1654. the Prime is required, add 1 to it, and then the number is 1655, to be divided by 19, which divisor place on the left hand of it, parted with a line both from thence, and from the Quotient to be found, as in the example; then under the first 5 make a point, for although 1 is contained in 1, 9 cannot be had in 6, this done see how often 1 will be found in 16, if I take 9 (which is the greatest number can be had at one time) 'twill be too great, for 9 times 1 out of 16, there will remain but 7, when 9 times 9 will be 81 out of 75, therefore set 8 in the Quotient, and say 8 times 9 is 72, taken from 75, and 3 will remain, which subscribe under the point, then say 8 times 1 is 8, and 7 decimals in mind is 15, which taken from 16, and 1 remains, that subscribed, make another point against 3, under the next place, & see how often 19 will be contained in 135, or 1 in 13 (the rest of the Divisor considered) and you will find 7 the nearest, which orderly place in the Quotient, and with that multiply always by the first figure in the Divisor upon the right hand, as 7 times 9 produceth 63, which taken from 65; or which is all one, take 3 out of 5 and 2 remains, which place under the point, then say 7 times 1 is 7, and 6 decimals in mind makes 13, which taken from 13 nothing remains; so there is only 2 remaining for the Golden number this present year; and 87 revolutions of this Cycle have been completed, and 1 year more since the Nativity of Christ.   

To find the Cycle of the Sun for the Dominical Letter any year. Example 8.  
This is a number proceeding by unites from 1, 2, 3, etc. And so to 28; it hath no relation to the motion of the Sun, but to the Sunday Letter, which are the first 7 of the Alphaber, serving by turns in a retrograde order, and every fourth year hath 2 Dominical Letters, so 4 times 7 is the period of this number, every letter having served in course, both in the common, and Leap years, and when this number hath had a period of 28, the next year it gins again at 1: the World's Redeemer was born in the ninth year of this number; therefore add 9 to the year of our Lord, in which it is required, divide the sum by 28, and if nothing remains 'tis the number sought, and if there does, that reremainder is the number required for the year currant in the Julian Calendar; and the Quotient will show the revolutions past, as in this example, where the Cycle of the Sun is required for the year of Christ 1654.; unto which add 9, and the sum will be 1663. for the Dividend, and 28 the Divisor: make a point under the second 6 from 1, and then see how many times 2 may be contained in 16; 6 will be too great, for 6 multiplied by 28 the product will be 168, whereas the 3 first figures are but 166, therefore take 5 for the first Quotient, which in order multiplied by the former rules into the Divisor, and the product subtracted from the Dividend, the remainder is 26, before which make an other point under 3 the last figure of the Dividend, and find a new Quotient, which in this will be 9, that multiplied, and the product subtracted, the true remainder will be 11, the number desired for the year 1654. by which Cycle, the Dominical Letter will be found A, the Quotient 59, the number of those Circles revolutions since our Saviour's Birth: but here I will say no more of this, lest I have said too much already, intending here Arithmetic, and not the computation of Calendars; but these two questions or propositions I rather chose, to show how ready for use the remainder stands, being in these the thing chief required, whereas in other questions (excepting such as are of several denominations) the remainder will be unknown, without the knowledge of Fractions.   

A number of divers denominations given, for to be divided into any parts required. Example 9  
Day's Hours ′ ″ ‴ ' ' ' '  730 11 37 0 0 0 A The magnitude of two Years. A Solar Month contains 30 10 29 2 30  B  
Here is given the space of 2 years, and it is required for to know the greatness of a Solar month, vulgarly accounted the space of 30 days, a natural day containing 24 hours every hour 60 minutes, commonly noted with such a dash over it as thus, 60 of them making one Second noted with 2 dashes of a pen, every Second is divided into 60 parts called Thirds signed with 3 dashes; Fourths with 4 etc. There are in 2 year's space 24 Months, therefore the number given, divided into 24 equal parts solves the question here propounded; this may be effected by the fift Example, but better by the way following, in division of each particular part or denomination thus.  
The number of days in 2 years are 730, which must be divided by 24, the number of months contained in that space of time, having placed your numbers in order prescribed, make a point under 3 in the Dividend, which done you will find 3 for the Quotient, that multiplied into the Divisor, and the Product subtracted from the Dividend, the remainder will be found, then make an other point a place farther, as under the cipher, and being 24 is not to be had in 10, put a cipher in the Quotient, which is 30 days, and the remainder 10, set down 30 in the former Table under days.  
The last remainder was 10 day's, which convert into hours the next denomination lesse, by multiplying of it into 24 (the hours contained in a natural day) the Product will be 240, to which add 11 hours (being parts of the 2 years given) the sum will be 251 hours: which divide by 24, the Quotient will be 10 hours, and the remainder 11, place 10 in the first Table against B under the title of Hours.  
The last remainder was 11 hours, which multiplied by 60, the minutes in one hour, the Product is 660, to which add 37 minutes against A, the sum will be 697 for the Dividend, which divided by 24 the Quotient will be 29 minutes, and the remainder 1, set 29′ against B in the column of minutes.  
In the last operation there was but 1 minute remaining, which is 60 seconds, that divided by 24, the Quotient will be 2 seconds, and the remainder 12: place the 2, in the first Table against B, under the title of Seconds.  
The 12 Seconds which did last remain, multiplied into Thirds (the next denomination lesse) the Product will be 720, which divided by 24, the Quotient will be 30 Thirds, and nothing remaining, if any thing bad, you might have continued them in this manner to Fourth's &c. The last Quotient put into the Table, you will find against B, that a Solar Month contains 30 Days, 10 Hours, 29 Minutes, 2 Seconds, and 30 Thirds.   

Breviates, or compendious ways and observations in Division exemplified. Example 10.  
If you are to divide by  5754 7101  1   4  2877 789   13701 10,000  2   3  4567 2500  any significant figure only, it is unnecessary for to set down the Divisor, but to keep it in mind and the product as before, while subtraction's made, the Multiplier will be the Quotient in all such cases, as by these 4 Paradigmas will appear; the first is the year since the Creation of the World, for to be divided into two equal parts; make a point under 5, wherein 2 is twice contained and 1 remains; then point forward: 2 will be in 17, 8 times: then 2 in 15, 7 times: and lastly, 2 in 14, 7 times, the Quotient 2877. In the second Table there is given 13701 Nobles, one being in value 6 s. 8 d. 3 of them makes 20 s. and it is required to know how many pounds sterling it will make in all: divide by 3, and say 3 will be found in 13, 4 times; in 17, 5 times; in 20, 6 times; and in 21, 7 times; the Quotient is 4567 li. In the third Example 10,000 Crowns are received beyond sea, and it is required to know how many pounds sterling must be paid for them in England, 4 of them making 20 shillings, make a point under the cipher where 4 will be had in 10, 2 times; in 20, 5 times; and being nothing remains but 2 cyphers more towards the right hand, annex them to it, and it is done, the Quotient being 2500 li. Fourthly, there is 7101 li. for to be equally distributed unto 9 men; 9 cannot be had in 7, therefore make a point under the next place, and then 9 will be found in 71, 7 times; and next in 80, 8 times; and lastly, in 81, 9 times; the Quotient 789 li. And so much does every one of their shares come unto.  
When there is an unite with cyphers annexed unto a Divisor, cut off so many places upon the right hand of the Dividend, as there were cyphers in the Divisor: as for example, 120 li. is to be distributed equally to 10 men, for the cipher in the Divisor cut off one place in the Dividend, and 12 li. will be every one's part or the Quotient; or if 100 Soldiers were to receive for their pay 625 li. every one's share will be 6 li. and 25 li. over, which by the former rules converted into shillings, by being multiplied by 20, the product will be 500 s. from whence cut off the 2 cyphers, and there will remain 5 s. So 6 li. 5 s. every Soldier must receive. And so for any other of this kind: the reason is evident, for 1 divides nothing, and the Quotient must have so many places less than the Dividend, by the number of cyphers in the right hand of the Divisor, as by the common way of Division will plainly appear.  
If the Divisor consists of any significant figure or figures in the foremost place, and a cipher or cyphers to the right hand, leave out the cyphers in the Divisor, and cut off so many places upon the right hand of the Dividend, and with the residue divide, and when the division is done, annex the cyphers to the D visor, and to the remainder, the figures that were severed from the Dividend, both of them constituting a Fraction, or true part of the Integer.   

Example 11.  
Admit the distance from London to the City of York were 48,080 Poles or Pearches, and it is required to know how many miles they are a sunder: 40 Poles makes a Furlong, and 8 of them a Mile, or 320 Poles, which is here the Divisor, and cutting off the cipher on the right hand, I must do so in the Dividend, which will be then 4808 for to be divided by 32, which accordingly done the Quotient will be 150 Miles, and 8 remaining, to which annex the cipher cut off it will be 80 Poles, or a quarter of a Mile, for the distance desired.   

An Examine of Multiplication and Division.  
These two species do try one the other, as Addition and Subtraction did: for in any number that is multiplied, if the Product be divided by the Multiplier, the Quotient will be the Multiplicand, as before: and so likewise any number that is divided, if you multiply the Quotient and Divisor together, and to the Product add in the Remainder (if there be any) the sum or Product will be the Dividend again, if your work be true.   

Example 12.  
The Julian, or old account An. Dom.. Bissextiles 1654. 413  1652 413 2 Rem. 2 1654. 1 2 did make the magnitude of the year for to consist of 365 days and 6 hours; 24 making a natural day, for which cause every fourth year contained 366 days, commonly called Leap-yeare or Bissextile; in one of those was our sacred Redeemer borne; now to find this Bissextile for any year since, or to come (according to the old Calendar) divide the year given by 4, the Quotient shows the revolution of those Leap-yeares, since His happy Birth, the remainder are the years elapsed since the last; and if nothing remains it is Leap-yeare; in this example is given the year 1654., which divided according to the tenth Example, the Quotient is 413, the number of Leap-yeares past, since the blessed Virgin Mary was a Mother: and the Remainder is 2, and so many years are elapsed since the last Bissextile, as by the first example in the margin; the second shows, whether the division be right, or no; the Quotient is 413 the number of Bissextiles, which now I make the Multiplicand, and 4 (which was the Divisor) the Multiplier, whose product is 1652, unto which add the Remainder 2 in the Division, the sum will be 1654., as before, in the second Table of the margin is evident. And thus is Division tried by Multiplication.   

Example 13.  
By the third example of Multiplication, Minutes 525960   Quotient 8766 Hours 8766 it was desired to know the number of minutes in a vulgar year containing 365 days and 6 hours, 24 making a natural day, that is 8766 hours in a year, which was there the Multiplicand, and 60, the minutes in an hour, was the Multiplier, which here I make the Divisor and the Product 525960 minutes, the Dividend which by the tenth example of Division may be divided by 6 cutting off a cipher upon the right hand of the Dividend, and then the Quotient will be found 8766 hours, the Multiplicand as before, which proves the Multiplication true; and so the like of any other: if any thing had remained, the last place cut off in the Dividend must have been restored unto it, and the cipher likewise to the Divisor.   

A Memorandum.  
Observe in this last way of Division, that how many points there be in the Dividend, so many figures, or cyphers there must be in the Quotient, and that every remainder must be less than the Divisor, otherwise the Quotient is too little, or the operations wrong, and when any number is given for to be divided, if you can find a number, that will divide both the Dividend and Divisor, without leaving any remainder, they will remain in the same proportion, as when cyphers are cut off from either, and if their Quotients do divide one an other, they will produce an other Quotient equal to the first, and their remainder (if there be any) shall have still the same proportion, as for example, if 48 were to be divided by 12 the Quotient would be 4, and 'twill be so if you take the half of these, as 24 to be divided by 6, or 12 by 3; the Quotient will be 4, as you shall see more at large hereafter, in all ways of Division, if the Dividend ends with an odd number in the place of unites, viz. 1, 3, 5, 7, 9, and the Divisor even, there must be a remainder when the division is done; for any odd number Multiplied by one that is even, the product will be even, although the Quotient be odd, all numbers may be divided into 2 equal parts, if the figure be even in the unite place, if odd it cannot without a fraction; if any number hath 5 or 0 in the unite place, 5 will divide it, without any remainder; but when any thing does remain, after Division is ended, although it were a part of the Dividend, yet as it hath relation to the Divisor, it must be of the same denomination with the Quotient as it is a fraction, or part of an integer; as if 10 s. were to be divided betwixt 4 men, the Quotient will be 2 and the remainder 2 s. but as it hath relation to four men it is but 6 d. or two fourth's of a shilling, the Quotients' denomination, and should be annexed unto it as a fraction according to the next Section and Paragraphs instructions.     

The Second SECTION treats of broken numbers, or parts of integers; divided into 5 Paragraphs, demonstrating Reduction, Addition, Subtraction, Multiplication, and Divisions, both of proper and improper Fractions.  

Paragraph 1.  
Explicating the definition, terms, value and qualities of Fractions, and how to reduce them from one denomination to another, as Fractions to Integers, and the contrary.  

Definitions and Terms.  
WHen the Dividend is less than the Divisor it is said to be a part of an Integer, or the whole, called a Fraction, or broken number, subscribed underneath one another with a line drawn between them in this manner ½ Fractions are proper, improper, or mixed.  
The terms proper to Fractions are usually these, the uppermost of the two is commonly called the Numerator of the Fraction, the 1 Numerator. 2 Denominator. other Denominator, and thus, as in the margin.   

Fractions demonstrated.  
 
A Fraction being defined a broken number, or part of any integer; the Denominator shows into how many parts the Integer is to be divided, the Numerator denotes the number of those equal parts for to be taken, and for the farther illustration of it, admit the lines A, B, C, D, E, F, for to be all alike in length, each containing a Foot by the Standard, and if it were required to know the value of this fraction 1/12 of a Foot, the answer will be 1 inch; for according to the Denominator, a foot is to be divided into 12 parts, and the Numerator being an unite, the fraction is one of those equal parts of the line A, G, containing 1 inch, and ¼ part of a foot is 3 inches; for the foot (according to this fraction) is for to be divided into 4 equal parts as B, H, and one of those divisions is to be taken, and ⅓ of a foot is 4 inches; 12 inches being divided into 3 parts as C, I, than one of those parts will be 4 inches, and ½ will be 6 inches, as in D, K, and ¾ will be 9 inches as by the pricks under the lines does evidently appear in E, L, and so in the rest, being hitherto simple fractions, whose Numerators are always less than their Denominators.   

Compound broken numbers or Fractions of Fractions Demonstrated.  
A compound Fraction is a part of an integer subdivided into other parts, and often those again; and usually subscribed in this manner ⅓ of ¾, by the Denominator of the second fraction, the integer is for to be divided into 4 parts, and according to the Numerator 3 of those parts must be taken, and by the first fraction ⅓ part of that is to be taken as in the former example ¾ of E, L, 12 was 9 inches, and that subdivided into 3 parts, one of them will be 3 inches, as by the divisions above the line, with 2 pricks does appear: again ½ of ⅕ of ⅚ are the compounded fractions given, whose value is required in parts of a foot: F, M, is divided by 2 inches into 6 parts, 5 of those divisions is 10 inches; ⅕ of that, is 2 inches; and the half of that is 1 inch, as is evidently proved.   

Reduction of broken numbers into one single Fraction demonstratively proved.  
In all questions of this kind, there is nothing more to do, then to multiply one Numerator into another, and that product into the next, and so proceed (if there be any more compound fractions) as in Multiplication continued, and thus you will produce in fine a common Numerator, and working in this manner you may find a single Denominator to it, as by this example following.   

Paradigma 1.  
Here are 5 compound ¼ of ⅔ of ⅔ of 9/10 of ●/6 Numerator 180 Denominator 2160 fractions given of one Integer, and those for to be reduced into a single fraction, having but one Numerator and Denominator: begin to multiply where you please, or upon the right hand, 5 times 9 is 45, twice that is 90, and 2 times 90 produces 180 a common Numerator, and the Denominator to it will be 2160, as thus 180/2160.  

How for to reduce Fractions into their least denominations. Axiom 1. Fractions that are proportional in their Numerators and Denominators are equal in themselves.  
To prove this general Rule, and explain it farther, suppose 80/100 be a fraction given for to be reduced: the Numerator is half the Denominator; take away a cipher from either, and they are divided by 10, and their Quotients are 8/16, which are still the half one to an other, and so if divided by 2 they would be 4/8, and again by 4 the Fraction would be ½ which is all one with 80/160, and so in the last Question 180/1260 may be reduced by 10 unto 18/126; and both divided again by 6 unto 3/21; and these by 3 will be reduced unto 1/7, so the Fractions ¼ of ⅔ of ⅔ of 9/10 of ⅚ reduced into a single Fraction were 180/2160, or which is all one in the least denomination will be 1/12 or 1 inch: for if the Numerator and Denominator of the former fraction were divided by 10, and that Quotient 18/216 by 6 it will be 3/36; and both again by 3 it will be 1/12, or 2160 by 180, this 1/12 may be proved by the former Demonstration, in the line F, M, as thus.   

An illustration.  
Begin first with the Fractions upon the right hand, where you will find ⅚ of 12 inches for to be 10 inches, than 9/10 of that will be 9 inches: ⅔ of 9 inches is 6 inches, and ⅔ again of that is 4 inches: and lastly, ¼ of 4 inches is 1 inch, the quantity of this compounded Fraction, ¼ of ⅔ of ⅔ of 9/10 of ⅚; all which when reduced into a single fraction, and that again into the least denomination, is but 1/12 part of a foot, or 1 inch, the thing required for to be demonstrated.    

To find the greatest number that will divide the Numerator and Denominator of any Fraction, without leaving any remainder. Paradigma 2.  
Subtract the Numerator of Denominator 144 Numerator 96 Remainder 48 Remainder 48 Remaining 0● the Fraction out of the Denominator, and so continue on subtracting the lesser from the greater until nothing remains, so you will find 2 Remainders equal, which are the greatest numbers that can divide the Numerator and Denominator of the Fraction without leaving any Remainder, as in this Paradigma 96/144 is the Fraction given for to be reduced into the least denomination, which to effect, a common number is to be found, that will divide them both, as thus: subtract 96 out of 144 and 48 will remain, which subtracted from 96 the Remainder will be 48, which will be a common measure to them both: for 96 divided by 48, the Quotient will be 2, and 144 by 48 the Quotient will be 3, and the Fraction ⅔ equal by the last Axiom unto 96/144; if an unite had remained, they could not have been reduced: this is demonstrated by Euclid, but too tedious for practice.   

A more compendious way to find a common measure betwixt the Numerator and Denominator of any Fraction. Paradigma 3.  
The quantities of Fractions Divisor 96 144 Quotient 1 48 Divisor 48 96 Quotient 2 00 are according to the proportion of the Numerator unto the Denominator, whose common measure will be thus found; divide the greater number by the less, and then the Divisor by the Remainder (if there be any) and so continue on dividing, making the last Remainder the next Divisor, and the last Divisor Dividend until nothing remains, and then it is evident the last Divisor is the common measure; but if an unite remains then they are in their least denominations, as in the last Example 144 divided by 96 the Remainder will be 48, with which divide 96 nothing will remain, which shows 48 to be the common measure between them: again, in the first Paradigma 180/2160 was a fraction which is thus reduced, divide 2160 by 180, the Quotient will be 12, and no Remainder, which shows at first that 18 was the common measure, which will reduce that fraction to 1/12, as before is said and proved.   

The shortest way for to reduce fractions into their least denominations. Paradigma 4  
The two last ways of 360 36 18 9 3 840 84 42 21 7 reducing Fractions, I have shown more for variety and young beginners, then for practitioners in the Art of Numbers; for those who are conversant in Arithmetic, cannot be to seek a number, whereby to reduce fractions into their least denominations: but if any should, let them take a number that will divide both Numerator & Denominator, and those divide again by some other number until you reduce them unto their least denominations: and observe, that if there be a cipher or cyphers, both in the Numerator and Denominator as 100/2000 cut off an equal number on the right hand and it will be 10/200, or which is all one 1/20 for it is the same with dividing of them by 10: if one of the ten hath a cipher on the right hand, and the other 5, they may be reduced by 5, and any even number by 2, as in this example ●60/840 continue on the line between them, cut off their cyphers, and then the fraction is 36/84, divide them by 12, or by 4 you may; but admit by 2 and then 'tis 18/42, and that again by 2 is 9/21 which may be reduced by 3 unto 3/7 which is in the least denomination, equal in quantity to 360/840: and observe, as you do reduce them, draw a line betwixt the former fraction and that reduced, as in this Example, where you may also find 360/840 reduced by the last Paradigma, and the common measure found 120, with which divide the former fraction, and the Quotients will be 3 and 7, or 3/7 as before.   

Of improper and mixed Fractions. Paradigma 5.  
These by an  li. li.  s. s. s. 1 48375 1125 2 20 10 5  43 1  4 2 1  s. s.   d. d.  3 42 21 3 ½ 4 10 5 2 ½  12 6  4 2 apt name are called improper, for they are not fractions but in respect of the form, they serve as broken numbers, when they are perfect Integers or mixed, with fractions and whole numbers together, being oppugnant to the definition of Fractions, the Numerators in these being always greater than their Denominators, as shall be made evident here, and their great use hereafter: in the margin stands 4 Examples, whereof the first 2 are improper fractions, the other mixed: these, or any other of this kind may be reduced by the former Rules (if they be not in their least denominations) and thus 48375/43 in pounds, wherein 43 will be found the common measure to both terms, the Quotient will be 1125/1, or so many whole pounds: the second Example is 20/4 of a shilling; if the Numerator of this fraction 20, be divided by 4 the Denominator, the quotient will produce 5 s. or Integers: or by the fourth Paradigma reduced by 2 you will find 10/2 or 5 shillings as before: the third Example is a mixed fraction, as 42/12 which may be reduced by the former Rules and by 6 unto 7/2, or by 2 to 21/6 then divide 21 the Numerator, by 6 the fractions Denominator, the quotient will be 3 3/6 that is 3 Integers or 3 shillings: then 3/6 or which is all one reduced to ½ is 6 d. the value of the mixed fraction: the fourth Example is 10/4 of a penny, which reduced is ●/2 or 2 ½ d. which is 2 d. half penny: this reduction of fractions is convenient, but of no necessity, and so I will proceed to other questions of more consequence.   

An Integer, or mixed number being given for to be reduced into an improper Fraction. Paradigma 6.  
Reduction of Integers into improper Days Weeks 365 52 1/7 1 365  7 Fractions is performed by drawing of a line, and placing an unite underneath it, in form of a fraction, as if the days in a vulgar year were to be made an improper fraction, it will be 365/1 as in the margin; if an Integer and a Fraction were to be reduced into one Fraction, equal to the mixed number propounded, multiply the Integer by the Denominator of the Fraction given, and to the Product add the Numerator, the sum will be the Numerator of a mixed improper Fraction, whose Denominator was the former Multiplier: there are 52 weeks, and 1 day in a vulgar year, which mixed number is to be reduced into a Fraction, set down the number given 52 1/7 as in the margin, then multiply 52 by 7 (the Denominator of the fraction) the quotient will be 364, unto which add 1 the Numerator, the sum is 365; under which draw a line and subscribe the Denominator 7, there will be 365/7 a mixed improper fraction, equal unto 52 1/7 as in the Table does appear, or reduced by the last Paradigma in dividing the Numerator by the Denominator.   

Reduction of Integers into an improper Fraction, that hath a Denominator assigned. Paradigma 7.  
In all such cases there is no more to 1 2 15 21 135 210 9 10 be done than for to multiply the Integer propounded by the Denominator that is assigned, which must be subscribed under the Product, as in the first Example where 15 is required to be made an improper fraction, with 9 a Denominator, which multiplied by 15 the Product will be 135, under which having drawn a line subscribe the Multiplier, which is the Denominator given: again if 21 Integers were required for to be made a decimal improper fraction add a cipher to it, and subscribe 10 underneath it, so you will have an improper decimal fraction, as in the second Table, and in both equal to their Integers: for by the former Rules, if 135 were divided by 9 the quotient will be 15 Integers, and from 210 cut off the cipher, and 'tis divided by 10, and reduced again to 21 whole numbers, and so for any other.   

Reduction of Fractions that have unequal Denominators, unto others that are equal both in value and Denominators. Paradigma 8.  
For to effect 1 5/6 7/8 9/10 1 4/8 6/8 2 400/480 420/480 432/480 2 2/4 ¾ 3 100/120 105/120 108/120 3 any question of this kind, it is convenient when there are but two fractions given, for to draw a short line from the Numerator of the one, unto the Denominator of the other like a St. Andrews Cross, or the Roman letter X, as in the Example where ½ and ¾ are given, having several Denominators, which multiplied by the Numerators crosse-wise, as 1 by 4, and 3 by 2, their Products are 4 & 6, for two were Numerators: then multiply the Denominators together, for a common Denominator: so now the fraction is 4/8 & 6/8, the second operation in the first Table, which in the third (by the former Rules) is reduced to a less denomination as 2/4 & ¾ equal in value unto the fraction given; for ¾ is the same as 'twas at first; and 2/4 or ½ is all one in value or quantity, as is evident if reduced; the Numerator being half the Denominator in each fraction: the reason of this Paradigma is perspicuous either fractions containing both denominations. If many fractions of several denominations were given for to be reduced into a common Denominator, observe to multiply all the Numerators into every one of the Denominators continued but its own, this continual multiplication will produce new Numerators: then multiply all the Denominators together, which continued multiplication produceth a Denominator common to all, the value and proportion of all the fractions still reserved, as by the last Table in the margin where ⅚, ⅞, 9/10, were the fractions given, to be reduced unto a common denomination thus, 5 multiplied by 8 and that product by 10 will be 400; then 7 by 6 and by 10 produceth 420, than 9 by 8 and 6 makes 432, for the 3 new Numerators, whose common Denominator is 480, produced by multiplying of 6, 8, & 10 continually, as in the second operation, in the 2 Table appears, where in the third operation, they are reduced to 100/120, 105/120, & 108/120, and all these equal in proportion to the value of the first, and by the former Rules may be reduced to the same predicament, and so may improper fractions being all connexed into one denomination.   

To find whole numbers that shall have proportion one to another, as any given Fractions have to themselves. Paradigma 9  
Numerators of 
●/3 ●/3 so will 1 be to 2. (2) 15/20 to 16/20 so will 15 be to 16.  Fractions, that have one common Denominator, are in proportion one to another, as are the Numerators made Integers, so in this first Example as ⅓ is in proportion to ⅔ so shall 1 be to 2, as in the demonstration of fractions is evident where ⅓ of a foot is 4 inches, and ⅔ is 8 inches, one double to the other, as 2 is to 1. In the second example ¾ & ⅘ of a pound sterling is given, and 'tis required to find two Integers that shall have such proportion one to another as their fractions are in: their Denominators being unequal, reduce them by the last Paradigma to 15/20 & 16/20 which must be in proportion as 15 to 16: or which is all one, as ¾ is in proportion unto ⅘, so shall 15 be to 16: their Denominators being taken away and so the Numerators made Integers: the thing is evident ¾ of 20 s. being 15 s. and ⅘ of 20 s. is 16 s. and thus may you find whole numbers to proper or improper fractions, and when found, multiply them both by any one number, and their proportions will remain the same, as was said before and proved by Reduction.  

The proportion of Fractions, as to their Integers and parts.  
1] As the Denominator of any Fraction  
Is to the Numerator of the same,  
So will the Integer be in proportion  
Unto the parts of the same Integer.   

Or thus,  
2] As the Denominator of a Fraction  
Is in proportion unto the Integer,  
So will the Numerator of the Fraction  
Be in proportion to the parts required.   

An illustration.  
To exemplify this, The Fraction given ⅔. As 3 is to 2, so 12 to 8. Or, As 3 to 12, so is 2 to 8. admit ⅔ of a foot were the fraction propounded, which Integer is divided into 12 inches; and according to the first demonstration of fractions ⅔ of a foot is 8 inches, and by these last two Rules, as 3 is to 2, and 12 inches is to 8 inches; or as 3 is to 12 inches, so will 2 the Numerator be to 8 inches; for the two means multiplied together, will be equal unto the product of the two extremes, that is, 2 times 12 is 24, and so will be 3 times 8, or as 3 to 12 so 2, which second and third number multiplied together, and the product divided by the first, the quotient will be 8; for 12 multiplied by 2 will make 24, and divided by 3 the Denominator of the fraction, the Quotient is 8, the part of the fraction required, and are proportional, as in Lib. 2. Parag. 7. Axiom 11.    

To find the value of a single Fraction according to the proportional parts of any known Integer. Paradigma 10.  
To effect any thing of this kind, observe to multiply the Numerator of the fraction, by the next inferior known parts of the Integer, the product divide by the fractions Denominator; if nothing remains, then is the Quotient the true value of the Fraction required, but if any remainder happen, you must proceed to the next inferior denomination; and if any thing remains, descend to the next, multiplying the Numerator by the known parts of that Integer and dividing by the Denominator, or abbreviate them as you please, and if any remainder be after the least denomination, subscribe it, as a fraction or part of that Integer, as by these 3 Examples are evident, wherein the first is given ⅚ of a foot, consisting of 12 inches or equal parts; which 12, multiplied by 5 will produce 60, that divided by 6, the Quotient is 10 inches, the value of the fraction required: In the second Table 5/16 of a foot is the fraction propounded, 12 inches multiplied by 5 and divided by 16, the Quotient is 3 and the Remainder 12/16, which reduced into the least terms is ¾, so the value of 5/16 is 3 inches and ¾, that is, 3 quarters of an inch. In the third Table, there is 53/66 of a chain for Land measure, whose length is 11 yards, every yard containing 3 feet, and every foot 2 links of the chain, and the value of this fraction is required; first 53 the Numerator multiplied by 11 produceth, 583 which divided by 66 the fractions Denominator, the Quotient is 8 yards, the Remainder 55; the Numerator of a fraction as 55/66 which multiplied by 3 the next denomination lesse, the Quotient is 165, or thus 165/66 which may be reduced by 3 to 55/22, as in the Table, which fraction proves 2 feet, and 11/22 remaining, and being that 12 inches is the next lest denomination, multiplied into 11 the Numerator, the product is 132/22 which reduced is 66/11, that is 6 inches or one link, but here you are to note that 11/22 might have been reduced to a ½ before the multiplication, for after it 'twill be reduced no lower; and thus the fraction 53/66 is reduced to 8 yards, 2 feet, and 6 inches, as in the Table: and thus 7/9 of a pound sterling will be reduced to 15 s. 6 d. 2 q ⅔: and ⅚ of a pound Averdupois will be 13 ounces, 2 drams, and 2 scruples: and thus in time 41/42 of an hour for the parts of this Integer. See the 9 Example, Sect. 1. Parag. 5. the Reduction thus, 58′. 34″. 17‴. 8 ' ' ' ' 4/7. the true value of 41/42 of an hour; but to lose time, and trouble you here with scruples, were impertinent; Examples are often troublesome, and practise pleasant, so I will proceed.  

Reduction of Integers, from one Species or denomination to another, into parts required.  
By this reduction of Integers, is understood the converting of one Species into another, as into a less or greater part, and hath relation unto Number, Weight, and Measure, viz: as in reducing pounds sterling into shillings, those into pence, and them again in farthings, and the contrary; and so pounds into ounces, drams, scruples, grains, etc. & so in measures, and all other things, whose lesser parts are usually known; all which parts are Integers as in respect of their own denomination, place, or column; but in relation as to the next denomination greater, they are no more than fractions to that Species; as a farthing in respect of his place is an Integer, but in relation to a penny 'tis a fraction of it as ¼, and so are pence to shillings, and those again to pounds, and so are all other divisible things in the world.    

How to reduce Integers of any Denomination, into Integers of a less Species. Paradigma 11.  
Multiply the Integers given, by the known parts; that is, the number of that denomination, which made that an Integer, the product will be the thing required: as for example, 1145 li. is to be reduced into farthings; but first into crowns, 4 making one pound sterling; then 1145 li. will be reduced to 4580; one crown being 5 shillings, multiply that by 5; or 1145 li. the sum given by 20 s. the product in either will be 22900, that multiplied by 12 produceth 274800 pence and that by 4, the number of farthings in a penny the total product is 1099200; the number of farthings contained in 1145 li. which if it had been multiplied by 960, the farthings in 1 li. sterling it must have produced the same, and so the like of any others at one operation, if you please.   

How to reduce Integers of mixed numbers, or several denominations to the least. Paradigma 12.  
This differs nothing from the last, if you do but add to every product, the Integers (if there be any) in every particular, and inferior denomination, as you orderly do descend from one to another; as admit 14 li. 13 s. 4 d 3 q were a sum of money given to be reduced into the least denomination, first begin with the greatest 14 li. which multiplied by 20 makes 280 s. to which product add 13 s. the sum w●ll be 293 s. that denomination multiplied by 12 produceth 3516 d. unto which add 4 d. the sum is 3520 d. that multiplied by 4, the number of farthings in a penny, the product is 14080 q. to which add 3 q. the sum will be 14083 q. the number of farthings in 14 li. 13 s. 4 d. 3 q. reduced into the least denomination: and thus 12 lb. 10 Oun. 8 Pen. 6 Gra. Troy weight will be reduced into their least denomination making 74118 Grains, and 2 Pearches, 4 Yards, 2 Feet, & 10 Inches, will be thus reduced into 574 Inches; and so the like for any other.   

How to reduce Integers of the least denomination into the greatest. Paradigma 13.  
Integers of any less denomination, are reduced into a greater by Division, for the number given must be made less, the species being greater, and contrary to the last; one proving the other, as now the examples, and let 1099200 be a number given in farthings to be reduced into pounds sterling, or their greatest denominations, reduce them first into pence, 4 being the Integer, so 1099200 divided by 4, the Quotient will be 274800 d. these reduced into shillings by 12, the Quotient will be 22900, and these divided by 20 s. the Quotient will be 1145 li. and no Remainder, the former sum reduced, as in the 11 Paradigma.   

How to reduce Integers that are mixed numbers, from their least denomination to their greatest. Paradigma 14.  
This is performed as the last by Division, and where any thing remains, set that Remainder in its proper column, and divide the Quotient again by the parts of the next denomination greater: as admit 14083 q. the mixed number given, as in the 12 Paradigma: the next denomination greater are pence, one penny containing 4 farthings, therefore 14083 q. divided by 4, the Quotient will be 3520 pence, and 3 farthings remaining which place by itself, and then divide again 3520 d. by 12, the parts of a shilling, the Quotient will be 293 s. the Remainder 4 d. which set down, and divide again 293 s. by 20 s. the denomination of a pound sterling and the Quotient will be 14 li. and 13 s remaining; so 14083 farthings in their several denominations reduced will be the sum of 14 li. 13 s. 4 d. 3 q. as in the 12 Paradigma, and so the like of any other mixed sum or quantity in either Number, Weight or Measure, etc.   

Reduction of Integers that are of several denominations into a single Fraction of the greatest denomination. Paradigma 15.  
This may be performed by what hath been said already, yet lest you should be to seek when there is use of such a question, observe this, reduce the Integers of several denominations, into the least (by the former Rules) which shall be a Numerator to the fraction, whose Denominator shall be the next denomination greater reduced to the least, as 4 s. 4 d. 2 q. in the least denomination is 210 q. for the Numerator, whose Denominator will be 960 q. that is, 1 li. reduced to q. so the fraction of 4 s. 4 d. 2 q. is 210/960 which reduced is 21/96 or 7/32 of a pound sterling; and thus 2 ℥. 3 ʒ. 1 ℈. if reduced to a single fraction in the denomination of a pound Averdupois the fraction will be 58/384 or reduced 29/192: and so for Measure, or any other question of this kind.   

To reduce an Integer of any denomination into a single Fraction of a greater. Paradigma 16.  
All questions of this kind are but fractions of fractions, and so accordingly to be reduced into any greater denomination: for if 1 farthing were to be made a fraction of a shilling, or a pound sterling, it must be thus expressed: ¼ of a penny, and in the next denomination ¼ of 1/12, that is, 1/48 of 1 s & that again 1/48 of 1/20 will be 1/960 of a li. & so likewise ⅓ of a ℈ is ⅓ of ⅓ to the next denomination, that is 1/9 of a ʒ: and that again 1/72 of an ℥ is 1/1152 of a lb Averdupois; and 2 ℈ is ⅔ of ⅛ of ⅙ that is 1/192 lb. and 1 inch is 1/12 of ⅓ of ½ of a Fathom, or 1 Inch is ⅙ of 1/33 of a Perch or Pole, according to the Statute that is 1/198 of that measure, 16 ½ feet to the Perch: and so you may reduce any other Integer, as a fraction of a greater or less denomination.   

To express or set down an Unite as an Integer to a Fraction, whose value is unknown. Paradigma 17.  
The number of Unites contained in the Denominator of any Fraction, are equal in value to the parts of the Integer, as in the demonstration of fractions was evidently proved before; which granted, there needs no more in this case then to place the Denominator of the given fraction by itself, in manner of a broken number, and make also the Denominator the Numerator, containing the other once: as for example, ⅔ the Integer of it is 3/3, representing an Unite, as 'tis a fraction; and so in 9/10, the Integer or unite to it is 10/10 being a whole one; for the Numerator of a proper fraction is less than an Unite, by so much as the Denominator is greater than the Numerator, for 19/20 wants 1/20 of an Unite: and the contrary, both in less and greater numbers.  
I have been tedious in explicating the most difficult questions in reduction of Fractions: and if I have trespassed upon your patience I will endeavour a satisfaction to your labours, by making the rest which follows (and most Questions in Fractions) short and easy, if my dictates be rightly understood; and all of them useful to the ingenious Practitioner, as you shall find hereafter, and so I will proceed.    

Paragraph II. Addition of proper and improper Fractions.  

A Demonstration Arithmetical and Geometicall; proving the addition of Fractions or broken numbers.  
 
ALL fractions that are to be added together (whether proper or improper) must be of one denomination, or reduced by the former Rules, and then the Denominator will be common to the fractions, or to the sum of the Numerators added together; as for example, admit the line A, D, were a foot in length, divided into 3 equal parts, as A, B, C, D, then is A, B ⅓ part of a foot, or 4 inches; ad A, C ⅔ parts of the whole length A, D, or 8 inches: now if A, B, were added to A, C, that is ⅓ to 2/● their sum will be 3/3 or one Integer as A, D; for C, D, is equal to A, B, which was added to A, C. Again the line E, I, is a foot, divided into 4 equal parts, viz.: E, F, G, H, I, every equal part containing 3 inches: then will E, F, be ¼ of a foot, or 3 inches, which added to E, G 2/4 or 6 inches, the sum is ¾ or 9 inches E, H; for G, H, is but 3 inches equal to E, F, which was added to E, G: in like manner if E, F ¼ were added to E, H ¾, the sum will be 4/4 the Integer E, I, but when fractions have several Denominators they must be reduced unto one denomination, before they can be added; as if A, B ⅓. of a foot were to be added unto E, F ¼, they must be first reduced as by the 8 Paradigma before, as A, B, into fourth's, and E, F, into thirds, making the fractions 4/12 & 3/12 which added together is 7/12 of a foot or 7 inches: for the Integer divided into 12 equal parts (according to the Denominator of the fraction) 7 of those parts must be taken, for the sum of the 2 fractions: admit K, L, (equal in length to A, D, or E, I,) were a line divided into 12 equal parts or inches; and the line M, O, equal to K, L; then take A, B, 4 inches, and E, F, 3 inches, they will extend to 7 inches in the line K, L; represented in M, N, being 7/12 of M, O, and so the like of improper fractions, as if A, C ⅔ or eight inches were to be added unto E, H, ¾ or nine inches, they will be reduced by the 8 Paradigma before unto 8/12 & 9/12 and being added, their sum is 17/12 or 1 5/12 that is 1 foot or Integer, as K, L, or M O, more by the fraction N, O, 5/12, and so the like of all other single fractions, whether they be proper, or improper: And if they do consist of many figures, you may reduce them if you please by the 2, 3, or 4 Paradigma above into their least denominations with facility.   

To add a single Fraction unto a compound Fraction assigned. Paradigma 1.  
In this Example there are 4 columns or rows, whereof the first is a compound fraction as ⅖ & 3/6 of ⅘ to be added together into one sum, the ⅜ of ⅘ being a fraction of a fraction must be reduced into a single fraction, as by the 2 Sect. Parad: 1. Paragr: 1. will be found 12/40 and reduced to 3/10, which I place in the second column for to be added to ⅖, which will be effected by the 8 Parad: as in this third column 20/50 & ●5/50, and added is 35/50 & reduced is equal in value to ●7/10, as in the fourth column does appear: and for the farther illustration of this, admit these fractions were parts of a pound sterling, then would ⅘ be 16 s. and ⅜ of that must be 6 s. for to be added to ⅖ of 20 s. that is 8 s. the sum is 14 s. and so will 7/10 of a pound sterling be 14 shillings, as was requ red to be proved.   

Addition of mixed Fractions, in their several denominations given. Paradigma 2.  
In this example there is given ⅘ of a pound sterling, ⅔ of a shilling, and ¾ of a penny, which Fractions of several denominations are to be added into one sum: first reduce ⅔ by the 16 Paradig: that is ⅔ of 1/20 of a pound, which will be 2/60 or 1/30, as in the second column; and ¾ of a penny will be 1/320; for it is ¾ of 1/12 of 1/20 of a pound, which makes 3/960 or 1/320, which by the 8 Parad: added to 1/30 the sum is 350/9600 which may be reduced as in the third column to 7/192; which is now to be added unto ⅘ of a pound sterling, by the 8 Parad: whose sum as in the fourth column is made 803/960 as in the 5 column does also appear; the total sum of ⅘ li. ⅔ s. and ¾ d. which is thus proved by the 10 Parad: multiply 803 by 20 (the parts of a pound sterling) the product will be 16060; this divided by 960 or 96 the Quotient produceth 16 s. and 70/96 remaining, or 35/48, which Numerator multiplied by 12 the Product is 420, that divided by 48 the Quotient will be 8 d. and the Remainder is 36/48 or ¾ of a penny, that is 3 farthings: so the total in their several denominations is 16 s. 8 d. 3 q. and ⅘ of 20 s. is 16 s. the second fraction was ⅔ of a shilling, that is 8 d. and ¾ of a penny is 3 q. which proves the total of all the fractions evidently true.   

Addition of numbers mixed with Integers and Fractions together. Paradigma 3.  
If the mixed number be an improper Fraction, and the Denominators alike; add the Numerators together (as in the Demonstration) if they be not alike, reduce them, as in Sect. 2. Parad: 8. and then divide the sum of their Numerators by their Denominators; as if 11/4 were added to 19/4, the sum will be 30/4, and 30 divided by 4 the Quotient is 7½ the sum of 11/4 & 19/4. Again, admit 10/3 & 18/4 were for to be added together by the 8 Parad: Sect: 1. they will be reduced to 40/12 & 54/12, and by the 3 or 4 Parad: Sect: 1. they may be made in their lesser denom: 20/6 & 27/6 the sum 47/6, that is 7 ⅚, or for more brevity in many cases of improper fractions, having found their Integers, add them together, and the Fractions by themselves: as in these last examples 11/4 & 19/4 may be reduced to mixed numbers, as 2¾ & 4 ¾; which Integers make 6 and their Fractions 6/4 or 1½ which added to 6, the sum is 7½ as before: and so 10/3 & 17/4 or 9/2, are 3⅓ & 4½ the sum of which Integers make 7, and then ⅓ & ½ reduced to one denomination will be 2/6 & 3/6 their sum ⅚, so the total of the Fractions or mixed numbers 3 3/1 & 4½ proves 7 ⅚ as before, and so 3⅓ & 4¾ & 5 ⅖ added together in their Integers first make 12 and the fractions 89/60 that is 1 29/60 so the total is 13 29/60. And so for any other in this kind.   

Compendious and useful Rules in addition of Fractions. To add unto a Fraction any part or parts of a given number or a Fraction to an Integer. Paradigma 4.  
To add a Fraction unto an Integer, or the contrary, needs no operation, more than prescribing one before the other; as to 1 s. to add ¼ is 1 ¼ s. that is 15 d. or to ¼ d. to add 9 Integers, it will be 9 ¼ d. or 9 d. 1 q. and so to add a Fraction unto any part of a given number, or the contrary is the same; for it is but taking the part of the Integer given, and annex the Fraction to it, as thus: admit ½ of a shilling were to be added unto ⅖ of a pound, the sum is 8 ⅓ of a shilling, or 8 s. 4 d. but in case the number given were not divisible without more fractions, add an unite to the fractions Numerator according to the 17 Parad: Sect: 2. Parag: 1. and so having made it an improper fraction subscribe an Unite under the Integer, then multiply the Numerators, and Denominators together, their products shall be the number required, containing the sum of the Integers given, and the part of them which was to be added. As for example, let 12 be a number propounded, and it is required to find an other number, that shall be 1/7 more, add an unite to this fraction, and it will be 8/7; then make the other 12/1, these multiplied according to my prescribed order, will be 96/7 or 13 5/7 the number desired, this will be performed by the former Parad: but more compendious this way, and as evident; for an unite being added to the Numerator of the fraction, and multiplied by the whole number given, can increase it, nor produce any more than itself once, and the fraction, by which part, the number sought is to be greater: for 8 the Numerator, in respect of the Denominator 7 is but an unite and 1/7. And so for any other of these kinds.   

To add a part of a given number to any parts of it required. Paradigma 5.  
In all such cases add together all the fractions or parts given, and make the number propounded an improper fraction, then multiply the Numerators and Denominators by one another, reduce their products and the work is ended: as if 20 were a number given, and that 1/15 of it were required to be added unto ⅗ of the whole number, ⅗ & 1/15 is 50/75, which you may reduce into a less denomination as ⅔, which multiplied by 20/1 or by the 10 Parad: Sect: 2. Parag: 1. will be 13 ⅓, the thing is manifest, the several proportions of the fractions being collected into one sum: admit the number propounded had been 20 s. ⅗ of it is 12 s. and 1/15 of 20 is 1 s. 5/15 or ⅓ which is 4 d. so the total is 13 s. 4 d. and so the other proved 13⅓ of 20, to confirm the truth.    

Paragraph III. Subtraction of proper and improper Fractions.  

A Demonstration Arithmetically and Geometrically proving subtraction of Fractions, or broken numbers.  
 
ALL broken numbers that are to be subtracted, must be single fractions, or reduced unto it, with a common Denominator, which done subtract the lesser Numerator out of the greater, and the remainder will be the difference betwixt those fractions, under which difference subscribe the common Denominator, and then reduce both terms unto their least denominations if you please: and for the farther illustration of this, the line A, D admit a foot, divided into 3 equal parts, from whence ⅓ part is to be subtracted as A, B, the Integer is 3/3; from whence take ⅓ and the remainder will be ⅔, that is B, D, or A, C, and is the same in number, for A, B, 4 inches subtracted from A, D twelve inches, the remainder is B, D, eight inches. Again, the line E, I is a foot divided into 4 equal parts, as by E, F, G, H, I, then is E, F 3 inches, or ¼ of the whole line, which subtracted from E, H ¾ or 9 inches, the remainder will be E, G 2/4 which reduced is ½ or 6 inches; and so if E, F were to be subtracted from E, I a foot, that is ¼ from 4/4 or 3 inches out of 12 inches, the remainder will prove E, H ¾, that is 9 inches: but when fractions to be subtracted, have unequal Denominators, they must be reduced as by Sect: 2. Parag: 1. Parad: 8. and then subtract the lesser Numerator from the greater, as before; and for the illustration of this, M, O is a foot divided into 2 unequal parts, as in N, the part M, N containing 7 inches or 7/12 from whence A, B ⅓ of a foot or 4 inches is to be subtracted: these fractions by the 8 Parad: reduced, viz: ⅓ & 7/12 will be 12/36 & 21/36, then take 12 from 21, and the remainder will be 9/36, or reduced to ¼. so if A, B 4 inches should be subtracted from M, N 7 inches, the remainder would be ¼ or E F, equal to 3 inches; or in the line K, L 12 inches, the line M, N will extend to 7 inches, and the line A, B to 4 inches, where will be found 3 inches, equal to E, F. and in like manner M, N, 7/12 subtracted from the Integer 12/12 the remainder will be 5/12 N, O. I did take purposely these examples, to show how Addition of Fractions may be proved by Subtraction, and the contrary, in Broken numbers, as in Integers if the Rules of Fractions be carefully observed.   

To subtract a single fraction from any compound broken number given. Paradigma 1.  
Here is represented to your view an Example containing four columns, the first is 7/40 & ⅝ of ⅕, to be subtracted from 7/10: first reduce the compound broken number into a single fraction, Sect: 2. Parad: 1. and this will be 5/40 or ⅛, as in the second column; this ⅛ must be added to 7/40, as in this 2 Parag: Parad: 1. and by the 8 Parad: Sect: 2. will be reduced to 56/320 & 60/320, as in the 3 column; which added, will be 96/320, reduced to 3/10; or 56/320 & 40/320 may be reduced unto 7/40 & 5/40 whose sum is 12/40 or 3/10 as before, the sum of these is to be subtracted from 7/10: and being their Denominators are equal, subtract their Numerators, and there will remain 4/10 or ⅖: to prove this, admit these fractions, parts of a pound sterling; then will 7/40 be 3 s. 6 d. unto which add ⅛ that is 2 s. 6 d. the sum is 6 s. which subtracted from 7/10 that is 14 s. the remainder will be 8 s. and so ⅖ of 20 s. is also 8 s. the subtraction proved.   

Subtraction of mixed Fractions from several Denominations. Paradigma 2.  
Here in this 1 1/10 li. & 2/3 s. from 4/5 of a li. 2 1/10 x 1/●0 2/3 of 1/20 is ●/60 or ●●/30 3 ●0/300 ●●/300 or 3/30 & ●/30 added is ●/30 4 2/15 from 12/1● remains 10/15 or 2/3 Table, is exposed unto your view 1/10 of a pound sterl: & ⅔ of a shilling, which mixed numbers are to be subtracted from ⅘ of a li. first by the 16 Parad: Sect: 2: reduce ⅔ of a shilling into the Denomination of a pound, as in the second row, which will be 1/30, that added to 1/10 of a pound will be 4/30 as in the operation of the third row, and reduced to the least denomination is 2/15, as in the fourth row or column, which is to be subtracted from ⅘ of a pound, which by the 8 Parad: Sect: 2. will be 60/75 & 10/75 which subtracted is ●0/75 and reduced is ⅔ of a pound; but in all such cases, if you can make the Denominators alike, either by Multiplication or Division it will be the same in effect: as in this Example 2/15 is to be subtracted from ⅘, which if multiplied by 3 the product will be 12/15 of the same denomination and proportion with 2/15 which subtracted is 10/15 or ⅔, as before, which ⅔ of a pound sterling is 13 s. 4 d. and now to prove this subtraction true, you may add to it again 1/10 of a pound, and ⅔ of a shilling, which is 2/15 of a pound, and ⅔ of a pound, their sum by the last Parag: and 2 Parad: will be found 36/45 or ⅘ as before; or try them in their several species thus, 1/10 of a pound is 2 s. and ⅔ of a shilling is 8 d. this 2 s. 8 d. taken from ⅘ of a li. that is 16 s. the remainder is 13 s. 4 d. as was the former.   

Subtraction of mixed numbers, when Integers are annexed unto Fractions given. Paradigma 3.  
When the number given is an improper fraction to be subtracted from another of the same denomination, take the less Numerator from the greater, and beneath the remainder subscribe the common Denominator, as in the last Demonstration: if not alike, reduce them, as by Sect. 2. Parad: 8. and then divide the remaining Numerator by the former Denominator: as for example, take 11/4 from 30/4 the remainder is 19/4, that is 4¾; or take 19/4 from 30/4 the remainder will be 11/4 bs before, that is 2 ¾; or reduce the improper fractions to mixed numbers, and then subtract them, as 2¾ from 4¾, their terms being alike; subtract the Integers, the remainder will be only 2; and so will 11/4 taken from 19/4 be 8/4 or two Integers.   

Compendious ways and useful Rules in subtraction of Fractions mixed or compounded. Paradigma 4.  
In all improper Fractions reduce them unto compound numbers by annexing the fractions to their Integers respectively, and if the greater number hath annexed unto it the greater fraction, subtract the lesser Integer from the greater, and then the fractions from one another, as in the last Examples: if the fractions be of several denominations, reduce them by the former Rules to a common Denominator, and then subtract the lesser Integer from the greater, and so likewise the fraction annexed to it if there be any, as 6 d. ¼ from 9 d. ¾, the remainder will be 3 d. 2/4 or ½: if the lesser number had no fraction the remainder would have been only 3 ¾ if any number were given for to be subtracted from a greater number with a fraction annexed to it: subtract one Integer from the other, and annex the fraction to the remainder; as 8 s. from 13 s. ¼ the remainder will be 5 s. ¼, when the lesser number for to be subtracted hath a fraction, and the greater number none, subtract first the fraction from an unite of the other number made a fraction as by the 17 Parad: Sect: 2. Parag: 1. and having set down the remainder, add the unite unto the lesser number given to be subtracted, and so proceed as in subtraction of whole numbers, and for example, take 13 s. ⅓ out of 3 li. or 60 s. First take the fraction ⅓ from 3/3, the remainder is ⅔, then say 1 borrowed and 3 is 4, which taken out of 10 the remainder is 6; then 1 and 1 makes 2, which subtracted from 6 the decimal, the remainder is 4: the total remaining is 46 s. ⅔, or 2 li. 6 s. ⅔ and so likewise if a fraction only were to be taken out of an Integer, borrow an unite of the whole number, which makes a fraction of the same denomination with that given to be subtracted; as from 14 to subtract ⅔, the remainder will be 13 ⅓; and if in case the fraction were greater than the fraction annexed to the greater number, take an unite from the greater Integer, and make the broken number annexed unto it an improper fraction, and then subtract by the former rules; as admit 6 ⅔ s. were to be subtracted from 11 ¼ s. the greater number, yet hath the lesser fraction; therefore take an Integer from 11, and make the fraction 5/4, then by the 8 Parad: Sect: 2. Parag: 1. 5/4 & ⅔ will be reduced to 15/●2 & 8/12 the difference is 7/12, than the unite borrowed, and 6 will be 7, which taken from 11 s. the remainder is 4 7/12 s. and so will 6 s. 8 d. taken from 11 s. 3 d. be 4 s. 7 d. if the fraction of the lesser number had been less, although of another denomination, reduce the fractions to equal Denominators, and then subtract them, and their Integers, the less from the greater, as before: but if you should be doubtful in any fraction or mixed number, (of several denominations) which is the greater fraction, Reduction makes it evident, and practice will make the most rugged part of Arithmetic plain, pleasant, and easy.   

To subtract any part of a given number, or fraction, from the same broken number propounded. Paradigma 5.  
How to subtract Fractions from Integers, or broken numbers, I have shown already, and so may these be performed, but yet more concisely thus: if there be many fractions given, either mixed or compounded, reduce them into a single fraction by the former Rules, and having subtracted the fraction from an unite, multiply it by the number or fraction given, whose parts these were required; the product will bring to light the fraction or part inquired: as for example to subtract ⅔ parts from ¾ of a shilling take ⅔ from the Integer, and there will remain ⅓, as by the 17 Parad: Sect: 2. Parag: 1. then multiply ⅓ by ¾ the product will be 3/12 or ¼ s. which is thus proved; ¾ s. or 9 d. is the fraction propounded, and 'tis required to take ⅔ parts of the same fraction from it, which ⅔ parts is 6 d. subtracted out of 9 d. the difference is 3 d. or ¼ s. as before. Again 9/10 li. is a fraction given from whence ¼ of ⅔ parts of the same number is to be taken: this mixed fraction is reduced to 2/12 or ⅙, that taken out of the Integer, the remainder is ⅚, which multiplied by 9/10 produceth 45/60, and reduced is ¾ li. or 15 s. the fraction given was 9/10 which is 18 s. from whence take ¼ of ⅔, the remainder is 15 s. as before: for ⅔ of 18 is 12, and ¼ of 12 is 3, which subtracted from 18, there will remain 15, if ⅓ & ⅖ parts of ¾ li. and the difference is required, reduce the parts given into a single fraction, than ⅓ & ⅖ will be 11/13: and taken from the Integer are 4/15, those multiplied by ¾ li. the given fraction produceth 12/60 or ⅕ li. the true diff rinse required; for ¾ of a pound sterling is 15 s. whereof ⅓ is 5 s. and ⅖ is 6 s. the sum 11 s. which subtracted from 15 s. the remainder will be 4 s. and so was ⅕ of a li. as formerly; when more fractions are given, they must be reduced to a single fraction; and then the operation is the same with these last, which have relation properly to the fraction from whence they are to be subtracted; as for a farther instance ⅔ of ⅖ of a li. is 8 s. as a fraction of a fraction; But in these cases ⅔ of ⅗ is to be subtracted from ⅗, as 8 s. from 12 s. the difference is 4 s. and so ⅓ multiplied by ⅗ will produce ⅗ li. or 4 s. the true remainder.   

To subtract a part of any given number, or any parts from a part or parts of the same number or fraction propounded. Paradigma 6.  
In all such cases as these, take the lesser fraction from the greater, and then multiply the remainder into the Fraction or Integer given (from whence these parts were to be subtracted) and the product will be the remainder: as for example, admit from ⅗ li. you were for to subtract ¼ li. take ¼ from ⅗ and there will remain 7/20, as by the last demonstration: this remainder multiplied by 20 or 2●/1 the number given, the product will be 140/20 or 7 s. and is thus proved; ⅗ li. is 12 s. from whence 2/4 of the same number is to be subtracted, that is 5 s. and then the remainder will be 7 s. Again, from ¾ to take the 1/18 part of ½, that is, from ½ of ¾ to take the 1/18 part of the same number given, that is ¾; subtract first the parts as 1/18 from ½, the remainder is 16/36 or 4/9, which multiplied by ●/4 makes 12/36 ot ⅓, the true remainder: and for trial of it, let the fraction given be part of a pound sterling, than ¾ will make 15 s. the half of it 7 s. 6 d. from whence take 1/18 part of the same fraction, that is 4, or 15 s. and the remainder is 10 d. which subtracted from 7 s. 6 d. there will remain 6 s. 8 d. which is ⅓ part of a li. as before: if 1/12 were to be taken from ⅔ & ⅙ of 24; reduce the ⅔ & ⅙ into a single fraction, as 15/18, or ⅚, from whence subtract 1/12 and there will remain ¾, which multiply by 24 or 2●/● the number given, the product is 72/4 or 18 the difference required: for ⅔ of 24 is 16 and ⅙ of 24 is 4, the sum of those fractions 20, from whence take 1/12 of 24, that is 2, and then the true remainder will be 18, as before: admit ⅓ of ¼ of 72 were to he subtracted from ⅛ part of the same 72: first reduce the mixed number ⅓ of ¼ into a single fraction, that is 1/12, which take from ⅛, the remainder will be 4/96 or 1/24, that multiplied by 72 or 72/1 the given number, the Product will be 7●/24 or three Integers: for ¼ of 72 is 18, & ⅓ of that is 6, which taken from 9 that is 1/0 of 72, the remainder is 3, as before; if there had been more fractions, thus to be subtracted, reduce them into one, and so proceed.    

Paragraph iv Multiplication of proper and improper Fractions.  

A Demonstration Arithmetical and Geometicall; proving the multiplication of Fractions, in mixed, and broken numbers.  
Multiplication of Fractions.  
As the square made of the Numerators  
Is in proportion to the square of their Denominators,  
So shall the superficial square of the Fraction  
Be in proportion unto the square of the Integer.   

An Illustration. Paradigma 1.  
In the former Scheme there is described a Geometrical figure or square, as A, B, C, D; every one of the four sides is divided into twelve equal part● or inches; so if A, B, 12 were multiplied by A, C, 12 the product would be 144, the number of square inches contained in a superficial Foot, as by the figure will evidently appear; admit a fraction propounded as ⅔ of the line A, B, to be multiplied by ¾ of A, C, multiply their Numerators by one another, the product will be 6 for the new Numerator, & 3 times 4 is 12 for the Denominator, so now the fraction is 6/12 or ½ which is less than e●ther of the terms given, that is ¾ or ⅔, for ¾ of A, C, is 9 inches, and ⅔ of A, B, is 8 inches, which multiplied together produceth 72 square inches, the half of 144, and so was the fraction, and their proportions thus, viz: as the square of the Numerators 6 is to the square of their Denominators 12, so will the square of the fraction 72 be in proportion to the integer or whole square 144 inches, that is 1 foot. Again, the 3 numbers given were 6, 12, 72; and as 6 the half of 12, so is 72 the half of 144: and consequently 72 multiplied by 12 produceth 864, which divided by 6 the Quotient will be 144: and so likewise in all the 4 numbers, the square of the two means, viz: 12 & 72 will be 864, and so much will the 2 extremes make, 144 multiplied by 6, Lib: 2. Parag: 7. Axiom 11. In the same manner ½ multiplied by ½ makes ¼; and so half of the line A, B, 6 inches multiplied by half A, C, likewise 6 inches, the product will be 36 inches, which is but ¼ part of a foot, containing 144 square inches: and in the same manner and proportion 1/12 of A, B, multiplied by 1/12 of A, C, will produce 1/144, and so the little square made of the Fraction representing 1 inch is but 144 part of a square foot, which is the Integer.   

To multiply compound broken numbers, or fractions of fractions. Paradigma. 2.  
Having reduced the mixed or compound numbers into a single fraction, the rest of your work is the same with the last, as admit ⅓ of ¾ were to be multiplied by ⅖ of ⅚: first reduce the fractions, as by Sect: 2. Parag: 1. Parad: 1. and you will find ⅓ of ¾ to be 3/12 or ¼ and ⅖ of ⅚ will be 10/30 or ⅓, which multiplied by ¼ makes 1/12, the true product of the fractions given; and being reduced are thus proved, ⅓ part of A, B, is 4 inches; and ¼ of A, C, is 3 inches, which multiplied together produceth 12 square inches; which is but 1/12 part of a foot, or of the square A, D, containing 144 inches.   

Divers proper Fractions for to be multiplied into one Product. Paradigma 3.  
In all such cases you are to add into one sum the fraction for the Multiplicand, and also the Multiplier; then multiply the single fractions together, as was said before, and the work is done: as admit ⅖ & 1/10 were fractions propounded for to be multiplied by ¼ & ⅙: first by Sect: 2. Parag: 2. add ⅖ to 1/10 the sum is 25/50 or ½, then add ¼ to ⅙ their sum is 10/24 or ●/12 to be multiplied by ½ the product will be 5/24, the thing required, which may be proved by the former demonstration.   

Mixed or improper Fractions given for to be multiplied. Paradigma 4.  
If mixed numbers be given for to be multiplied, (as Integers and Fractions) you must make them improper fractions, as by Sect: 2. Parag: 1. Parad: 6. and not multiply the Integers by themselves, and then their fractions, because the square of all broken numbers are greater or less, according to the square of the whole numbers to which they are annexed: as 2 ⅓ & 3 ¼ is more than 6 1/12, therefore make them improper fractions as 7/3 & 13/4; and then their products will be 91/2 that is 7 7/12. Again 2 ½ is a fraction propounded for to be multiplied by 1 ⅗ these reduced to an improper fraction, will be 5/2 & ⅕; their product is 40/10 or 4 Integers: and in the same manner must the operarion be, if one of the terms be a mixed number; and the other a single fraction or an Integer only, as 3 ½ to be multiplied by 4; they will stand thus 7/2 & 4/1 their product 28/2 or 14: if an Integer and a fraction only, as 20 to be multiplied by ●/4; place them thus 20/1 & ●/4: their product 6●/4 or 15 Integers: or 12 by ⅔ thus 12/1 ⅔ the product 24/3 or 8; yet here you are to note, that these last products if several denominations are but fractions, in respect of the square made of the Integer, which is the Denominator to it; as 8/12 or ⅔; and so A, B, 12/1 inches multiplied by ⅔ of A, C, a foot, viz: A, G, 8 inches, the product will be 96 or 9●/144 to the whole square A, B, C, D, 144, which reduced unto the least denomination, is likewise ⅔, as by the former Demonstration is made evident, in A, B, E, G, unto A, B, C, D; yet 12 by ⅔ is 8 Integers, if of one denomination. If any whole number be given for to be multiplied by a mixed number, add unto the product of the Integers, the fractional part of the entire whole number, the sum will be the product required: As for example, 12 ¾ is a mixed number propounded for to be multiplied by 16, which increased by 12 produceth 192, which is defective by ¾ of 16 the whole number, that is 12, the sum 204, equal unto the product of 12 ¾ or 51/4 multiplied by 16, according to the Rule of Fractions: for if the mixed number were made an unite more instead of the fraction, the product would contain the Integer once, and so much too great, as the fraction was less than an unite: thus 14 multiplied by 12 ¾ will produce 178 ½; and so for any other number of this kind.   

To find any part or parts of a given number or Fraction. Paradigma 5.  
In all cases of this kind, if there be many fractions given, reduce them by the former Rules into a single fraction, then multiply them together and the work is done. As for example, admit ⅓ of ¾ of 12 were required, the fractions reduced are 3/12 or ¼ which multiplied by 12/1 (the number given) the product is 12/4 or 3 Integers; which is thus proved, ¾ of 12 is 9 and ⅓ of 9 is 3 the number sought. Again, let the fractions given be ¼ & ⅛ parts of 4/5, first reduce the broken number into a single fraction, then will ¼ & ⅛ be 12/32 or ⅜ which multiplied by ⅘ produceth 12/40 or 3/10 the true numher required: suppose 20 were the Integer, then were ●/5; of it 16 whereof ¼ part is 4, and ⅛ part is 2 the sum of them 6, and 3/10 of 20 will be also 6, these differ nothing from the former prescribed Rules, so needs no more Examples to explain it.    

Paragraph V Division of proper and improper Fractions.  

A Demonstration Arithmetical and Geometrical, manifesting division of Fractions as well in mixed, as broken numbers.  
Division of Fractions.  
WHen broken numbers are to be divided, either proper, improper, mixed, or compounded, they must be reduced to single Fractions both for the Divisor and Dividend; then place that fraction, which is the Divisor, upon your left hand; and that for the Dividend towards the right; this done, make a Roman X betwixt them, as in Addition or Subtraction of Fractions that have several Denominators: then multiply the Denominator of the Divisor by the Numerator of the Dividend, the product will be a new Numerator; then multiply the Denominator of the Dividend by the Numerator of the Divisor, and the product will be a new Denominator, unto the fractions given which were thus divided, and this new fraction the Quotient; but directly contrary unto Division in whole numbers, where the Quotient is ever less than the Dividend, and in fractions divided, the quotient will be greater than the terms given, and the Divisor propounded may be greater or less than the Dividend, as by the Examples following shall appear, they being in proportion thus:  
As the Divisor of any fraction propounded  
Shall be in proportion unto the Dividend  
So will 1 Integer, or an unite  
Be in proportion unto the Quotient.   

An Illustration. Paradigma 1.  
In the last Scheme is described a Geometrical square figure, as A, B, C, D, representing a Fathom, every side containing 6 feet, and consequently the content of the whole square 36 square feet, now admit ⅙ part of A, B, were to be divided by ⅙ part of A, C, it will stand thus, ⅙ X ⅙ and multiplied crosswise the new Numerator will be 6, and the new Denominator 6, so the Quotient will be 6/6 or 1, the side of one square being 12 inches, will be contained in 12 inches once: so ½ is in ½, and ¼ contained in ¾ three times demonstrated by the first and second little squares, and ⅓ of A, B, divided by ⅓ of A, C, will be likewise 3/3 that is an unite, viz: E, C, by E, F, for if A, B, and the line A, C, were equally divided into 3 parts, one of those parts must be 2 Foot, or 24 Inches: so 2 will be contained in 2 once, or 24 in 24, also one time; and so ⅓ divided by ⅓, must be 3/3 or 1; and consequently in all fractions, as in Integers, when the Divisor is equal unto the Dividend, whether the fractions consist in Numbers, Lines, or in any Geometrical figures, the Quotient is an unite.   

Divers fractions being given for to be divided by a single fraction, or the contrary. Paradigma 2.  
Here is ⅕ & 3/10 for to be diviced by ¼: first reduce ⅕ & 3/10 into a single fraction by the former Rules in Reduction, and they will be 25/50 or ½ for the Dividend, and then set down ¼ the Divisor, which observe to place always towards your left hand; and having made a Roman X between them, multiply the Numerator of the Dividend crosswise by the Divisors Denominator, which product will be 4 for the new Numerator: then multiply the Denominator of the Dividend by the Numerator of the Divisor, the product will be 2 in this, the new Denominator, so now the Quotient is 4/2 or 2. Again, in the second Example, the Divisor is made the Dividend; and the Dividend, Divisor; and being multiplied crosswise (according to the order of Division) you will produce a Quotient of 2/4 or ½: for proof of these, in the last Demonstration A, B, is a Fathom or 6 Feet, which multiplied by A, C, 6, produceth the whole square A, D, 36 square feet: now in the first Example of this Paradig: the half of it was for to be divided by a fourth part of the same square, that is 18 feet, by 9 feet, the Quotient must be 2 Foot, that is 2 of those little squares; and this according to the proportion stated, viz: as the Divisor 9, shall be unto 18 the Dividend, so will 1 be unto 2: Or in the second Example, as 18 is to 9, so 1 to ½: and so in any other thing, as in Coin, admit 12 Pence the Integer, and the ½ of it, were to be divided by ¼ part of the same Integer, than I say 3 d. would be contained in 6 d. twice; and on the contrary if ¼ were to be divided by the ½ of the same Integer, that is 3 d. by 6 d. the Quotient must be 3/6 or ½ of a penny: in the same manner if ½ of ⅓ were a compound fraction given, for to divide ⅔ parts of the same Integer; reduce first ½ of ⅓ to ⅙, with which divide ⅔, the Quotient will be 12/3 or 4: and on the contrary make ⅔ the Divisor, the Quotient will be 3/12 or ¼, that is by the Demonstration as 6 is to 24, so 1 unto 4. And secondly, as 24 is to 6 so 1 unto ¼ part of a foot, as in the second square of the Scheme you may see: and so in money ⅙ part of 12 d. is 2 d. with which if you divide ⅔ of 12 d. that is 8, the Quotient will be 4 d. and on the contrary, if you would divide 2 d. by 8, the Quotient will be 2/8 or 2/4, according to the Demonstration.   

A whole number being propounded for to be divided by any Fraction, or Fractions given. Paradigma 3.  
When there are many fractions given, of necessity they must be made a single fraction, before you can divide any number with them, and 'tis necessary to reduce them into the least denomination: in the first column of this Scheme there is given ⅖ & 1/10; by which fractions, 20 must must be divided: the broken numbers are made a single fraction, by the former rules of Reduction, as 25/50, and reduced to ½, as in that column appears: In the second column I place ½, the Divisor, and then against that stands the Dividend made an improper fraction, as 20/●: between them I place the letter X, and having drawn a line beneath them all and multiplying then the Divisor  by the Dividend, the product is the Quotient, which here is 40, which is evident, if the nature or force of Fractions be well understood, for if 20 s. were to be divided by 1 s. the Quotient must be 20 s. and consequently by ½ it must be 40; for the fraction being but 6 d. 'twill be contained in 20 s. 40 times, and ½ divided by 20 will be 1/●●.   

A whole, or a mixed number being given to be divided, by any mixed number propounded. Paradigma 4.  
In the first column there is given 36 to be divided by 4 ½ which is the Divisor, and made an improper fraction will be 9/2, the Dividend 36/1, which multiplied by 2 (the Divisors Denominator) produceth 72, for the new Numerator, than 1 multiplied by 9 is but 9 for the Denominator, so the Quotient is 72/9 or 8 Integers; now if 9/2 should be divided by 36/1 the Quotient would have been a fraction of 9/72 or ⅛, as for these it is evident, that 4 ½ the Divisor multiplied by 8 the Quotient, the product will be 36, the Dividend as before: and so likewise 36 the Divisor, multiplied by ⅛ the last Quotient the product will be 36/8, that is 4 ½ the Dividend. In the second column there are two mixed numbers given to be divided, viz: 3 ⅓ & 6 ⅔, which made improper fractions, will be 10/3 & 20/3: and if 10/3 be the Divisor, the Quotient will be 60/30 or 2 whole numbers, as in respect of themselves, and by the operation in the Scheme appears, for 6 ⅔ contains 3 ⅓ twice, and is evident in themselves, and consequently if 20/3 were to divide 10/3, the quotient would be 30/60 or ½, the Divisor being twice the Dividend.   

Any part, or parts of a number given, to find the Integer. Paradigma 5.  
Admit 6 were a number given, which is the ¾ part of the number required: in all such cases divide the number given, by the fraction, the Quotient will be the number required: as in this Example, with ¾ divide 6/1 the Quotient is 24/3 that is 8: for ¾ of 8 is 6, the thing required. Or in any Question of this kind, you may multiply the number given, by the Denominator of the fraction propounded, and divide the product by the Numerator of the same fraction, the Quotient will be your desire; as 6 if multiplied by 4 produceth 24, and divided by 3 the Quotient sought as before: and so if 10 were the ⅚ part of a number desired, by either way 12 must be the number found: and so for any other question of this kind.      

The Epilogue to the first Book.  
HEre I conclude my first Book of Natural Arithmetic, having conducted you through most of the rugged ways both in the Theory and Practice, in Whole numbers, as in Fractions: yet but the foundation properly, upon which the ingenious Arithmetitian must build: I do not mean confined to my Works, or Rules, any more, than young practitioners shall think fit, as they please, or their fancies prompt them: but yet it is necessary in all Arts and Sciences, to state Principles and Elements, which well understood, it will be easy, and pleasant for the ingenious to proceed, and if these my labours can assist them in their progress, I shall be joyful of it, if slighted for my endeavours, I shall comfort myself with the thoughts of many Associates, since few or none, are quite exempt from censure, and usually the weakest sort of intelligible creatures are most capricious: but I being encouraged by some that are esteemed both wise, learned, and expert in divers Arts, I am from thence armed with resolution to proceed: and here in the Conclusion of this Book, for the benefit of the Reader in perusing History, ancient dates of Deeds and Records of antiquity, I will insert the Characters, or 7 Numeral letters used by the Romans, and divers Countries under their subjection, and in many things frequently continued to these days, and are briefly these, viz: I, 1. V, 5. X, 10. L, 50. C, 100 D, 500 M, 1000 And sometimes thus: D, 500 & M, 1000 And observe I before V is 4, as thus IV. & before X is 9, thus IX & X before an L is 40, as thus XL. all other numbers are made by reiteration of these letters, as TWO, 2. III, 3. VI, 6. VII, 7. VIII, 8. XI, 11. XII, 12. etc. then CC, 200. CCC, 300. M M, 2000 M M M, 3000, etc. but no otherwise increasing, or decreasing, according to their places, as Arithmetical Characters do, which are in number only 10, for to express all numbers known to men; and is undoubtedly the best and readiest way, for Computation, of any: but those who do condemn it out of ignorance, (as illiterate men do learning) let them use Pebble-stones and Chalk, like Conjuring characters behind their doors; or reckon upon their fingers, as Juvenall in his tenth satire describes an old man: 
— At que suos jam dextra computat annos.  
By counting of his Joints, his Age appears:  
Thus at his Finger's Ends, he hath his Years.   
The end of the first Book.   

THE SECOND BOOK SHOWING THE EXTRACTING OF SQVARE & CUBIQVE ROOTS, etc.  
Arithmetical and Geometrical Progressions: Universal Axioms, and Canons in Arithmetic: WITH The Similitude and Proportion of Numbers, in relation unto their Quantities and Qualities: from whose speculation, all Rules in Number are originally derived.  
By Thomas Willsford.  
LONDON, Printed by J. G. for Nath: Brook at the Angel in Cornhill, 1656.   

THE SECOND BOOK, Showing the extracting of Square and Cubique Roots, etc.  

Paragraph I. The definition of a Square Root.  
A Square is a Geometrical figure of the second quantity, composed of right lines, multiplied one into another: and the Root is one of those sides, as in Multiplication and Division was shown before, so to find the Square root of any number, is nothing else but to find the side, upon which the Square was composed: or to find a number, which multiplied in itself, shall produce the number given.  
Of these there are 3 several, or distinct species, viz: Single, Compound, and Irrational numbers.  
I. Single, are all Squares which are made of any of the 9 significant figures only, viz: 1 times 1 is but 1, but 2 times 2 is 4, and the Square of 3 is 9, the Square of 4 is 16, and of 5 is 25, etc.  
II. Compound, are all such Squares that are made of more figures than one; as if 10 were the Root, the Square were 100; the Square of 11 would be 121, and 12 squared proves 144, etc.  
III. Irrational numbers, are all such Squares whose Roots cannot exactly be discovered by Art, neither in whole numbers, nor in fractions, but some error will remain, there being no proportion known betwixt the Square and the side or Root, viz: of 2, 3, 5, 6, 7, 8, 10, etc.  

To find the Square made of any of the 9 significant figures, or their Squares given to find their Roots. A Demonstration in extracting of Square Roots.  
 
Every number propounded (whose Root is to be extracted) is conceived to be a Square number, and hath a Root or side on which the Square was made by connexing of lines parallel, or multiplication of one number by another, and so comprehending divers other little Squares, as by this Scheme appears, A, B, C, D, the side A, B, & A, C, is divided into 9 equal parts, containing the 9 significant figures; and all the Squares made of them, in each common angle: and first the Root of 1 is but 1 for its square, because it multiplies nothing; if 2 were the Root, then is the Square of it 4, as in the common angle, and so many little Squares are made of that Root: if 9 were a Square given, then would the Root be 3; the Root of 16 is 4, of 25 will be 5; of 36, 6; of 49, 7; of 64, 8; and the Root of 81 is 9; Or the Square made of the Root A, B, or A, C, 9, will be 81 little Squares; and so of all the rest; for as the multiplication of the Root or side, produceth the Square, so the extracting of a Root from a Square number propounded, is to find the side; but if any whole number were propounded less than 100, and greater than an unite, (yet is not found in this Table) the number is irrational to man's understanding, whose art is to imbecile in all such cases, and cannot find a true and exact Root, which multiplied in itself, will produce the Square number given: so in this defect of humane art, take a Root that's less, yet nearest to it, whose Square subtracted from the Square given, the Remainder make a Numerator of a fraction, whose Denominator shall be the Root doubled, and an unite added to it: As for example, admit 10 were a number propounded, whose Root is required; in the Table it is not, therefore I take 3 for the Root, whose Square is 9 the nearest to it, and less, which take from 10, there will remain 1: the Root doubled is 6, and 1 added to it makes 7; so the Root is thus expressed 3 1/7. Again, the Root of 80 is not found in the Table, 9 being too great, therefore take 8, whose Square is 64, the difference 16; so the Root of 80 express thus 8 16/17, but this is not axactly true, nor yet so near the truth, as I will show hereafter; for this squared is but 79 273/289, and should have been 80 the difference 16/289, which is but a small error in ordinary things, and so not to be rejected.   

A Square number given that is greater than 100, the number for extraction must be prepared after this manner. Example 1.  
The number propounded here is 144, whose Square Root is desired; having set down your number under the first figure upon the right hand make a prick, or point with your pen, and from thence under every second figure towards the left hand: as here in this number given 144, the unite and the hundreth place hath points, and so many, as there shall happen in any number given, the Square Root will consist of so many figures; this done, make a place upon the right hand of the number given, to set the Roots in, like the Quotient in Division, then find a number, which multiplied in itself, shall be equal to the number over the first point, and the figure before it (if there be any) and in case there is no such number in the former Table, take a Root next less unto it, which multiplied in itself, and then subtract that Square from the number over the first point, and set the remainder above that, & if there be none, a cipher; as here 1 stands over the point and 1 is a Square equal to it, which Root place in the Quotient, subtract it from 1 and put a cipher over it, then double the Quotient, and place it betwixt the next two points, then inquire how many times that figure will be contained in the number over it, for a new Root, whose Product and Square must be always equal, or the nearest less, to all the number unto the next point, as here 'twill be 2, which place in the Quotient, and likewise under the next point, then take the Square of the last found Root, and the product of that Root in the other number, which here makes 44, that subtracted from the number over it nothing will remain; so 12 is the Square Root of 144, which is always tried by multiplying the Quadrat Root in itself as in this, 12 multiplied by 12 produceth 144 as before; if there had been any Remainder it must have been added unto it, and then the sum would have been the same with the number propounded, otherwise the work is false.   

To find the Quadrat of any number consisting of four places or more. Example 2.  
Admit the number propounded were 7056, according to the last directions I make a point under 6, & likewise under the cipher, then find a number which multiplied by itself shall be 70 or the nearest less; if I should take 9 it is too great, because the Square of it is 81, then take 8, which place in the Quotient, that squared will be 64, which subtracted from 70, there will remain 6, which set over the cipher: then double the Root by saying 2 times 8 is 16, place the last figure on the right hand (which here is 6) betwixt the two points, and the next figure in order towards the left hand, as under the point; then see how many times 1 will be contained in 6 if I take 5, 'tis too great, because 5 times 6 cannot be had in 15, nor the square of 5 in the last 6: therefore take 4, which place in the Quotient, and likewise under the next point: then multiply this last Root in itself, and in the former Root which was doubled, that is here in 4 and by 16, saying 4 times 4 is 16, that is 6 and go 1, 4 times 6 is 24 & 1 makes 25, that is 5 and 2 decimals, then say 4 times 1 is 4, and 2 decimals in mind will be 6, which 656 subtracted out of the figures uncancelled, that is 656 and nothing remains, so the Quadrat Root of 7056 is 84; for the square Root 84 multiplied in itself, will produce 7056 as before; and thus you may proceed in any other number; yet I will show one more example of a Square number, where a cipher happens in the Root.   

To find or extract a Quadrat Root, consisting of 5 places. Example 3.  
The Square number given here to be subtracted is 43264, whose Quadrat Root will consist of 3 places, denoted by the 3 points, viz: under 4 first on the right hand, then under 2 in the place of hundreds, and beneath 4 in the decimal place of thousands: this done find a Root whose Square shall be equal, or the nearest less unto the number over the first point on the left hand, which here will be 2, this set down in the place of Roots, but shall not need to write down the first number underneath those figures to be squared, but take the Quadrat of it, which subtract from the first figures; as the square of 2 is 4 which taken from 4 nothing will remain; double then the Root, and place the first figure of it betwixt the next points, as 2 times 2 is 4, which set under 3, now 4 cannot be had in 3, therefore put a cipher in the Quotient, and likewise under the next point beneath 2, to supply the place only, and having canceled them both, double the Root, saying, 2 times nothing is nothing, for which put a cipher betwixt the next two points, then double the 2, and place that 4 one degree to the left hand of it, as under 2, then see how many times, 4 will be found in 32, which will now be 8 times, place it in the Quotient, and also under the last point, then multiply this by 8, that is in itself, the product is 64, which taken out of 64 nothing remains, than 8 times nothing is nothing, but the 6 decimals borrowed makes 6, which subtract from 6 and nothing will remain: and lastly, 8 times 4 is 32, which taken from 32, and nothing will remain: so the Square Root of 43264 is found 208 whose Square will produce 43264 the number propounded, which proves the extraction to be exactly true.   

To extract a Quadrat Root out of any irrational number; and to find the Root with a fraction in any proportion assigned, as ⅓ or ¼ etc. Example 4.  
Admit the number given (whos's Quadrat Root to be extracted) were 5676, which number being irrational, the Root is required in such proportion, as that the Denominator of the fraction shall be 3: to perform this, or the like square the number given, in whose parts the Root is required, then multiply that Square by the number propounded, from whence extract the Quadrat Root, and then divide the Root found by the Denominator of the fraction given: then note what points the number propounded would have had, and the other number, or numbers shall be the Numerator of a fraction, unto the Denominator given: As in this example 5676 is the number propounded, 3 the proportion given which multiplied in itself is 9, and that by 5676 produceth 51084, as in the margin, which number will bear 3 points as under 4, the cipher, and beneath the figure of 5, whose nearest Root is 2, which being the first Root, you shall not need for to subscribe under 5, but set it in the Quotient, which squared is 4, and subtracted from 5 the remainder is 1, which place over it, and cancel the 5; that done, double the Root as 2 times 2 is 4, which place betwixt the next two points; then see how many times 4 will be contained in 11, which will be twice, then set 2 in the Quotient, and under the next point beneath the cipher, then say 2 times 2 is 4, which place beneath it, and also multiply the Root 2 by 4, whose product is 8; which 84 subtract from 110 over it, the remainder will be 26, which place overhead (as in Division) then double the two Roots found, and always place the first betwixt the next two points, and so proceed in order towards the left hand; then look how many times 4 is in 26 over it, and you will find 6, which put in the Quotient, and also under the next point, towards the right hand; then take the square of 6, that is, multiply 6 by 6 the product is 36, that is 6 and go 3 decimals, then say 6 times 4 is 24 and 3 in mind is 27, that is 7 and go 2: then multiply the last Root by the next 4, which produceth 24 and 2 in mind makes 26, which writ down, being there is no more, and subtract them from the figures above them uncancelled, as 6 out of 14 there will remain 8, than 1 borrowed and 7 maketh 8, which taken from 8 nothing remains, and so likewise 6 from 6 and 2 out of 2: now the Root is 226. which being extracted in third●, divide it by 3, the Quotient will be 75⅓ for the Quadrat Root of 5676; and extracted according unto the Demonstration, the Root would have been 75 51/1519 but neither of them exactly true, for their Squares will not produce the number given.   

To extract the Quadrat Root from any irrational number, with a Fraction, whose Denominator shall be a decimal, as 1/10, 1/100, 1/1000, 1/10000, 1/100000, etc. Example 5.  
In extracting of a Quadrat Root from an Irrational number, where no Denominator is assigned, but the nearest unto truth is required, this way is better than the first, more compendious than the second, and much more exact than either: As for example, let 168 be a number given, whose Square Root is required in Decimals: all that is now to be done, is only to take any even number of cyphers as you please, annex those to the number given, on the right hand, as in this Paradigma, where 168 by annexing of cyphers is made 1680000, the Root 12 96/100, observe what points there will be under the cyphers, so many figures or places must be severed from the Root extracted, cut off from the right hand, to make the Numerator of a fraction, whose Denominator shall be an unite, with as many cyphers annexed unto it, as the Numerator hath figures or places: the reason of this is, first wherefore the cyphers are in couples is evident, they being Square numbers, as 10 times 10 is 100, which hath 2 cyphers, the square of 100 hath 4 cyphers, 1000 will have 6 places, 10000 must possess 8 places, 100000 will have 10 places, etc. Now these Squares multiplied into any number propounded, can increase it but so many cyphers, as by the fourth Example in Multiplication of Whole numbers: neither shall you need for to divide the Root extracted by the proportion given, as by 10, by 100, 1000, etc. as in my Breviates of Division is explained; but sever so many figures in the Quotient, as there were points under the cyphers of the Square annexed, as was abovesaid; so according to the Demonstration & the first Example, the Square Root of 168 will be 12, and the Remainder 24, for the Numerator of a fraction, whose Denominator is double the Root with an unite added to it, that is 25, so 'twill stand thus 12 24/25, to try if the Root be truly extracted, the Square of 12 is 144 to which add the remainder 24, the sum will be 168, as before; if the Root had been extracted in Tenths, it would have been 12 9/10, which is less than the former; but here it was extracted in 1/100, by annexing 4 cyphers to it, & then it was found as in the Scheme 1296, that is 12 96/100, or reduced 12 24/25, as before; if 6 cyphers had been annexed to it, than it had been 168000000, and the Quadrat Root 12961, that is 12 961/1000, which comes nearer the truth, erring not an unite in 1000, and so the greater number of cyphers you take in this kind, the nearer you will approach unto the true Root; yet not without error in all irrational numbers, the offspring of works merely humane, and be satisfied with this.   

How to extract a Quadrat Root from any Fraction or broken number propounded. Example 6.  
In this Square given A, B, C, D, each side suppose an unite, divided into four equal parts, and consequently those parts, or fractions, multiplied in themselves, will constitute sixteen little Squares, as by the Quadrat of 4, or by the Geometrical Demonstration is evident, and by construction the Square A, E, containing 4 little Squares, is ¼ part of the whole Square A, C, 16, and the Quadrat Root of this fraction is required, that is the side A, G, or A, F; now to find the Square Root of any broken number, differs nothing from the former Rules but only in this, that in Fractions there are two terms given, viz: Numerator and Denominator, and the Quadrate Root must be extracted from both; As here in this Example of ¼ the Root of 1 is but 1, and the Root 4 is 2, so the Square Root of ¼ is ½, that is, the Root of the Square A, E, (which ¼ of A, C,) is the line A, F, or A, G, half the Root, or side of the great Square A, B, C, D, and in the same manner A, H, is 1/16 of the Integer A, C, whose Quadrat Root is ¼ the side of the little Square A, H, that is A, I, or A, K, which is a fourth part of the whole Root, or great Square A, B, or A, D, and the Square Root of 16/81 is 4/9, and here observe, that Fractions may be irrational in one denomination, and yet perfect Square numbers in another: as admit the fraction given were 200/392, which are irrational in both terms, and yet reduce them as to 10●/196, and then the Square Root will be found 10/14 or 5/7, but in all such cases it is best for to reduce the fraction into the least denomination as 200/392 will be 25/49 whose Quadrat root is 5/7, as before.   

To extract the nearest Quadrat Root from any Fraction, that is irrational in both terms, yet reduced to their least denomination. Example 7.  
The former Rules which I have delivered might effect what ordinary use can require, yet lest you should be to seek when more exactness is desired, observe this general Rule: Annex unto the Numerator of the fraction given so many cyphers as you please in pairs, and those divide by the Denominator, the Quotient shall be the Numerator of a new fraction, whose Denominator will be an unite with so many cyphers as the Numerator hath places; and for the illustration of this admit 17/20 were the fraction given whose Quadrat Root is required, annex two cyphers to the Numerator 17, and then 'twill be 1700, which divide by 20 the Quotient is 85 for the Numerator of a fraction, whose Denominator is 100, which fraction 85/100 is equal in value to 17/20, the Quadrat Root of 85/100 is 9/10, but not exactly true; if 4 cyphers had been annexed, it would have been 8500/10000, and the Square Root 92/100: and so for any other of this kind.   

To find the Quadrat Root of any mixed or compound number, when the whole number, the terms of the Fraction, or either of them are rational or irrational. Example 8.  
In extracting the Square Root from any compound broken number, you must convert it into an improper fraction, although the whole number hath a perfect Root, and also the fraction rational in both terms; as 4 9/16 the Root of this is not 2¾, but the Radix of 73/16 is required, for the fractions of Square numbers are greater, or less, according unto the quantity of the Magnitude, of which the fraction is a part; and therefore they must be reduced into improper fractions, whereby they may have one denomination: and for the more illustrating of this, admit 10 9/16 were a Square number given, whose Roo● is required; this made an improper fraction will be 169/16, the Root of 169 is 13, and the Square Root of 16 will be 4, so the Root of ●69/16 is 13/4 or 3 ¼, and so the like of any other; and all improper fractions, as proper, would be reduced unto their least denominations, before their Roots are extracted; and if their terms prove then irrational, annex to the Numerator so many cyphers as you please in pairs, and so make it a decimal fraction, and then find the Quadrat Root, as by the 5, 6, & 7 Example of this Paragraph, which here shall have a period; only observing that these compound fractions, will have a Square Root consisting both of an Integer and a broken number, therefore note well what Square you take of Decimals; As in this Example, where the Root of 73/16 is required, & in 1/100, the Square of a 100 is 10000, which multiplied by 73, or annex the cyphers to it, the sum will be 730000, under these cyphers there will be 2 points, and consequently 100 for the fractions Denominator; As in the 5 Example. But to return, divide 730000 by 16, the Quotient will be 45625, the Quadrat Root 213/100, that is 2 13/100 the Root of 4 9/●6, the compound number assigned.    

Paragraph II. The Definition of a Cube.  
A Cube is a Geometrical figure of the third quantity, composed of several superficies added or connext together, and proceeds from right Lines multiplied in themselves, so constituting a superficies called the second Quantity; that superficies, or Product, multiplied by the Line given, (which was the first quantity) produceth a Cube number, representing a body, consisting of 6 equal sides, having these dimensions, viz: Length, Breadth and Depth.  
The Root of any number perfect cubical, is a right Line of a Solid body containing 6 equal sides, which constitutes as many square superficies, or a number multiplied twice in its self, as was said before: and the extraction of this Root, is the finding out of the side, or first number, which makes divers little Cubes, comprehended with a great one, according unto the first line or quantity propounded.  
All cubical numbers are either single, compounded, or irrational.  
All those which are called single, are to be understood such Cubes as are made of any one sinificant figure, multiplied twice in itself, viz: 1 multiplies nothing, and so is both Root and Cube: 2 times 2 is 4, and 2 times 4 is 8; thus the Cube of 3 will be 27: the Cube of 4 is 64, and of 5 will be 125, etc. always less than 1000  
Compounded, are all such Cubes, whose Roots do consist of more figures than one, and are never less than 1000 the Cube of 10; the Cube of 11 is 1331, the Cube of 12 will be 1728, etc.  
Irrational, are such Cubes that want a known proportion to their Roots, that is all such Cubes, whose Roots cannot exactly be discovered by humane Art, either in whole numbers or fractions, as are the Cubes of 2, 3, 4, 5, 6, 7, 9, 10, etc.  

To find a Cube made of any significant figure; or the Cubes of them given to find the Root.  
A Demonstration For the extracting of cubical Roots from all the 9 significant figures. The figure of a Cube.  
This Geometrical figure is a solid body of the third quantity, having these 3 dimensions, viz: Length, Breadth, and Depth; as the sides A, B, or B, C. Secondly, C, D, or D, E. Thirdly, E, F, or A, F, the sides of this Figure are all equal, and each divided into 9 parts alike, demonstrating the Cubes made particularly of all the 9 significant figures: and first for the Superficies A, C, and likewise the squa●e C, E, the sides are divided into 9 equal parts, by 1, 2, 3, etc. containing the superficial square, made of all the significant figures simply of themselves, as from an unite to 81 at D, and the square A, C, shows all the perfect cubical numbers, or solid bodies, made by the multiplication of any one figure twice in itself: or the product of a superficies, and any significant figure: the square A, E, contains the 9 numeral Arithmetical characters, to direct your Optic sense from any Root, or superficies given unto the Cube comprehended by the greater, and also showing what number of little Cubes it does contain, as by the Scheme is evident.   

Example 1.  
First, the Cube of 1 is but 1, as it is an Inch, a Foot, a Yard, etc. but as the Root or side is divided into parts, it does admit of many, as the Cube of A, C; suppose the Root A, B, or A, F, one quarter of a Yard, whose Cube is A, C, A, F, & C, E, in 3 sides, bounded with parallel lines, and so consequently the other 3 sides must be equal: but as it is divided into 9 inches, the superficial square C, E, contains 81, and the whole Cube 729 of square Inches or little Cubes, whereof 1 Inch is here the first: the superficial square of 2 is 4, as in E, C, that multiplied by the Root 2, is 8; as upon the plain A, C, is evident; or number the little Cubes, and you may discover 8; the Cube of 3 is 27: and that of 4 is 64, and 5 multiplied cubically will make 125, and so of all the rest as by the figures do appear: and as the multiplication of any number twice in itself produceth the Cube of it; so the extracting of any Cubique Root is nothing else but to find the side by which the Cube was made; as the Root of 216 is 6: and if the Cube were 343 the side or Root must be 7, of 512 it will be 8, and the Root extracted from the Cube of 729, will produce the side 9, as A, F, equal to all the other sides of the whole Cube made of the 9 significant figures; and if any whole number less than 1000 were propounded, and the Cube required, but not found upon the plain A, C, and yet not less than an unite, the Root is termed irrational, as wanting a true proportion to the Cube, or Man understanding in the perfection of Art, or the secrets of Nature.   

To extract the nearest Root out of any number, whose Cube was less than 1000 Example 2.  
Find first a number which multiplied Cubically in itself, shall be equal unto the number given, or the nearest less; then subtract the products of the multiplications from the number given, the remainder will be a Numerator to a fraction, whose Denominator shall be the Cubique Root tripled or multiplied by 3, with an unite added unto the product, and these added unto the square of the Root tripled; the total sum is the Denominator of the fraction: and for the better illustration of this, all single cubical numbers that have perfect Roots under 1000 are expressed in the Demonstration, and admit here 800 were the number given, whose Cube is required: the Root of this is 9, and the Cube is 729, which take from 800 there will remain 71 for the Numerator of a fraction, the Root tripled is 27; to which add an unite, 'tis 28; then take the Square made of the Root, which here was 9, and squared is 81, that tripled is 243, to which add 28, the sum is 271 for the Denominator: so the fraction stands thus 71/271, and the Cube Root of 800 in this manner, as 9 71/271: and the cubical Root of 999 will thus be found for to be 9 270/271. And in this manner you may make a fraction to any compound Irrational number; yet note this Rule is not exempt from error, nor so exact as you will find the following Rules: yet these approach so near the truth, as is necessary in common practice, or at least will be required in things of ordinary use.   

To extract a cubical Root from any number greater than 1000, and how to prepare the number, and point out the figures for the Root which is to be extracted. Example 3.  
The Cube Root of 1000 is 10, and all perfect Roots under it, are made evident in the Demonstration already, and also those irregular, extracted with a fraction; so that hitherto there needs no farther explanation; but to extract the Root from a Cube greater than 1000, the number must be prepared after this manner, First inscribe your number, as here at A 1728, supposed cubical inches, or what you please; and the side or Root from whence this number proceeds (containing these little Cubes) is the thing required: under the first figure or cipher, on the right hand make a point, and so underneath every fourth figure inclusive, as here in this Example under 8, and 1; and so mark them on, if there be more figures, leaving two figures or places between all the points; whose number is always equal to the places in the Cubique Root, as here in this, there are 2 points; and so many figures will be in the Quotient, which make upon the right hand of your number propounded, as you did in extracting of the Quadrat Root; the number thus prepared observe these general Rules.  

The general Rules and Canons in extracting of all Cubique Roots.  
I. The number thus prepared as before is specified at A, take a significant figure, whose Cube shall be equal, or the nearest less, to the number over the first point upon the left hand, with the figure, or figures before that, if there be any.  
II. Place this significant figure in the Quotient, as the first Cubique Root found, then subtract its Cubique number from the figures over it (if it consists of more places than one, or that you can commit to memory, subscribe them) under the first point towards the left hand in order, then make a subtraction, and write the remainders over them, in their proper places, cancel the other figures and draw a line under all.  
III. Multiply the Root or Roots by 3, and place the first figure of this tripled number one degree from the next point towards the left hand, according to the orderly succession of the figures, as B, 3.  
iv Then multiply the whole Quotient by the Triple, and write the Product underneath the tripled number, as C, 3, with the unite place of it, beneath the Triples' decimal, and then draw a line under both, this I call the Index, from pointing out the next Root.  
V In the next place find how many times the first figure upon the left hand of this Index will be contained in the figure or figures over it: but here observe for to take such a number as shall be equal or the nearest less, to the remainder over it (from whence the Root is to be extracted) after all the several multiplications and products; this number found place in the Quotient as the next Root.  
VI Multiply this last Root cubically in itself, and having drawn another line under the last Triple and Index, subscribe this Cube under the next point, as against the letter D, 8.  
VII. Next multiply this Root squared in itself, by the Triple, and subscribe the unite place of it one degree towards the left hand from the Cube, that is, under its decimal, if the Root be more than 2; and the other figures in order, as E, 12.  
VIII. This done, multiply the last Cube Root found, by the Index only, which must be subscribed beneath the last number, with the unite place under its decimal, that is one degree more to the left hand, as F, 6.  
IX. Draw a line under these 3 numbers, and subscribe the sum of them, as 728 G, which subtract from the number over it, (whose cubical Root is required) and cancel the rest; if there be no more points, make a fraction of the remainder, if any, and thus you have the Cube Root of the given number.  
X. If the number given requires a Root of more places, set down the figures (not extracted) in a new place, with the point or points under them, to which number, in order annex the last remainder unpointed, and so proceed, according to these Rules excepting the first and second, which only serves for the beginning of any Cube Roots extraction.   

An Arithmetical illustration of the former Rules and Canons.  
Admit the number propounded be 1728 cubical inches, as before, whose side or Root of this great Cube is required, and prepared, as in this last Example, with a point under the figures of 8 and 1, with two places between them, as in the number at A; which having two points, will require 2 figures for the Root, and in this, the first will be 1, whose Cube can be but 1; which subtracted from the figure over the point upon the left hand, which being but 1 nothing will remain, so cancel the figure, and place this Root in the Quotient, which tripled is 3, subscribe this under 2, against B, and multiply it by the Root, which being but 1, the product is also 3 for the Index, subscribe this beneath the Triple one degree towards the left hand, as under 7 and against C: here draw a line, and find how many times this last Index 3 will be contained in the figure over it, as in 7; which will be found twice, then set 2 in the Quotient, and multiply it cubically, which subscribe under the next point, as against D: then square 2 the last Root, which 4 multiplied by the Triple produceth 12, which subscribe beneath the last Cube, one degree towards the left hand as against E, then multiply the last Root 2 by the Index, and that produceth 6, which subscribe beneath the rest, one degree more towards the left hand, as at F, the sum of these is 728 as at G, and that subtracted from the remaining Cube at A 728, there will be no remainder, as you see at H; so cancel all the other figures, and the operation is ended, 12 being the Root found, which multiplied cubically in itself, produceth 1728: as 12 by 12 will be 144, and that again by 12 brings forth the former number, the Cubique Inches in a cubical Foot as was required.    

To extract a Root out of a Cubique number consisting of 5 places. Example 4.  
The extracting of this or any other Cubique root differs nothing from the former prescribed Rules: but yet since Man is better instructed with a few examples than many words, for the case of young beginners, I will briefly insert more of these, viz: A is here a Cube propounded, as 46656, whose Root is required: and first having pointed it by my former direction, find the Root of 46, or the nearest less as 3, which place in the Quotient as the first Root, whose Cube is 27, and that subtracted from 46 the remainder will be 19, which writ over the figures extracted, as in the Example; then triple the Root 3, and place that 9 under 5 against B. Secondly, multiply the Root by the Triple, the Product is 27 for the Index as against C, under these draw a line, and find a new Root by examining how often the foremost figure of the Index will be contained in the remaining figures over it, as in this, how often 2 is in 19; provided always you take no greater a Root than that the following numbers may be extracted according to my former Rules, which if you doubt, make the experiment first in a void piece of paper, for here the Index 2 will be in 19 but 6 times; which 6 place in the Quotient as the second Root, whose Cube 216 subscribe beneath the line, the unite place under the next point, the rest towards D, than the Square of this last Root 6 is 36, which multiplied by the Triple makes 324, that subscribe against E, one degree towards the left hand Lastly, multiply the Root 6 by the Index only, as here 27, the result will be 162, which subscribe under the last number, one degree more to the left hand towards F, the sum of all is 19656 as G, this subtracted from the number uncancelled above, there will nothing remain, as at H, the Root required is 36, which multiplied cubically in itself, will produce 46656 the former number propounded.   

How to extract a Root from any Cube number that consists of six places. Example 5.  
Admit the Cube number propounded be 970299 as at A, whose Root is desired: having pointed it by my former directions, as under the unite and the thousand place, find the Cubique Root of 970 which will be 9 being the nearest less, whose Cube is 729, that subtracted from 970 there will remain 241, which place over the three cancelled figures: this done the Quotient or Root 9, tripled is 27, which place one degree from the next point towards B; this Triple number multiply by the Root 9 does produce the Index 243, which subscribe beneath the Triple, with the unite place one degree more towards C; here draw a line under all, and by the Index find a new Root, by ask how many times 2 is contained in 24, or 243 in 2412, the answer will be 9, which set in the Quotient, as the second Root, whose Cube is 729, which place against D, the last Root squared, and multiplied into the Triple will produce 2187, which subscribe with the unite place one degree more towards E, then multiply the Root 9 by the Index, the product will be 2187, which subscribe one degree more towards F, the sum of those 3 numbers is 241299 as at G; and if subtracted from the uncancelled figures over them, nothing will remain as at H. so 99 is the Root, whose Cube will be 970299 as before was given.   

To extract a Root from any perfect Cubique number consisting of significant figures and cyphers in what number of places soever. Example 6.  
The Cube number here given consists of 9 places, viz: 128024064, & pointed by my former directions, the Root must consist of three figures or places, and the first will be 5, whose Cube is 125, which subtracted from the numbers to the first point on the left hand, that is out of 128, the remainder will be 3; this done triple the Quotient, the product will be 15, which place one degree from the next point towards the left hand, then multiply the Quotient by the Triple, the product is 75 for the Index, which cannot be contained in 30, therefore put a cipher in the Quotient: and here it is convenient to remove the figures, and so in finding every Root (but the first) according to this Example.  
Having found the second Root, the operation is in the same manner as was exemplified before; but here the second Root proving a cipher, it assumes a place in the Quotient only (as in Division) but of itself effects nothing: so here I remove the Root 50 and Remainder with one point as 3024064 at A: next triple the Root 50, the product is 150, which place against B, one degree from the next point; this found, multiply the Quotient 50 by the Triple 150, and they will produce 7500 for the Index, which place one degree more towards C, then draw a line under all, and find by the Index a new Root, as by looking how many times 7 is contained in 30, which will be found 4, for the next Root; this put in the Quotient, multiply it cubically, it will be 64, which place under the next point towards D, then multiply the Square of it (that is 16) by 150 the Triple, there will be produced 2400, which place underneath the last number, one degree more towards E Thirdly, multiply the last Root 4 by the Index, whose product will make 30000, which place in order under the last one degree more towards F, the sum of these 3 numbers amounts unto 3024064, equal to the remaining numbers over it, from whence subtracted, there can be no remainder, this was a perfect Cube, whose Root is found 504, which multiplied cubically in itself, produceth the former Magnitude, viz: 12802●064, the thing required.   

From any irrational Cubique number to extract the Root, or the nearest to it, with a fract on in any given proportion as ⅓ or ¼ etc. Example 7.  
The number here propounded is 44 whose Root is required, and being the number is irrational to the Root demanded in fourth's: first take the Cube of the proportion assigned, which is 4, and the Cube of it 64 must be multiplied by the number given, which here is 44, the result will be 2816 as A; which point by the former Rules, and find the Root of 2, or the nearest less, which is 1, whose Cube is also 1, and that taken from 2, the remainder will be 1, which writ over 2 being canceled; this done triple the Quotient, whose product is 3, to be subscribed next the second point towards B; and multiplied by the Quotient, is but 3 for the Index C, by which find the next Root, which will be 4, whose Cube is 64; but first here draw a line underneath the Triple and Index as before, and then subscribe 64 under 16 as against D, the last Root 4 being squared is 16, and multiplied by the former Triple produceth 48, which subscribe as against E, then multiply the last Root 4 by the Index, the product is 12, which subscribe as against F; the sum of these 3 numbers is 1744, which subtracted from 1816 (as yet uncancelled) the remainder is 72, which is but part of ¼; and 14 is the Root of this number, 2816, which was extracted in fourth's, therefore divide 14 by 4 the Quotient is 3 ½ the Cubique Root required; and according to the second Example, the Cube of 44 would be 3 17/37 which is less than ½; and if the Root had been extracted in any greater proportion, it would have nearer approached the truth; but every thing should be performed according unto the state of the question; yet if at liberty, observe the following Example, being the truest and most facile way.   

To extract the Cubique Root from any number that is irrational, and to produce the Root in a decimal Fraction, viz. in 1/10 1/100 in 1/1000 in 1/10000 etc. Example 8.  
There is a Cube consisting of forty little Cubes, and the Root of this greater body is required, to be extracted in the hundred part of an unite; observe here to annex unto the number given, cyphers in trines, or by three, because the Cube of 10 is 1000, of 100 it is 1000000, and the unite prefixed before them multiplies nothing: so here the Root of 40 being required in hundreds, I do annex six cyphers unto 40, making it 40000000 as at A, and being pointed, I find 3 places in the Root, whereof the first is 3, whose Cube is 27, and that subtracted from 40, there will remain 13: now find a new Triple and an Index, viz: 9 & 27, B & C; by this Index 27 find another Root as 4, But here note that the Index 27 would have been contained 5 times in 136; but then an unite of that number had only remained, from whence the product of the Root found, and the Index should be subtracted, besides the following numbers: but to proceed, the Cube of 4 is 64, which subscribe against D, the Square of 4, that 16 multiplied by the Triple 9 produceth 144 as against E, and the product of 4 (the last Root) and the Index makes 108 as F, the total G 12304, which subtracted from the number uncancelled over it, viz: 13000, the remainder will be 696; but the total of the Cube remaining is 696000.  
The number uncancelled being transcribed as at A 696000, and having made the Quotient, and inserted the 2 Roots last found, viz: 34, I find the Triple of it 102 as against B, and the Index 3468 as C, and the new Root 1, whose Cube is also 1 as against D, under the last point. Secondly, the Square of it is but 1, which multiplied by the Triple makes but 102 as E, and that multiplied by the Index is but the same for F 3468; the Totall of these is G 347821, which subtracted is 348179 as H for the Remainder; and the true Root of 40 is 3 41/100: for if all the Root 341 were divided by 100, the proportion assigned, the Quotient would be 3, and the remainder the Numerator of a fraction, whose Denominator here is 100; so in all such cases, it is but cutting off so many places in the Root as there were cyphers in the proportion given: had this Root been extracted in tenths it would have been 3 4/10, and by the second Example in this Parag: but 3 23/37, which is not so exact as in tenths, nor yet that, as in hundreds: although sometimes they will not differ, when little remains after the extracting of a former Root, as admit 36, the Cubique root in tenths will be 3 3/10, and in hundreds but 3 ●0/100 which if reduced is all one: and so in whole numbers let this suffice.  

To extract a Cubique Root from any single fraction or broken number propounded.  

A Demonstration of a cubical Fraction.   
Admit A, B, were equal to A, C, and likewise C, D, with all the other sides divided into 3 feet, and so constituting a cubical yard, consisting of 27 Feet or little Cubes, and so every fraction of this kind is to be understood a solid body having Length, Breadth and Depth, and part of a greater Cube: and so one of these supposed Feet or Cubes is 1/27 part of the whole body, and ⅓ part of any one side or Root; for in all these kind of fractions, the Cube root of the Numerator will be a Numerator unto a new fraction, and so likewise the Root of the Denominator: as here in 1/27 the Cube root of 1 is but 1, and the cubical root of 27 the Denominator is 3: so the Root or side of this little Cube is ⅓ part of the cubical yard, viz: of A, B, of A, C, or C, D; and so the Root of 8/27 will be found ⅔ of the whole side A, B, or A, C: yet the fraction of itself as a Cube contains 8 solid feet, the body being divided according to the Denominator, into 27 little Cubes; and according to the Numerator, 8 of them must be taken: so the extracting of any cubical root from a fraction, differs nothing from the former Rules, but only observing this, always to reduce the fraction propounded into its least denomination, otherwise they will not be commensurable; as admit the cubical root of 81/192 were required; herein is included a perfect Cube, yet the Root cannot be found or extracted in these numbers, without another fraction; therefore reduce both Numerator and Denominator into their least denominations, viz: 81/192 will be reduced by 3 unto 27/64, from which cubical fraction extract the Root as the Root of 27 is 3 for a new Numerator, and the Cubique root of 64 is 4, so the true Root or side of the fraction 81/192 or 27/64 is ¾, and so the like of any other that is commensurable.    

To extract the Cubique root from any irrational fraction, when either the Numerator, or Denominator, or both, are incommensurable. Example 9  
The fraction here given is ⅔ and the Cubique root of it is required in Centesmes or hundreds: which to effect, the Cube of 100 is 1000000; this multiplied by 2 the Numerator of the fraction, or which is all one, annex the 6 cyphers unto 2, the sum is 2000000, which divided by 3 the Denominator, the Quotient will be 666666, as A, the number thus prepared and pointed, find the Root of 666, which is 8, whose Cube is 512, the nearest less unto 666, the difference being 154, cancel the 3 first figures, viz: 666, and write the remainder over them: then triple the Quotient 8, and it will be 24 as B, and multiplied by the Triple produceth the Index 192, which place as against C; under these draw a line, and by the Index find a new Root as 7, whose Cube is is 343 as against D, the Square of 7, which is 49 multiplied by the Triple is 1176, which subscribe as at E. Thirdly, multiply the Index 192 by the last Root 7, the product is 1344 as F, the sum of these is 146503, which subtract from the remaining Cube at A, the remainder at H will be 8163, the Root 87, the Numerator of a fraction, whose Denominator is 100: so it is thus 87/100, the cubical root of ⅔ required in a centesme fraction; and so in a greater, or less proportion, or in any other simple fraction.   

To extract a Cubique root from any mixed or compounded fraction, when either Numerator, or Denominator, or both are incommensurable, or the improper fraction a perfect Cube. Example 10.  
The mixed or compound Cube here propounded is 4492 ⅛, which if reduced into an improper Fraction will be 35937/8 a Cubique number whose Root is required, inscribe the Denominator as 35937 at A, and having pointed it, find the Root of 35, which will be 3, set it in the Quotient, and take the Cube of it 27 out of 35 the remainder will be 8; this done triple the Root 3 and find the Index, as 9 & 27 against B, & C, under these draw a line, and find a new Root as 3 again, whose Cube is 27 against D, the Square of it multiplied by the Triple is 81 as at E. Thirdly, the Root and Index multiplied together, viz: 3 & 27 produceth also 81 as F, the total G 8937, subtracted from the remaining Cube 8937 nothing will remain in the Numerator of this fraction, whose Denominator was 8, and the Cubique root of it is 2, so the true Root of 4492 ⅛, or which is all one, this improper fraction 35937/8 will be 33/2 or 16 ½; and if supposed feet, it is the length of a statute Pole, whose Cube made upon this Root or side is 4492 ⅛ as before.   

Example 11.  
In this there is given a mixed fraction, viz: 1 ¼, whose Cubique root is required, and being the number is irrational, it is desired in a decimal fraction, and in this proportion, as an unite is to 100, whose Cube is 1,000,000; make the number given an improper fraction, viz: 5/4, this Numerator multiplied by 1000000 will be 5000000, and divided by the Denominator 4 proves 1250000, as A, this pointed shows 3 figures in the Quotient or Root, and the first an unite only, whose Cube being 1 and subtracted, nothing remains; the Triple and Index will be alike, viz: 3, which cannot be had in 2, the figure over it, therefore put a cipher in the Quotient, as for a place in the second Root.  
The remaining part of the Cube being removed, viz: 250000, and 10 placed in the quotient, which tripled is 30 B, and then the Index will be 300 C, and the third Root found 7; draw a line under all, and subscribe the Cube of 7, viz: 343 against D, the last Root squared, and multiplied by the Triple makes 1470 as E. Thirdly, the Root multiplied by the Index produceth 2100 as against F, the total at G, is 225043, and remaining as at H 24957, the Root found is 107/100 that is 1 ●7/100 and the Cubique root of 1 ¼ according to the proportion assigned; but whether truly extracted, or no, you may prove by multiplication of the Root cubically, and adding the Remainder unto the Cube produced, as by the former Examples may be demonstrated.  

The reason wherefore are taken the Triples, Squares, and Cubes of particular Roots, whereby to extract or find the total Root of any number propounded.  
In any Cubique number assigned (whose Root is to be extracted) there is required no more than such a number, as multiplied twice in itself shall produce the Cube, viz: 2, is the Root to 8, and so will 3 be to 27, and 4 unto 64, etc. but when a number propounded must have more figures than one in the Root, it seems almost unexplicable to humane discovery how to effect it at one operation: upon which our wise Forefathers who laid the groundworks of all our Liberal Sciences, have given us Principles and Demonstrations for to guide us by, but these selfconceited times have neglected their Dictates, both in what is Humane or Divine, where through neglect, many things are lost, the abortive Saints of these days, having the opinion of our Predecessors, as children have of old men, thinking them to be fools, when old men know that they are so indeed: but leaving them, and turning to the Ingenious, whom my endeavour is to assist, I will (according to my ability) show them some light of reason in this    

Example 12.  
Divide any number Numbers 10 & 2. The Triples 30 & 6. The Squares 100 & 4. The Cubes 1000 & 8. The total Cubes 1008. The triples product 600. 120. The total is 1728. given into what parts you please, then add to their Cubes the Square of one part multiplied by the Triple of the other interchangeably, the sum of them will be equal unto the Cube of the whole number, as for a farther illustration, let 12 be a number given, which here I will divide vide into 2 parts, viz: 10 & 2, their Triples 30 & 6, their Squares 100 & 4, their Cubes 1000 & 8, their numbers thus prepared, 100 multiplied by 6 produceth 600, and 30 by 4 makes 120, which products added to the sum of their Cubes, viz: 1008 the total will be 1728, and so is the Cube of 12; and as the Cube here is made of a number divided into several parts, so may the Root likewise be extracted in parts, or any other number, as 125 divided into 3 parts, viz: 100, 20, 5, the Triples of these are 300, 60, 15, their Squares are 10000, 400, 25, their particular Cubes 1000000, 8000, 125, the sum of their Cubes is 1008125, next the Square of 100 is 10000, which multiplied by 60 (the Triple of 20) the product is 600000. Secondly, the Square of 20, that is 400, multiplied by 300 produceth 120000. Thirdly, add the two last numbers together, their sum will be 120, the Square of it, 14400; which multiplied by the Triple of the third number 5 that is by 15, the product is 216000. Lastly, the Triple of 120 that is 360 multiplied by the Square of 5, viz: 25 produceth 9000, all these numbers are 1008125, 600000, 120000, 216000, & 9000, whose total sum is 1953125, equal to the Cube of 125, which is also 1953125: and as these Cube numbers are made in several parts by their Cubes, Squares, and Triples, in the contrary manner are the Roots extracted.    

Paragraph III. How for to extract all other Roots composed of these, viz: a squared Square: a squared Cube: a squared squared Square: a Cubique Cube, etc.  

The extraction of a Biquadrat Root.  
THese Roots do all depend upon the former, and of little use in Natural Arithmetic; yet to satisfy the curiosity's of some, I will briefly show the manner of their extraction, both in whole numbers and fractions, either rational or irrational, as thus: a Biquadrate number or a Squared square is nothing else but a number composed of a Square multiplied in itself; and the Root is thus found, extract the Quadrat Root from the number propounded, and the Square Root of that extraction is the Root required: As for example, the Biquadrat Root of 625 is required, the Square Root of it according to our former Rules will be found 25; and the Square Root of 25 is 5, the Byquadrat Root of 625 required.   

To extract a Root from any Squared Cube.  
A Squared Cubique number is composed of any Cubique number squared or multiplied by itself; and the Root of any such number will be discovered by extracting the Square Root from the number given, the Cubique Root of that extraction will be the Root required: As for example, let 729 be the number given, whose Quadrat Root is 27; and the Cubique Root of 27 is 3; the true Root of the Squared Cube required.   

To extract a Root from a Biquadrat number squared.  
A Biquadrat squared is a Square number multiplied in itself, and the product of that squared again is called a Biquadrat squared, whose Root is thus extracted, from any number given extract the Quadrat Root (which is a Biquadrat) whose Root will be discovered as the last: and by this example, 1679616 is the number propounded whole Square Root is 1296, the Quadrat Root of that is 36, and the Square Root of that is 6, the Biquadrat squared Root of 1679616 as was required.   

To extract a Root from any Cubique Cube number.  
A Cubique Cube is a number whose Cube is multiplied Cubically in itself, whose Root will be thus drawn out from any number propounded, and as by this example 5159780352 is a number given, whose Cubique Root extracted by the former Rules will be 1728, and the Cubique Root of that will be 12, the true Root of 5159780352 as was desired: in any of these, to try whether the operation be right, multiply the Roots extracted according unto the quantity given, and to the total product, add the remainder (if there were any) the sum will then prove the number propounded, as in this Example, 12 was the Root found by extraction, whose Cube is 1728●, and the Cube of that again will produce 5159780352, the Cubique Cube number given.   

To extract any Root out of an irrational number in a decimal fraction, or in any other proportion propounded, either in whole or broken numbers.  
Admit the number given were 5754, and the Biquadrat Root of it is required to 1/10 of an unite the Biquadrat of 10 (the Denominator of the fraction propounded) will be 10000, which multiplied by the number given, or annex the 4 cyphers to it, the number will be 57540000 whose Square Root is 7585, neglecting the remainder, extract the Square Root again out of 7585, whose Root will be 87/10, that is 8 7/10 the Biquadrat Root of 5754 in tenths; if it had been required in any other proportion as in ½, or ⅓, or ¼, or ⅕, etc. multiply the Denominator of the fraction to the Quantity given, and those products into the number given, from whence extract the Root required, and that Root must be divided by the proportion assigned, according to the seventh Example of this Parag:   

To extract any Root from a simple fraction, or any compounded number that shall be assigned, and in any proportion, as 1/10, or 1/100, or 1/1000, etc.  
Let 3/7 be a fraction given, whose Biquadrat Root is required in tenths: the Biquadrat of 10 is 10000, therefore annex 4 cyphers unto 3, the fraction's Numerator it will be 30000 which divided by the Denominator 7 the Quotient will be 4285, whose Square Root is 65, and the Quadrat Root of that is 8, the Numerator of a fraction, whose Denominator is 10, so the Biquadrat Root required is 8/10 or ⅘, if it had been a compound fraction, it must have been reduced to a single fraction: As for example, to extract the Biquadrat Root from 16 ⅔, this made an improper fraction will be 50/3 whose Root is required in a decimal of 100; therefore annex 8 cyphers to 50 and then it will be 5000000000, which divided by 3, the Denominator, the Quotient will be 1666666666, whose Quadrat Root is 40824, and the Square Root of this 202, that is 202/100 or 2 2/100 the Biquadrat Root of 16 ⅔ the thing required.  
Observe here that fractions in any of these quantities may be commensurable in their Roots without reduction, but yet being reduced they are the sooner discovered: As for example, to this fraction 32/512, you will not find the Biquadrat Root, reduce it therefore by 2, it is 16/256 whose Root is 2/4 or ½, and if you reduce 16/256 to their least denomination, you will find 1/16; whose Biquadrat Root is ½: and so for all other Roots these Examples well understood may suffice the ingenious.    

Paragraph IU. How Numbers have relation one to another in respect of their Quantities and Qualities.  

The definition of Quantity in Numbers.  
ALL Numbers are said to differ in Quantity, as in respect of the excess or difference betwixt those numbers, and of this kind there are two species, viz: the Greater and the Less, usually called the Antecedent and Consequent; as admit the proportion were 6 & 7, or 3 to 9; here 6 i● the Antecedent, and 7 the Consequent, and so likewise 3 & 9, this is the lesser proportion unto the greater; and the contrary is when the greater is Antecedent, as 7 to 6, or 9 to 3, etc.  
The relation of numbers one to another (as in respect of their Quantities) is the difference betwixt those numbers found by subtracting the less from the greater, whether Antecedent or Consequent; as 3 & 9 admit were terms compounded, subtract the lesser from the greater, the remainder or difference is 6, and this is the relation of numbers, as in respect of their Quantities.  
The proportion betwixt any Numbers in Quantity is the Quotient of the Antecedent when divided by the Consequent, as the proportion of 4 to 2 is Double, 9 unto 3 Triple, 16 to 4 Quadruple, etc. and the contrary, as 2 to 4 is ½, and 3 unto 9 is ⅓, or 4 to 16 is ¼, etc. that is the Antecedent divided by the Consequent, as 4 divided by 2, or 2 by 4.  
This proportion of Numbers is defined to be equal, or unequal; Those said to be equal which are of the same quantity, as 2 to 2, or 5 to 5, or 10 to 10, etc. And the other is the proportion of unequal Numbers, one to another, and the greater to the less, or the less unto the greater, viz: 4 to 2, or 2 unto 4, as before: and where the greater is Antecedent, the Quotient must be more than an Unite; and where the lesser Number is Antecedent the Quotient is a fraction always less than an Unite, as in the former Examples.  
These kinds of unequal proportion, are subdivided into 5 several species, whereof 3 are simple, and the other 2 are mixed.  
The three simple species are these, viz: 1 Multiplex: 2 Superparticulare: and 3 Superpartiens: and the two compounded species are, viz: 1 Multiplex-superparticulare: and 2 Multiplex-superpartiens, which are thus explained.  
I. Multiplex, or manifold proportion, is when the Consequent is contained in the Antecedent more than once, and exactly without any remainder, as 10 unto 5, a double proportion: 18 to 6, a triple etc. and the contrary to this, is the proportion of the less to the greater, and is usually called Submanifold, viz: 5 to 10, double proportion, and 6 to 18 triple etc.  
II. Superparticular proportion, is when the Antecedent contains the Consequent but once with a fraction, whose Numerator is ever an Unite, or may be reduced unto it, viz: 3 to 2 is 1 ½, and 5 to 4 is 1 ¼, and so likewise 12 to 8 is 1 ½, & 15 to 12 is 1 ¼; so 20 to 16, or 10 to 8, is as 5 to 4. The contrary to this is Sub-superparticular, as the less to the greater, viz: 2 to 3, or 4 to 5, etc. and you may find in many Writers the fractions thus expressed according unto the Denominator, and in Latin, with Sesqui added to it, as ½ Proportio Sesquialtera: ⅓ Sesquitertia: ¼ Sesquiquarta: 11/10 Sesquidecima, that is in proportion as 11 to 10, which will be 1 1/10, and so of any other.  
III. Superpartiens, is a proportion when the Antecedent contains the Consequent once with a fraction, whose Numerator is always more than an unite, as 5 to 3 is 1 ⅔, or 9 to 7 is 1 2/9: the contrary to this is when the Antecedent is the lesser number, and is usually called Subsuperpartient, viz: as 3 to 5, or 7 to 9 Unto these proportions is always added Super, the middle is derived from the Numerator of the fraction which must be 2 at least: and the word is terminated with the Denominator of the same fraction, as 5 to 3 is in proportion 1 ⅔, and is usually called Superdupartiens tres; and 7 to 4, is 7 divided by 4, and will be 1 ¾, that is Supertripartiens-quartas: and so of all the rest.  
iv Multiplex Superparticularis, is a proportion betwixt two numbers, when the Antecedent contains the Consequent twice at least, with a fraction whose Numerator never exceeds an unite, this proportion is derived from the two former, and thence denominated, as 9 to 4 is 2 ¼, and is called Duplasesqui-quarta: and 9 to 2 is 4 ½, that is Quadruplasesquialtera: and 26 unto 5 is 5 ⅕, that is Quintupla Sesquiquinta: and so for any other. The contrary to this is when the Antecedent is less than the Consequent, and is denominated sub Multiplex superparticularis.  
V Multiplex-superpartiens, is when the Antecedent contains the Consequent twice at least, with a fraction, whose Numerator must ever exceed an unite, as 8 to 3 is 2 ⅔, and this proportion is called Dupla superdupartiens tertia; and 19 to 5 is 3 ⅘, and is termed Tripla super-quadripartiens quinta: and the contrary to this, is when the Consequent is greater, than is the Antecedent, but in the same proportion (and so is to be understood of the rest) as 3 to 8, or 5 to 19, etc. and is usually termed sub Multiplex superpartiens.  
In these 5 species are comprehended all the varieties of proportions betwixt any two numbers whatsoever; and as for fractions, they are all included within these 5 kinds, observing (as in whole numbers) for to divide one by another, according to the 5 Parag: & 2 Sect: of my first Book: and so also in mixed and compounded Fractions, viz: 3 to ⅔, that is 3/1 to ⅔, which if divided according unto the Rules of Fractions is 9/2, that is 4 ½, the proportion is Quadrupla Sesquialtera, or ½ divided by ⅓ is 1 ½, that is Proportio Sesquialtera; and so of any other.   

Addition of these 5 Species or unequal Proportions.  
Of these Proportions there is great use both in Geometry and Music, and are called by some Harmonical Proportions, whose excellency I will leave to the learned of that Art, and here only explicate the practic part, in the addition, and subtraction of them; that is, how to express in one sum, any two or more of these proportions, or how to subtract one from another, which is easily performed by stating the question according to common fractions: and first for addition, multiply their Numerators together, their product will be a new Numerator, and so likewise their Denominators, by this means producing one sum, comprehending both the former proportions, of which if there be many, having multiplied the first into the second, and that product into the third proportion, or term, and so to the fourth etc. as by this following Example shall be perspicuous.   

An Example of Addition in these Proportions.  
Admit the Proportions stated were 2, 4, 6, the first and second numbers are in a double proportion: the second and third, viz: 4 & 6, Sesquialtera, that is 1 ½, which place thus 2/1 & 3/2, which multiplied, as in Reduction of compound Fractions, Lib: 1. Sect: 2. Parad: 1. the product is 6/2 or 3, which is a Triple proportion.  
All numbers whatsoever differing in quantity or quality, if increased, or diminished, multiplied, or divided, by any one common number, I say the differences betwixt their sums, remainders, products, or quotients, will continue in the same proportion as were the differences of the numbers propounded, as admit 4, 12, & 20, 36, the difference of 4 & 12 is 8, and the difference between 20 & 36 is 16, double unto the former difference 8, and so will the difference of these numbers be, if increased or diminished by any one, or common number; as add or subtract 3 from them all, the sum or remainder will be 7, 15 & 23, 39 or 1, 9 & 17, 33; whose differences are in the same proportion as before, and so likewise if they were multiplied or divided by any common number.    

Paragraph V Treating of Natural Progressions and Arithmetical Proportions, with the addition of them.  
Natural Progression consists always of more than two numbers of equal difference or proportion betwixt them, either ascending or descending in order, as 1, 2, 3, or 3, 2, 1, the common difference is 1; or 2, 4, 6, 8, the difference is 2; or in this Progression 1, 4, 7, 10, or 3, 6, 9, etc. wherein the difference is 3; and so likewise 10, 20, 30, etc. or 100, 200, 300, the one proceeding by 10, the other by 100: and so of any others where the difference is equal, and this Progression of Numbers is called Arithmetical Proportion continued.  
Proportion interrupted is when there are at least 4 numbers propounded, but the Progression interrupted or broken off, and yet then proceeds again, as 1, 3, 5,— 6, 8, 10, the difference betwixt 1 & 3 is 2, or 3 & 5; and so likewise in 6, 8, 10, the difference proceeds by 2; yet where it is broken off the difference is but 1, as betwixt 5 & 6.  

Theorem 1.  
Any three numbers given in Natural Progression continued, the middle number doubled will be equal un●o the sum made of the extremes, viz: 2, 5, 8, the difference of these three numbers is 3, the sum of 2 & 8 is 10, and so is twice 5 also 10. Again 3, 7, 11, which proceeds by 4, if 7 be doubled it will be 14 equal unto the sum of 3 & 11; and the reason is evident, because the middle number is the mean proportional betwixt the two extremes, being so much less than the third number, as it doth exceed the first in a Natural Progression, or in any Arithmetical Proportion: in which if two numbers were given for to find a mean, half the sum of the two extremes will be the mean proportional number sought, as 10 & 20, two extremes, their sum is 30, the half is 15; and so it is 10, 15, 20, proceeding by 5: the mean betwixt 4 & 9 is 6 ½, so it is 4, 6 ½, 9; and if the mean and one extreme were given, and the other required, add the difference betwixt the two numbers to the greater, and the sum will be a third proportional ascending, and subtracted from the lesser the remainder will be a third descending, as 10 & 13 the difference is 3, which added unto 13 makes 16 a third proportional, viz: 10, 13, 16, or 3 taken from 10 the lesser, the remainder will be 7, the lesser mean, viz: 7, 10, 13, and so the like of any other.   

Theorem 2.  
Any four numbers propounded in Natural Progression, the sum of the two means shall be equal unto the sum of the extremes, whether the Proportion be continued or interrupted: as for example, 1, 2, 3, 4, the sum of the two means, that is 2 & 3 makes 5, and so is 4 & 1: and likewise 10, 20, 30, 40, the extremes make 50, and so 20 & 30 added together will be 50: and so in Progression interrupted, or broken off, as 4, 10, 20, 26, the sum of the two means is 30, and so 4 and 26 added together is 30, the reason is the same with the former; for as 10 exceeds 4 the lesser extreme, so is 20 less than the greater extreme, from whence consequently the sum of the two means is equal unto both their extremes.  

Addition of Numbers proceeding in Arithmetical proportion continued. Example 1.  
All numbers that are in Arithmetical proportion may be collected into one sum, as by Addition of common numbers, in the first species of my Arithmetic; but more expeditely thus: add the first and last Progression together; and then either multiply that sum by half the terms or progressions; or else with the progressions, multiply half the sum made of the two extremes, which will be thus explained: as admit it were demanded the number of orderly strokes that a Clock strikes in 24 hours; 12 of them being in Arithmetical Progression, as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, the sum of the first and last is 13, which multiplied by half the progressions, or number of terms, that is 13 by 6 produceth 78; or wh●ch is all one, 12 multiplied by 6 ½ will produce also 78 as before, the number of strokes in 12 hours, which doubled will be 156, the number of strokes in 24 hours, as was required; and in any number equally proceeding, as 5, 9, 13, 17, 21, 25, the extreme are 30; and now it is indifferent whether you multiply 30 by 3, or 15 by 6, the sum will be 90 either way; and it is not material whether the Progression be ascending or descending, so it be continued; the reason of this proceeds from the two former Theorems.   

A distinction of Progressions continued. Example 2.  
Although all Progressions of this kind are numbers increasing or decreasing wi●h equal differences; yet divers Authors do use divers distinctions, which here I will comprise in two; the one Natural, viz: 1, 2, 3, etc. proceeding by unites only; and all others denominated from the difference between their terms, or progressions, viz: 1, 3, 5, 7, or 2, 4, 6, 8, etc. and is called by 2; and 1, 4, 7, 10, or 2, 5, 8, 11, Progressions by 3; and so the like of any other.   

One extreme being given with the order and number of the progressions, to find the other extreme. Example 3.  
In Natural progression (proceeding from an Unite, and so continuing) the number of them will be equal to the greater extreme; but if the Progression gins with any other figure, that is the Progressions term, then multiply that term by the number of Progressions, the product will be the extreme required: and the contrary, if the greater extreme be given, divide it by the number of Progressions, the Quotient will give the lesser extreme; as if 2 were the lesser extreme, and the Progression; what will the 6 place be? The answer to this is 12, & thus they will stand 2, 4, 6, 8, 10, 12; if by 3 what the 7 place, the answer will be 21, as it will thus appear 3, 6, 9, 12, 15, 18, 21, & if the greater extreme were given to find the less, as 32 whose Progression is to 8 places, therefore divide 32 by 8, the Quotient is 4, the less extreme, and the order of Progression, as will thus appear, 4, 8, 12, 16, 20, 24, 28, 32, or 30, the greater extreme in the sixth place, the lesser extreme was 5; but when the first term is not the difference of the Progression, then multiply the term of the Progression given by one place less than the number of Progressions, and to the product add the less extreme, the total will be the greater extreme; as if 3 were the less extreme of a Progression by 2 unto 7 places, then say 6 times 2 is 12, to which add 3 the less extreme, the sum will be 15 for the greater; and thus appears 3, 5, 7, 9, 11, 13, 15.  
The greater extreme, with the number of Progressions, and their difference being given, to find the lesser extreme, observe this as a general Rule: multiply the difference of Progressions, by one less than the number of the said Progressions, subtract the product from the greater extreme given, and the remainder will be the first number of that Progression continued: As for example, admit 44 were the extreme propounded, which had proceeded by 10 in five Progressions, 10 multiplied by 4 produceth 40, which taken from 44 there will remain 4, for the first number, and the lesser extreme, as by these numbers will appear, viz: 4, 14, 24, 34, 44. Again, admit 29 to be the greater extreme, proceeding by 2, in 13 Progressions, I say 12 multiplied by 2 produceth 24, which subtracted from 29 leaveth 5 for the first number, as 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, and so the like of any other continued Progression.   

With a mean proportional, and one extreme being given, to find the other extreme, and the number of progressions. Example 4.  
Any numbers in Arithmetical proportion, the sum of the extremes is double unto the mean, by the former Theorems; and therefore it is evident, that either of the extremes taken from the mean doubled will give the other extreme, as 4, 6, 8, and 6 doubled is 12, from whence take 4 and there will remain 8; or subtract 8 and there will remain 4: and consequently the two extremes being given, the mean is also included, being half the sum of two extremes; and to find the number of Progressions, subtract the lesser from the greater, and divide the remainder by the difference of Progressions, the Quotient will be always one less than the number, as if 5 & 29 were two extremes, whose difference was 2; the difference between them is 24, which divided by 2 (the difference of Progression) the Quotient will be 12, unto which add one, the number is 13, the whole series or order of the Progression, viz: 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29.  
For the clearer explanation of this, suppose two men were to go from London to Edinburgh, by computation 292 miles; they took horse together, one resolving for to travel 30 miles every day; the other 6 miles the first day, the second day 12, and so proceeding in Arithmetical Progression of 6 until he should arrive at his journey's end: and here it is required to know which of them came first to Edinburgh; or in how many days one will overtake the other: first observe that 30 is here the mean proportional, and 6 the lesser extreme, and the greater 54, the mean doubled is equal unto the sum of the extremes, that is 60, as by the former Rules, where the number of progressions will be found 9, and in so many days these Travellers will overtake one the other, having gone 270 miles, which is 9 times 30 equal to the other, as by these numbers 6, 12, 18, 24, 30, 36, 42, 48, 54, in this series you may observe 30 the mean, exceeding the others first day's journey, as it is exceeded by the last, and consequently first at his journey's end, for the next day will be 60 miles, wherein he rides as fast again as the other, and having but 22 miles to ride, I will leave them, and end this Paragraph, and proceed no farther in Natural Progressions.     

Paragraph VI Treating of Geometrical Progressions and Proportions, continued and interrupted, with the addition of those numbers.  
Geometrical Progression doth consist of more than two numbers, not of equal difference in numbers, but of like proportion in quality and quantity, viz: 1, 2, 4, 8, 16, 32, 64, etc. this is called a double proportion, every one being but half the succeeding number; or these 1, 3, 9, 27, or 2, 6, 18, 54, denominated a triple proportion, every number containing the precedent 3 times; so 2, 8, 32, 128, or 3, 12, 48, a quadruple proportion, every succeeding number being 4 times the precedent, and so of all other numbers, of what quality or quantity soever proceeding in this kind, and is nominated Geometrical proportion continued.  
Proportion interrupted or broken off, is when the Progression is discontinued in the proportion of the numbers, and consists of four places or progressions at the least, viz: 3, 6, 7, 14, wherein the second and the third number (that is 6 & 7) differs in the proportion, not as 3 to 6, or 7 to 14, which is a double proportion: and so 1, 3, 27, 81, is a Geometrical progression interrupted, for 1 & 3 & 27 & 81 are in a Triple proportion, but not the second and the third, 27 containing 3 nine times.  

Theorem 1.  
Geometrical progressions are either ascending or descending in the former proportions; and if continued, and proceeding from an unite: ●he first from 1 is called the Root or first quantity. The second is the Square or the second quantity. The third the Cube or third quantity. The fourth is Biquadrat, the Squared square or fourth quantity. The fift is called the Surde solid or fift quantity. The sixth is denominated the Squared Cube, or the sixth quantity, etc. all which are explained in these numbers, viz: 1 being neither Number nor Quantity: 2 the Root: 4 the Square: 8 the Cube: 16 the Biquadrat, made by the square of 4: next 32 is the Surde solid, made by the multiplication of the Biquadrat and the Root, or the product of the Square and the Cube: 64 is the squared Cube, that is the Square made of the Cube multiplied in itself; all which numbers proceeding from a Root, are found by multiplying the Root in its self, and then in the next product or progression, and so proceeding, the next proportion all above it will be produced, and so to what number of Progressions you please, making the Root always Multiplier; and the numbers count nued betwixt the first and the last, are called Mean proportionals, or Continued means, in Geometrical progression, as is evident in the last Example: to find the Root or number on which the Progression was made, divide any one of the Progressions, by the next inferior or less unto it, the Quotient will be your desire: As for example, 3, 6, 12, 24, etc. if you divide 6 by 3, or 12 by 6, or 24 by 12, the quotient will be 2 for the number by which the progression was made; and so 5, 20, 80, 320, I say any one of these divided by the next less will show the Progression was by 4; and so for any other.   

Theorem 2.  
Any three numbers in Geometrical progression, the Squa●e of the mean or middle number, is always equal unto the product, made by the multiplication of the first and third number: As for example, 2, 4, 8, are the three numbers given in a Geometrical proportion, where 4 is the mean, and being squared is 16, equal to the product of the extremes, viz: 2 & 8, which is also 16; and so in 3, 6, 12, the square of 6 the mean is 36, equal to the product of 3 and 12 being likewise 36; so 4, 12, 36, the square of 12 is 144, equal to 36 multiplied by 4, which produceth also 144, as before; so 3, 6, 12, are in the same proportion, and the square of the mean is 36, equal unto the product of the two extremes, viz: 3 & 12, which multiplied produceth also 36; and so the like in any three numbers where the first is contained in the second so many times as the third contains the second number, and s evident in itself, for what quantity or proportion the fi●st hath to the second, the same hath the second unto the third number, and so consequently the mean squared must be always equal to the product of the two extremes, otherwise those numbers are not in any proportion, or progression Geometrical.  
In the same manner if 4 numbers be in proportion, the product of the two means are equal unto the product of both the extremes: As for example, 2, 4, 8, 16, the product of 4 & 8 is 32, and so is 16 multiplied by 2: or 3, 6, 12, 24, the two means 6 & 12 will produce 72, equal unto 3 times 24, the two extremes; and from the former reason, the second being in proportion to the first, as the fourth number is to the third; and this is general, whether the Geometrical progression be continued or interrupted, as 4, 12, 14, 42: for as 12 contains 4 thrice, so is 14 (the third number) contained 3 times in 42, and the product of the two means, viz: 12 & 14 is 168, and so 42 multiplied by 4 will produce likewise 168: and so for any other four numbers that are thus in a Geometrical progression, as the first is to the second, so the third to the fourth are proportional numbers.   

Proposition 1. Any two numbers propounded, for to find a mean proportional between those numbers.  
Multiply the two given extremes one by anoother, the Quadrat Root extracted out of that product shall be a mean proportional between the two numbers propounded: As for example, 4 & 9, admit the numbers given, these multiplied together produceth 36, whose Square Root is 6, the mean proportional number desired, and is evident in itself, for it contains 4 the lesser mean 1 ½, and so does 9 contain 6; for by the last Theorem, the product of two extremes being equal unto the square of the mean, consequently the Square Root of that product, must be the mean proportional required. And for a farther explanation, suppose 12 & 1728 be two extremes, whose mean proportional is required; the product of 12 and 1728 is 20736, whose Square Root will be 144, but in case the product should prove irrational, that is, wanting a perfect Root, then extract the Quadrat Root with a fraction, as your occasion shall require by the 4 or 5 Example, Lib: 2. Parag: 1, or in Fractions, as followeth in the same Parag:   

Proposition 2. A mean proportional number being given, with either of the extremes, to find the other extreme.  
To effect this, or any Question of this kind, Square the number propounded for the mean, and then divide that Square by the extreme given, the Quotient will show the other extreme: As for example, admit 5 were an extreme propounded, and 10 the mean, whose Square is 100, which divided by 5 the product is 20 for the greater extreme required: this may be also proved by the last Theorem, for this mean proportional squared contains both the extremes multiplied together, then consequently if that Square or number be divided by one extreme, the quotient must needs produce the other. Or by the 2 Theorem find the Progression, and with that multiply the mean, for to find the greater extreme, or by division find the less, as 3, 9, 27, are proportional numbers, and admit 9 were known and one extreme, the other is easily found, the progression being discovered to be triple, 9 the mean divided by 3, the Quotient will be 3 for the lesser extreme, and 9 multiplied by 3 shows the greater extreme.   

Proposition 3. Betwixt any two numbers propounded, for to find two mean proportionals.  
To perform this Question, or the like, you must always observe to Square the lesser extreme given, and that multiply into the greater extreme, from which product extract the Cubique Root, which will be the least of the two Proportionals sought; the other must be a mean between that and the greater extreme, which will be discovered by the second Theorem, or by the first Proposition: and for the farther illustration of it admit 2 & 16 were two extremes, between which, two means are required; the Square of 2, viz: 4 multiplied in 16 produceth 64, whose Cube Root is 4 for the lesser mean, and by the progression 8 must be the other; so these proportional numbers found are 2, 4, 8, 16: so likewise between 3 & 24 to find two means, the Square of 3, viz: 9, multiplied in 24 produceth 216, whose Cube Root is 6 for the lesser mean, whose greater must be 12, and the 4 proportionals will stand thus 3, 6, 12, 24, the reason of this differs not from the former, for whereas the single mean is found by multiplication of the two extremes in their first quantity, the Square Root being the Medium, which is the second quantity, so in finding two mean proportional numbers the greater extreme given is to be multiplied by a term in the second quantity, that is by the lesser extreme squared, from which product the Cube Root is to be extracted, a term of the third quantity according to Geometrical progression, which by some succeeding Examples shall be illustrated.   

Proposition 4. Between any two Extremes to find three Mediums or Proportional numbers.  
The solution of this Proposition, or the like is thus, multiply the greater extreme by the Cube made of the lesser propounded, from thence extract the B●quadrat Root, which shall be the least of the three extremes required; the other two will be easily found by Progression: As for example, betwixt 2 & 32 there are three proportional numbers required, which will be thus discovered: the Cube of 2 (the lesser extreme) is 8, which multiplied by 32, the greater extreme produceth 256 whose Biquadrat Root (as by the third Paragr:) is 4, for the least mean, the rest will be discovered by Progression, as thus, 2, 4, 8, 16, 32, to find 3 Mediums between 3 & 48, the Cube of 3 is 27, that multiplied into 48 the greater extreme, produceth 1296, whose Biquadrate Root is 6 for the lesser mean, the rest in Progression will stand thus, 3, 6, 12, 24, 48, or more expedite in many Questions; as thus, where the Progressions consist of an odd number, as of 5 places, 7 places or more, multiply the two extremes given together, the Quadrat Root will be a mean proportional between those extremes, then with that Medium, and one of the extremes, find another Medium as by the 3 Proposition, and so you may continue extracting of Roots until all the Mediums are discovered: As for example, betwixt 4 & 2916, there be 5 mean proportional numbers required, the product of 4 & 2916, is 11664, whose Quadrat Root is 108, with which Medium and 4 the lesser extreme, find the two lesser mean proportionals by the 3 Propos: the Square of 4 is 16, which multiplied into 108 produceth 1728 whose Cube Root is 12 the lesser mean, and the Progression is by 3, so the whole order or series of them is 4, 12, 36, 108, 324, 972, 2916.  
Or thus, here being 5 mean proportionals required, multiply the less extreme 4 to a Surdesolid which is the fift quantity, and thus found: the Square and Cube of any number multiplied together will produce the Surdesolid, that multiplied by the greater extreme, from whose product extract the squared Cube, as by the 3 Parag: the Root is the least mean proportional: As for example, the two extremes, admit 4 & 2916 the Square of 4 is 16, and the Cube 64, their Product 1024, which multiplied by 2916 the greater extreme, produceth 2985984, whose Square Root is 1728, and the Cubique Root of that is 12 for the lesser mean proportional number, as before. Observe always in finding of mean proportionals, that the less extreme given be multiplied by such a quantity as is equal to the number of the mean proportionals, as if 2 Mediums be required, then square the lesser extreme; if 3 Mediums take the Cube etc. these multiplied by the greater extreme, must have a Root extracted of the next quantity, as if multiplied by a Square, extract a Cube; if by a Cube, then extract a Biquadrat Root, and so still proceeding in a Geometrical progression; but farther note that the Roots of all Surde numbers cannot be extracted.   

Proposition 5. To find so many mean proportional numbers between an unite and any number, as shall be required.  
To find continual Mediums betwixt an unite and any number given, is with facility effected, by continual extracting of the Quadrat Root, as by some few following Examples shall be made perspicuous: and first admit between 1 & 4294967296 a mean proportional be required, an unite multiplies nothing, therefore the Square Root of 4294967296 is 65536, and to find another Medium betwixt this and 1, the Quadrat Root of 65536 is 256; and a mean proportional betwixt 1 & 256 is 16, and between 1 & 16 is 4, and the Square Root of 4 is 2, and so the continual means are these, 1, 2, 4, 16, 256, 65536; 4294967296, and these are properly continued Mediums between an Unite and the last Root extracted; if in finding any mean proportion a true Root cannot be extracted, I refer you then to the first 3 Paragraphs of this Book, whereby you may extract any of them with a fraction as you please, or as the condition of your Question shall require. As for example:  
Admit 5 such Mediums were required between 1 & 2, annex what number of Ciphers you please in pairs, as admit 6, than the number will be 2000000 whose Quadrat Root is 1414 the Numerator of a fraction, whose Denominator is 1000, and so it will stand thus 1414/1000, annex both to this Numerator and Denominator two cyphers, then extract the Quadrat Root, which will be 376/316 annex to either, two 00 again, the Square Root of 37600/31600 will be 193/177; annex two cyphers more, than the fourth Medium will be 13●/133. Lastly, annex two cyphers more, and then from 13800/13300 extract the Quadrat Root which will be then 117/115, and so the five mean proportional numbers between 1 & 2 are these fractions viz: 1. 117/ 115, 138/133, 193/177, 376/316, 1414/1000. 2. These are all less than 2, and yet greater than 1. Here observe that 2 the greater extreme was made an improper fraction, as 2/1; and as I annexed 6 cyphers to either of them, so I had done in the rest but for your ease; for the more cyphers you connex, the less error will arise in extracting of the Roots whereby to find these Mediums.   

Proposition 6. In any number of Geometrical Progressions continued, to find the total sum of them.  
To effect any Proposition of this kind, the Geometrical progressions being continued, find first the term of the Progression, as by the first Theorem, with which multiply the last Progression or greatest extreme, from whose product subtract the lesser mean, the remainder divide by a number which is an unite less than the term by which the Progression was made, this last quotient will give you the total sum of all the Progressions: As for example, 4, 12, 36, 108, 324, 972, is a series or order of Progression, and by dividing any one by the next less, shows the Progression was by 3; then multiply by 3, the last or greatest extreme, which here is 972, the product will be 2916, from whence subtract the least extreme 4, the remainder will be 2912, which divide by 2 a number, an unite less than the term by which the Progression was made, the Quotient will be 1456, the total of all the Progressions, as you will find by adding them all together to be 1456, as before: so 4, 20, 100, 500, 2500, 12500, whose total is required; the Progression is found to be by 5, which multiplied into the greater extreme, viz: 12500 produceth 62500, from whence subtract 4 (the less extreme) the remainder is 62496, and being the Progression was by 5, divide 62496 by an unite less, viz: by 4, the Quotient is 15624, the total sum of the last proportional numbers.  
And when there shall happen an unite for the Divisor, the Dividend is the total required, as if the sum of 2, 4, 8, 16, were required, the Progression is by 2, with which multiply the greater extreme, viz: 16, the product is 32, from whence subtract the lesser extreme, viz: 2, there will remain 30, the total sum; for 1 which is an unite less than the Progression cannot divide: so 1, 2, 4, 8, 16, 32, 64, 128, the Progression is by 2, with which multiply 128, the product will be 256, from whence subtract the lesser extreme, viz: 1, the remainder will be 255, for the total sum of all the Progressions, 1, 2, 4, 8, 16, 32, 64, 128, the reason is, that every Progression in this kind, contains all the inferior numbers, & the lesser extreme, therefore one Progression more must be found, from whence the lesser extreme is to be taken, and if the Progression were by any other number than 2, as 3, 4, or 5, etc. it must be divided by an unite less, by which means it will be reduced unto the state of the last Question, where by 2 the Progression is made, and here I will end, and make no farther a progress in Progressions.    

Paragraph VII. Treating of Universal Axioms in Arithmetic, from whence (with the former Theory and grounds) proceeds the Practic part, and the explication of the Golden Rule.  

Axioms.  
I. EVery thing that is whole, is equal unto all the parts taken together upon which it consists.  
II. What things soever are equal to one and the same, are all equal amongst themselves, viz: if A be equal to B, and C equal to B, then shall A be equal to C.  
III. The half of any thing hath the same proportion to the half of another, as the whole had unto the whole, viz: as 12 is to 8, so is 6 to 4, or 3 to 2.  
iv If equal numbers or quantities be added unto equal things, their sums will be equal.  
V Equal things subtracted from equal numbers or quantities, their remainders will be equal.  
VI All equal numbers multiplied by like quantities or numbers, will have equal products.  
VII. Any equal numbers divided by equal quantities or numbers, their Quotients will be alike.  
VIII. All equal numbers added or subtracted, multiplied or divided by unequal numbers or quantities, their sums, remainders, products, and quotients, shall be all unequal.  
IX. Any two numbers in what proportion soever, the Square of the lesser number shall have the same proportion to the product of them, as the propounded have one to another, viz: 2 & 4 a double proportion, so is 4 & 8, or as 3 to 9, so 9 to 27.  
X. In any two numbers given, the proportion between the numbers multiplied into the Square of the lesser number, will be equal unto the product of both numbers, viz: 2 & 6, a triple proportion, and 3 times 4 is 12, equal to the product of 2 & 6.  
XI. In any three numbers propounded, if the product of the second and third number be divided by the first, the quotient will produce a fourth number in proportion to the second, as the first shall be unto the third; and the 2 means squared, are equal to the square of the two extremes.  
A B 15 to B C 9; so will A D 10 be to D E 6.  
As A E 8 to E F 4; so is A C 12 to C G 6.  Quae uni tertio conveniunt, inter sese conveniunt: Ergo universè sunt. 
As A B 15 to B G 3; so A D 10 to D F 2.  
Or thus,  
As A D 10 to D F 2; so A B 15 to B G 3.  
Or thus,  
As F D 2 to G B 3; so will E F 4 be to C G 6.  
Or as  
A F unto F D; so will A G be to G B.  
Or as  
A F to F E; so shall A G be to G C.  
XIII. In any numbers given, where there is a Divisor, a Multiplier and a Multiplicand, the Divisor divided by any number, and one of the other two by the same number, I say their proportions and quantities will still remain the same, viz: 24, 18, 8. by the 11 Axiom, the fourth proportional found will be 6, divide 24 & 18 by 3 the numbers will be 8, 6, 8, then by the 11 Axiom the fourth number will be 6. Or divide them again by 2, as 4, 3, 8, or 2, 3, 4, or 1, 2, 3, the fourth number produced will be 6; and in the second or third number, it is not material, whether of them is made Multiplier or Multiplicand, and as these numbers are diminished by Division, they may in the same manner be increased by Multiplication, the numbers reserving their proportions entire: as in 1, 5, 3, or 2, 5, 6, the fourth will be 15 in a triple proportion to 5 as 1 to 3, or as 2 to 6, and so will 4, 10, 6, or 12, 10, 18, etc. produce likewise 15.  
XIV. Where there As 4 19 8 — 114  3  9 As 12 19 72 — 114 As 1 19 6 Facit— 114 are five numbers propounded to find a sixth in proportion to the rest, as 4, 19, 8, 3, 9; this requires the 11 Axiom twice stated, viz: As 4 to 19, so 8. or Axiom 13. As 1 to 19, so 2 unto 38. Again, as 3 to 38, so 9 or as 1 to 38, so 3 unto 114; in this there are two Multiplyers, as 8 & 9; and two Dividers, as 4 & 3; and yet but one Medium to all: therefore in such cases, if the Multiplyers and Dividers respectively be increased by one another, the Products, and former Medium, will be in the same proportion unto a fourth number, as the other 5 were unto the sixth number before required, viz: 4, 19, 8, 3, 9, by the 11 Axiom, and twice repeating of that Rule of 3, there will be 114 produced, and so will the fourth proportional be of 12, 19, 72; or by the 13 Axiom, as 1 to 19, so 6 unto 114, from whence this proportion ariseth.  
As the Product of the two Multiplyers  
Is in proportion unto the given Medium,  
So will the Product of the two Dividers  
Be in proportion unto the number required.  
XV. As the total of any As 9 to 18, so 

2  
3  
4   4 to 6 8 thing shall be in proportion to the total of some other; so will the particular respective parts of the one be proportionable unto the parts of the other; as in 9 & 18 being in a double proportion, I then divide 9 into 3 parts, viz: 2, 3, 4, this done, by the 11 Axiom, you may find 3 proportionals, viz: 4, 6, 8, the total 18, and double unto 9 as the parts are: and the like in any other numbers.  
XVI. In any three numbers   Diff: Produ: A 9 B 12  2 24 C 7 3 21 45  5 45 given, or propounded, wherein the first is lesser than the second, and yet greater than the third, I say the difference betwixt the first and second, multiplied by the third; and the difference between the first and third multiplied by the second; the sum of those products will be equal unto the product of the first number, multiplied by the sum of the aforesaid differences, as admit A, B, C, were three numbers given, viz: A 9, B 12, C 7, the difference betwixt A & C, that is 2, multiplied by B 12 produceth 24, and the difference of A 9 & B 12 is 3, which multiplied by C 7 will produce 21; the sum 45: and so will the product of 5 (the sum of the differences) and A 9 increased by one another make 45; and as it is true in the connexion of two numbers, the same it will be in many, and is demonstrated in my Book of trigonometry.  
XVII. In any two numbers given, half the difference of them, added unto half the sum of both numbers, shall be equal unto the greater number: admit 20 & 8 were two numbers given, half the difference betwixt them is 6, the sum of them both is 28, the half 14, to which add 6, the sum is 20, equal unto the greater number.  
XVIII. From half the sum of any two numbers, take half the difference that was between them, and the remainder shall be equal unto the lesser number as by the last, where 20 & 8 are the numbers propounded, half their sum is 14, from whence take 6 half the difference and the remainder will be 8, for the lesser number.  
XIX. All Fractions whatsoever, if their Num●rators have one and the same proportion, unto their respective Denominators, they are all equal in their quantities, as in relation to their Integers, whose parts they are; viz: ½, 3/6, 5/10, etc. are halfs; 7/3, 3/9, 4/12, etc. are thirds, and therein alike.  
XX. Any number propounded if multiplied by two several numbers, the difference of their products will be equal unto the difference of their numbers multiplied by the given number; from this Axiom the following Canon is derived.   

In any number for to be multiplied by 9, the Product may be discovered by subtraction only, and thus.  
Annex, or suppose a cipher to be annexed unto the Multiplicand, or number propounded, as admit 45, it is then multiplied by 10, and will be 450; or thus 45 let the other number be an unite (whereby 9 may be the difference) which multiplies not, from whence subtract 45 (the number given) and there will remain 405, equal to the product of 45 increased by 9, the difference betwixt 1 & 10 the two Multiplyers propounded, and so likewise in all Multiplyers consisting of nine, viz: for 99 take 1 & 100, for 999 take 1 & 1000, etc. besides in these numbers your labour may be eased, as in 19 the Multiplier, or Multiplicand take 1 & 20: for 29 take 1 & 30, etc. the difference being the number propounded according to this Axiom.   

How to find two squared numbers, whose difference shall be any number propounded.  
 (1) (2) The sum and difference 12 & 8 9 & 1 The half of either is 6 & 4 4 ½ & ½ The squares of them are 36 & 16 81/4 & ¼ The difference in either 20 20  (3) The sum and difference 12 & 2— 10 The half of those are 6 & 1— 5 Their squares will be 36 & 1-10 ✚ 25 The true difference is— 20  
XXI. Divide the given difference by any number at pleasure (yet without fractions is best (if it may be) add and subtract the Divisor from the Quotient, the half of that sum and difference are the two Roots, whose Squares are the numbers sought, As by these 3 following Examples, of two squared numbers required, whose difference shall be 20, according to prescription and the first Table, I make 2 Divider, the Quotient will be 10, to which add and subtract 2, the sum and difference is 12 & 8, the half of them 6 & 4, their squares 36 & 16, the difference 20. In the second Table I take 20 again, and divide it by 4, the Quotient will be 5, the sum and difference 9 & 1, the half 4 ½ & ½, their squares 81/4 & ¼, and their difference 20, as before; if 10 had been Divider, and 20 the Dividend or difference given, the Quotient will be 2, to which add 10, the sum is 12, now to subtract 10 from 2 is not within the confines of Natural Arithmetic, yet by the assistance of Art, brought under her jurisdiction thus, 2-10 which is 2 less by 10, the half of either is 6 & 15, that is 4 less than nothing, their Squares 36 & 1-10 ✚ 25, the difference of their squares 20, as in the third Table, which is extracted out of Algebra.  
XXII. Several Dividers are made one, by Multiplication continued, and Fractions avoided, which will happen in every operation, if any thing remains, as if 500 groats were given for to be reduced to pounds sterling and parts, it should be divided by 3, and that Quotient by 20; the product of these is 60, with which divide 500 gr: the Quotient will be 8 ⅓ li. that is 8 li. 6 s. 8 d. Again, admit 1656 were for to be divided by 5, the Quotient and Fraction by 6, and that Quotient with the Remainder divided again by 7, the last Quotient & Remainder will be 7 31/35; whereas the 3 Dividers, viz: 5, 6, & 7, by multiplication continued will make 210 for a common Divisor, with which divide 1656, and at one operation the Quotient will be 7 186/210, which reduced will be 7 31/35, as before. See Lib. 1. Parag: 4. Exam: 9  
XXIII. The Multiplier of a number if divided by any significant figure, and likewise the Multiplicand, and if nothing remains in either number, the same figure will divide the product without any fraction; and if there happens to be a Remainder in both of them, their products divided by the same figure, and then that Reminder will be equal to the fraction of the whole product divided by the same significant figure, and in case nothing remains, you shall not find a Fraction in the first total Product.   

An illustration.  
Let A 69 be the Multiplicand  The proof A··69 B··39 621 207 2691 propounded, and B 39 the Multiplier; and for a common Divider take 9, with which divide 39; the Remainder will be 3, which for form, place upon the right hand of a cross, as by the Proof in the upper Table: this done, divide the Multiplicand A 69 by 9 (the common Divider) the Remainder will be 6, which place against the former, their product is 18, which divided by 9, nothing remains; put a cipher over the cross: then with 9 divide the product of A & B (which is 2691) and nothing will remain, for which place a cipher beneath the cross. Again in the second Proof, take any other figure for a common Divider, as admit 4, with which divide 39, the Remainder will be 3, which place on the right hand, then divide A 69 by 4, the Remainder will be 1, which place against it, their product is but 3, which place above the cross, and divide the total product 2691 by 4, and there will remain 3 to be placed beneath, equal to that above. This, some do use as a probat to Multiplication.  
XXIV. All Integers whatsoever A 1656 B 2444 C 4100 Proof 5-5 of several quantities, if divided by a common number, & the total of the Remainders divided by the same (if greater than the Divisor) what then remains will be equal to the Remainder of their total sum, divided by the common Divisor; as in the example A, B, C, wherein I take 7, and divide with it A 1656 the Remainder 4, which place upon the right hand of the cross: then divide B 2444 by 7, & 1 will remain, which place against the other; the sum 5 set above the cross. Lastly, C 4100 divided by 7, there will also 5 remain as in the Proof, which some uses for a trial of Addition.  
With these Axioms (as in a circular motion) I have returned to my first species of Addition, and so enclosed my work of Fundamentals, for all Rules in Natural Arithmetic, yet not to circumvent the Ingenious, nor charm their fancies within a circle of my own imagination, since there are many more Principles besides these, which I conceive of sufficient force to prove what is past, and justify the Rules to come, as a light exposed indifferently to both, by which means observing my Orders, and viewing the Tract behind, you may safely proceed without doubt or deviation.    

Paragraph VIII. The Golden Rule both direct and reverse demonstratively proved; upon which Rule of Proportion (as the foundation) are all the other Rules erected, in framing this Art of Numbers.  

A Definition of this Golden Rule of Proportion.  
THREE is all, the Christians Axiom; the sage Philosopher's Maxim; and the acute Arithmeticians Fundamental, from whence the Golden Rule is derived, and so denominated, from the excellent use thereof in numbers, being the Basis, on which the other Rules do stand: this is called the Rule of Three, it consisting properly of three numbers known, from whence a fourth in proportion (though unknown) may be discovered; from which sympathy it is also called the Rule of Proportion, as shall be demonstrated, and explicitly proved, with various examples, in several kinds in this following Treatise.   

The Golden Rule of Proportion in Numbers, Geometrically demonstrated by Lines. In any three right lines given, what proportion soever the first hath to the third, the second line will be in the same proportion to a fourth; and the Square of the two mean proportionals, shall be equal unto the long Square made of the two extremes. The Demonstration.  
Draw a right line as A, C, E, at the point A make an Angle at pleasure by drawing the line A, B, D, from A set on the given lines, which suppose to be A, B, 2. Secondly, A, C, 3. Thirdly, A, D, 4, by the two points at B & C draw the line B, C, and by the point D draw a parallel line, as D, E; then will A, E, be a fourth proportional line required: for D, E, being a parallel line to B, C, they make two equiangled triangles, viz: A, B, C, & A, D, E. then by the 19 Proposition of my trigonometry, the sides are all proportional, so it is as A, B, 2, to A, C, 3, so A, D, 4, unto A, E, 6; for as 4 the third number contains the first twice, so shall 5 the fourth number be in a duplicate proportion to 3 the second number; and the long Square made of the two extremes, viz: of A, B, 2, and A, E, 6, is equal to the Square made of the two mean proportionals, as A, C, 3, and A, D, 4, whose products are 12, as in the 9 Squares T, W, does evidently appear, the thing required for to be demonstrated.  

Canon 1.  
The Golden Rule is either direct, or reverse, and those single, or compounded of more numbers than 3, and consisting of more denominations than 2.   

Canon 2.  
Any three numbers propounded, a fourth proportional number may be found, according unto the 11 Axiom before cited in the last Parag: and by an ocular demonstration made apparent.   

Canon 3.  
The proper terms belonging to the single Rule of Three, must consist of two denominations, viz: a Multiplier and the Divisor alike; the other Multiplier of the same denomination with the Quotient; which is the fourth term required: As for example, 2 Yards cost 8 Shillings, what 10 Yards? The second number multiplied by the third, that is 8 by 10 produceth 80, which divided by 2 the first term, the Quotient will be 40 shillings, and the four terms will stand in this order:  
As 2 yards is unto 8; shillings; so will 10 yards be unto 40. shillings.   

Canon 4.  
The numbers in the Rule of Three direct, should be so ordered, that the known term (upon which the query is made) may possess the third place in the Rule: the other term which is of the same denomination, must stand in the first place, and consequently the other term betwixt them, which is known, and the same denomination with the fourth term required; as in the last example, the Question stated is upon the price of 2 Yards: and the query is, what any number of Yards will be worth in the same proportion; as here where the price of two Yards is known, and the price of 10 Yards is required: in this, the first and third term are both of one denomination, likewise the second term given and the fourth found, by multiplying the second with the third term, and dividing that product by the first; which is always required in the Rule of Three direct, as if 8 shillings bought 2 yards, what will 40 shillings buy? The answer will be 10 yards, or the contrary,  
As 40 shillings shall be to 10, yards so will 8 shillings be unto 2 yards.   

Canon 5.  
The third term in the Rule of Three direct proceeds from the first, as the fourth term required or found, proceeds from the second; so that what proportion there shall be betwixt the first and third term, the same proportion there will be between the second known, and the fourth found, As in the last Example, 40 s. contains 8 s. in a quintuple proportion, that is 5 times, and so many times doth 10 contain the fourth number found, viz: 2, and it is general both in whole numbers and in fractions.  
If 3 lb Sugar shall cost 5 shillings = what shall 300 lb Sugar?  
The product of the second and third term divided by the first produceth 500 s. where it is manifest, that the third term exceeds the first one hundred times, and so the fourth found exceeds the second known, from whence the third and fourth term proceeded: and so likewise if the terms be changed.  
As if 5 shillings bought 3 lb. sugar than 500 shillings will buy 300 lb. sugar.   

Canon 6. Sheweth the operation of this Golden Rule both in broken and compound numbers, with the explication of the most compendious and facile way.  
The Rule of Three in broken, or compound numbers is the same with Integers, for as in the Rule direct, the product of the second and third term, must be divided by the first, observing Multiplication and Division according to the former Rules of Fractions, and also in compound numbers, for to reduce the terms into improper fractions, whereof I will show you some examples: As if ½ yard cost ¼ of a pound sterling, what ⅔ of a yard; the product of ¼ & ⅔ is 2/12 or ⅙; which ⅙ divided by ½, the Quotient will be ⅓, that is 6 s. 8 d.  
As ½ yard to ¼ lib. so ⅔ yard will be unto ⅓, lib. that is 6 sol. ····8 d.  
Or as ⅓ lib. to ½ yard, so will ¼ lib. be unto ⅜ yard.  
Or,  
If ⅔ of a lb cost ¾ of a s. what ⅘ of a lb facit 9/10 s.  
Again,  
As 2 ¾ yards cost 3 ¼ lib. what shall 10 ½ yards.  
Or,  
As 11/4 yard's cost 13/4 lib. what 21/2 facit 12 9/22 lib.  
That is,  
As 11/4 y. to 13/4 l. so 21/2 y. unto 273/22 l. that is 12 l. 8 2/11 s.  
If 3 ¾ y. did cost 7 s. 6 d. what shall 24 ½ y. cost●, These made improper fractions will be thus:  
As 15/4 y. cost ⅜ l. what shall 49/2 y. cost? facit 2 l. 9 s. For the second and third term multiplied together, that is 49/2 by ⅜ the product will be 147/16 which divided by the first term 15/4 the Quotient will be 588/240 and reduced, thus 49/20 lib. that is 2 lib. 9 s.   

Canon 7. Any Question in Fractions by the Golden Rule direct, may be performed more compendiously at one operation, and thus:  
Multiply the Denominator of the first term by the Numerator of the second term, and that product by the Numerator of the third term, this last product will be a new Numerator, whose Denominator shall be the product of the Numerators first term, multiplied by the Denominator of the second term, and that product by the Denominator of the third term:  
As ⅔ s. ¾ lb ⅘ s. and a fourth proportional is required; say 3 times 3 is 9, and 4 times 9 is 36, for the new Numerator, and for its Denominator, multiply 2 by 4, the product is 8, and that multiplied again by 5 (the third terms Denominator) the last product is 40; so 36/40 or 9/10 is the fourth proportional number required, and so the like of any other; for it is evident, Multiplication and Division being both performed, but in a preposterous order, Division being before Multiplication: As for an example in whole numbers, as 2, 8, 10, the second term multiplied by the third term, and divided by the first will produce 40 for the fourth term; so if 8 be divided by 2 the Quotient is 4, which multiplied by 10 produceth 40, as before.   

Canon 8. An examine, or trial whether the Rule of Three be truly performed, either in whole numbers or fractions.  
Having found a fourth proportional number by the Rule of Three, you may easily discover whether the operation be right or no by the 11 Axiom, which affirms that if 4 numbers be proportional, the Square of the two means will be equal unto the Square made of the two extremes: As for example,  
If 3 lb shall cost 14, Crowns then 27 lb will cost 126. Crowns  
The square of the two extremes, viz: 3 & 126 will be 378, and so will the long Square made of the two mean proportionals (that is 14 & 27) be 378. And so likewise in the last of the 6 Canon, there were four proportionals found, viz: 15/4, ⅜, 49/2, & 49/20, the long Square made of the two means will be 147/16, and the long Square made of the two extremes, viz: 15/4 & 49/20 will be 735/80 equal to 147/16 for by the first Book, Sect: 2. Parag: 1. Parad: 4. the improper fraction 735/801 will be reduced by 5 unto 147/16, as before.   

Canon 9 The Rule of Three, or Proportion reversed, and how known from the Rule direct; with its operation exemplified both in whole numbers and fractions.  
This Rule differs not essentially from the former, but in the manner of operation only, for it consists of three numbers, and a fourth in proportion is required, proceeding from the second term, as the first does from the third term; whereas in the Rule direct the third term proceeds from the first, as by the 5 Canon, which is here a Multiplier, and was there the Divider: but in any question propounded where it seems ambiguous which way of these two to take, observe carefully the state of the question, and whether the fourth proportional number sought, should be greater or lesser than the second term; and if a greater be required, than the lesser of the two extremes given must be Divisor, and if the fourth number ought to be less than the second term, than the greater of the two extremes is Divisor.  

I. An example in whole numbers of the Rule of Three reversed.  
This is also the Golden Rule, and differs nothing from the former in the operation, but in making the third number Divisor, and the fourth proportional sought to proceed from the first; as if 3 Men in 18 Days did reap a certain number of Acres of Wheat, and it is required to know in how many Days would 12 Men have reaped the corn in the same field; it is evident that if 3 men could effect this work in 18 Days, a greater number of equal workmen must perform it in fewer days; than it is the Rule Three reversed, and by the last Canon, the greater extreme of the three numbers given must be the Divisor, which here is 12; then 18 multiplied by 3 produceth 54, which divided by 12, the Quotient is 4 ½ Day: so 12 Men may perform the same work in 4 Aequinoxiall days and 6 Hours, as the 3 Men did in 18 Days; for as 12 contains 3 just 4 times, so does 18 contain 4½, and the 4 proportional numbers will stand in the Rule thus:  
3 Men— 18 Days— 12 Men— 4 ½ Days.   

II. An example in compound numbers of the Golden Rule reversed.  
A Lawyer bought 4½ yards of Cloth for to make him a Gown; the cloth was 6 quarters wide, and it is required to know how much stuff would line it, that was but 3 quarters wide: The three numbers will stand thus, the cloth in breadth 1¾ of a yard, in length 4 ½ yard, the stuff in breadth ¾, and if made improper fractions thus, 7/4 B. 9/2 L. ¾ B. Here observe, the Stuff being narrower than the Cloth, there must be more yards of Stuffe than Cloth to make them equal; and being the fourth proportional number required must be greater, according to the 9 Canon, the lesser extreme of the three numbers given must be Divisor, which is here the third term; therefore this Question must be performed by the Rule of Three reversed; so the product of 7/4 & 9/2 is 63/●, which divided by ¾ (the third term) the Quotient will be 212/24, that is 21/2, which is 10 ½ yards of this Stuff equal to 4 ½ yards of the Cloth. This Question may be performed according to the 7 Canon, but with a contrary operation, as thus, Multiply the Denominator of the third term, by the Numerator of the second term, and that product by the Numerator of the first term will produce a new Numerator, whose Denominator will be found by multiplying the Numerators third term by the Denominator of the second term, and that result by the Denominator of the first term, as in this last Example 7/4 B. to 9/2 L. so ¾ B. by a reversed proportion unto 252/24, or 23/2, that is 10 ½ yards, as before: the Rule will stand thus, As 7/4 breadth unto 9/2 length, so will ¾ breath be unto 252/24 length or 21/2.    

Canon 10. An exact trial of the Golden Rule reversed both in whole numbers, as in fractions.  
The fourth proportional number found according to the Rule of Three reversed; multiply the third term by the fourth, and that result will be equal unto the product of the first and second term, if not alike the operation is wrong; as in the example of the 9 Canon in whole numbers, where the four proportional terms are these, 3, 18, 12, 4 ½, and the product of 12 & 4 ½, or 12/● & 9/2 is 108/2, that is 54 equal to 3 times 18, which is also 54. In the second Example of the same Canon the four proportionals are these viz: 7/4, 9/2, ¾, 21/2, and the product of ¾ & 21/2 is 63/8, equal to 9/2 multiplied by 7/4, which is also 93/8, as before. This is evident by the 11 Axiom where the Square of the two Means is equal to the Square of the two Extremes, for the fourth term sound in the Rule of Three reversed, proceeds from the first term given, and consequently belongs unto the first place, and so the third number given and the fourth term found are properly the two extremes, as in the former examples, Lib. 2. Parag: 8. Canon 9  
So as 12 Men to 18 Days, so 3 Men to 4 ½ Days.  
Again,  
As ¾ unto 9/2, so shall 7/4 be in proportion to 21/2.     

Paragraph IX. The Rules of Practice with the Definition, Theory and Practice of it, in the Rules of Three direct and reverse, both in whole numbers and fractions.  
THis Rule of Practice is an abstract of the Golden Rule, and the same in operatian, only it always hath an unite for the Divisor, or one of the Multiplyers, or such three proportional numbers as may be reduced unto one of them: this Rule is thus denominated from the practice, not consisting in many Questions either of any art or theory, but even Natures dictates, and commonly used in the whole Genealogy of Merchants down to the Pedlar, as I will instance in some Examples; the Coins, Weights, and Measures, supposed to be known, as they are unto Tradesmen.  

Question 1.  
If 1 lb of Pepper cost 1 s. 10 d. how much will 1 C & 8 lb come unto? facit 11 lb. First 1 C & 8 lb is 120 lb. then say 120 s. is 6 li. and 120 times 6 d. is half so much, that is 3 li. and 120 groats is 2 li. so the total is 11 li. and stands in order thus:    li. s. d. 1 s.  120 shillings makes 6 0 0 6 d.  120 times 6 pence comes to 3 0 0 4 d.  120 groats amounts unto 2 0 0   The total is 11 0 0   

Question 2.  
If 1 yard of black Cloth cost 1 lb 19 s. what shall 30 yards of this Cloth cost at the same rate? Here is 30 lb. 30 Angels, 30 Crowns, 30 half Crowns, 30 Shillings, and 30 Testers; which collected together is 58 lb. 10 s. as by this following example:   li. s. d. 1 li. 0 s. for the 30 yards there is 30 0 0 10 s. then 30 Angels 15 0 0 5 s. next 30 Crowns 7 10 0 2 s. 6 d then 30 half Crowns 3 15 0 1 s. and then 30 Shillings 1 10 0 6 d lastly, 30 times 6 pence 0 15 0  The total sum is 58 10 0   

Question 3.  
This Rule of Practice is less vulgar, lb L. lb 30 45 56 6 9 56 3 9 28 1 3 28 facit 84 L. yet more artificial, and of greater use than is the former, and solves divers questions, which the other cannot do without some mental reservation; for here in this Question 30 lb of Commodities cost 45 L sterling, and it is required what half a C or 56 pound weight will cost? According unto the Rule of Three direct, the product or 45 & 56 divided by 30 will produce for the fourth number 84; now by this Rule & the 13 Axiom this Question may be solved with less Multiplyers, and without a Divisor, that being first reduced unto an unite, as thus, divide 30 lb by 5, See Lib: 1. Sect: 1. Parag: 5. Exam: 10. & Lib: 2. Parag: 7. Axiom 13. And so likewise 45 L. being one of the Multiplyers, the reduction will be 6, 9, 56, then divide 6 & 56 by 2, the Quotient will be 3, 9, 28, then divide 3 & 9 by 3, and so you will produce these 3 numbers, viz: 1, 3, 28, in the same proportion that the Question was stated in, and 28 lb multiplied by 3 L. produceth 84 L. the solution of the Question, as was required.   

Question 4.  
In all Questions of this kind in the Rule of Three, having reduced the Divisor unto an unite, if the second number (on which the Proposition stated does depend) shall consist of several denominations, divide them into what parts you please, as in the first Quest: of this Parag: but best into such parts as may successively depend one upon another, either in number, weight, or measure, as in these following Examples: A man had made a piece of Cloth that stood him in 1 li. 12 s. 2 d. farthing a yard, and it is required to know what 30 yards would come unto at the same rate? In the first place take 1 li. then 10 s. as the half of that, than 2 s. as the ⅛ of the last: next 2 d. the 1/12 part of 2 s. and lastly, ⅛ of 2 d. for the Farthing, the total will be 48 li. 5 s. 7 d ½, or divide the price into these parts, according unto this example, L S D  L S D 1 0 0 In the first place there is— 30 0 0  6 8 Secondly, ⅓ part of the last is 10 0 0  3 4 Thirdly, take ½ the last as— 5 0 0  1 8 Fourthly, ½ the last again— 2 10 0  0 5 Fiftly, ¼ of the last which is 0 12 6  0 1 Sixtly, ⅕ taken of the former 0 2 6  0 0 ¼ Lastly, ¼ of the last before— 0 0 7 ½    The total — 48 5 7 ½   

Question 5. If 10 yards of Holland did cost 1 L. 3 S. 6 ½ D. What will be the price of 1 yard?  
In the Rule of Practise, where an Unite is the Multiplier, the other number is to be divided only by the first in the Rule direct, which is commonly an unite with cyphers annexed unto it, so there is nothing more to be done in such cases than to cut off so many figures upon the right hand of the first denomination, as there be cyphers in the Divisor, and if in case the Dividend does prove the lesser number, reduce it into the next denomination lesse, and having added in their parts; (if there be any) cut off so many places as before, and reduce that remainder unto the next, and so proceed: but to avoid mistakes, and to have your work before you, observe the solution to this question and form of the Table, where first I state the question again, as 10 Y. cost 1 L. 3 S. 6 D ½, what costs 1 Y? The third being an unite, and the first denomination of the middle number is 1 L. in which 10 the Divisor cannot be contained; reduce it to Shillings, and then you will find 23 S. place 3 S. on the right hand of the line drawn in the middle of the Table: this 3 S. is 36 D. & 6 pence added to it, the sum will be 42 D. place 2 D. on the right hand of the Table, and 4 D. on the left: the last remainder is 2 D. which is 8 Farthings, to that add ½ D. the sum is 10, place the cyper on the right hand, the question solved; and 1 Y. cost 2 S. 4 D. ¼ just, as in the Table. See Lib. 1. Sect: 1. Parag: 5. Example 10.   

Question 6. If 10 yards of Linen cloth cost 2 L. 14 S. 9 D ½: what will the price be of 6 yards?  
In this question 10 Y.— 2 L. 14 S. 9 ½ D.— 6 Y. Product L 1 6 L. 8 S. 9 D. S 12 8 D 10 5 Q 2 0 10 yards is the Divisor, and upon 6 yards is the query made, which multiply into all the several denominations of the middle numbers, which is the price propounded, the product will be 16 L 8 S. 9 D. and being 10 is the Divisor, place 6 upon the right hand of the line, and 1 L. on the left, which is the Quotient; then reduce the 6 L. (which remains) into Shillings, and add 8 S. unto it, the sum is 128, which subscribe with 8 S. placed upon the right hand of the line, those converted into pence with the 9 D. added to it, makes the sum of 105 pence, which set down, placing 5 D. upon the right hand of the line, which reduced to the next denomination less will be 20 Q. which divided by 10, as before, by placing the cipher upon the right side of the line, the Quotient is 2, and so if 10 yards cost 2 L. 14 S. 9 D ½. then 6 yards of the same cloth will cost 1 L. 12 S. 10 D ½, the question solved as was required.  
If an unite shall be one of the Multiplyers, and the Divisor a greater decimal, than so many figures or cyphers must have been placed upon the right hand of the line, as the Divisor shall have cyphers, but if they are not annexed unto an unite, but as 20, 30, 400, etc. or all significant figures; a division must be made by those past and the succeeding examples, this Rule of Practice will be sufficiently illustrated to the ingenious.   

Question 7. If any certain number of Men shall perform a piece of work within a known time; in what time would a greater or lesser number of men have performed the same?  
The Question here stated is 1 Men Days Men 28 12 8 2 7 12 2 3 7 6 1 4 Facit 42 Days. how 28 Men performed a piece of work in 12 Days, and it is required in what time 8 Men would have finished the same work; it is evident that the fewer Men must have the longer time, and consequently performed by the Rule reverse, 8 being made Divisor, which must be reduced unto an unite (if performed by the Rule of Practice) and may be thus reduced, 8 may be divided by 4; and also 28 (one of the Multiplyers.) So in the second row you will find 7, 12, 2, then divide 2 & 12 by 2, and then you will find in the third row these 3 numbers, viz: 7, 6, 1, the product of the two Multiplyers will be 42 Days, the Question solved as in the fourth row; if 8 Men should have performed the same work in 12 Days, and it had been required in what time 28 Men would have performed the same, the Question must have been solved by the Rule of Three reversed, because the greater number of Men would have required a shorter time, viz: 8, 12, 28; or divide 12 & 28 by 4, than it is as 8, 3, 7, than the first and last divide by 7 and it will be 1 1/7 or 8/7 3 & 1, that is 3 3/7 Days, and at 12 hours to the day, the time is 3 Days, 5 Hours, 8′, 34″ etc. But all Questions of this kind reduced into fractions are as readily performed by the Rule of Three, as by this of Practice; yet some I will here insert when an unite is Multiplier. See the 5 & 6 Examples.   

Question. 8. If part of any number, weight, or measure, did cost a part of any sum or integer of money, what shall any quantity of that commodity cost?  
In the first place here is ¾  Yard. L. yards. 1 ¾ ⅝ 78 2 24/32 20/32 78 3 24 20 78 4 6 5 78 5 1 5 13 6 Facit 65 L. 0 s. of a yard of cloth, which cost ⅝ L. and it is demanded, what 78 yards of the same cloth will cost, at the rate propounded? according to the former Rules in Fractions, reduce them unto one common Denominator, as in the second row, 24/●2 & 20/32 to 78. See Lib: 1. Sect: 2. Parag: 1. Parad: 8. now according to the former Rules, the Denominators being alike may be omitted; then in the third row, the proportion is as 24 unto 20, so will 78 be to a fourth proportional number: but to return, these 3 numbers may be reduced by dividing 24 the Divisor, and either of the Multiplyers by the same number, according to the 13 Axiom, Parag: 7. and here 24 & 20, divided by 4, the fourth row will be 6, 5, 78. then divide 6 & 78 by 6, the fift row will be 1, 5, 13, which 5 L. multiplied by 13 yards produceth 65 L. the solution of this question, as in the 6 row.   

Question 9 When the Divisor is a Fraction, or compound number in the Rule of Three direct or reverse; it may be reduced to the Rule of Practice by this following Example.  
The state of this Question  Broad. Long. Broad. 1 6/4 ⅞ ½ 2 ●/2 ⅞ ½ 3 3/1 ⅞ 1/● 4 facit 21/8 or 2 ⅝ yards. is how that cloth 6/4, or 6 quarters broad: ⅞ yard made a Child's coat, and it is required to know how much Stuff will make the same Child a coat when the Stuff is but ½ yard wide? which being narrower than the cloth, it is evident a greater quantity must be required, and consequently performed by the Rule reverse: the Question will stand as in the Example 6/4 or 3/2 requires ⅞ yar. what at ½ yard broad?    

Paragraph X. Shows the double Rule of Proportion both direct, reversed, or mixed, at two operations, or at one in whole numbers or fractions.  

A Directory unto the double Rule of Three.  
IN this double Rule, there are five numbers or terms propounded, and a sixth proportional number is required, which must be always of that denomination, on which the Question depends, which number in this Rule stands in the first Proposition simply of itself, without relation to any of the other 4 numbers: this Rule is performed with two operations at the most; the two extremes in either Rule must be of like denomination, the middle number in either Rule must be of one and the same denomination, and likewise with that, required, which the Question demands, as was said before; so the fourth proportional number discovered by the first Rule of Three, must be the mean proportional in the second Rule, and the fourth proportional then found is the true number required, as by Examples shall be made evident.   

Question 1. There was a Knight had 5 Sons, and 3 of these in 4 months had spent him equally 19 L. that is in all 57 L. and it is required what his 5 Sons would have spent at that rate in 12 months?  
In this double Rule of Three there are 5 terms or numbers given, whereof one is simply of itself, viz: the money spent, which denomination is the proportional number required, and in the Rule of Three direct, as by the Example; for if the 3 Sons did spend 57 L. then his 5 Sons would have spent a greater sum, as 95 L. and if the space of 4 months (as by the second Rule) required 95 L. then 12 months (in a direct proportion) will require 285 L. as in the Example, the Question solved; being the true expenses of the 5 Sons in 12 months, at the same rate, as the 3 Sons in 4 month's time did spend 57 L. and as in this, so in all other Questions of this kind, according to my former directions, it is not material whether of these two Rules are first, so long as the single term be made the mean proportional number in the first Rule: as 4 M. to 57 L. so 12 M. to 171 L. then in the second Rule it will be; as 3 is to 171 L. so 5 unto 285 L. as before.   

Question 2. If 12 Pecks of Provender did serve 4 Horses for 3 days, how long will 24 Pecks serve 3 Horses at the same daily allowance.  
In such Propositions two Rules of Three are contained, the one direct, the other reversed; as in this Example, the first Rule is in a direct proportion, viz: as 12 pecks to 3 days; so will 24 pecks, be in proportion unto 6 days, a mean proportional number for the second Rule, in which 4 Horses must be the first term whereon the Question depends, being of the same denomination with the third term, upon which the demand is made, and performed by the Rule of Three reversed, it is evident, because the fewer number of horses must require a longer time; so it will be as 4 horses to 6 days, so will 3 horses be in a reversed proportion unto 8 days; and so long time will 24 pecks of provender serve 3 horses; that is a peck allowance every day for a horse, the thing required.  
If this double Rule of Three (performed at two operations) shall fall in fractions or mixed numbers, either in the Rule direct or reverse, by these last Examples it may be solved, with help of the 8 Parag: to which I refer you; only observing in all double Rules of Three compounded (as in this last Example) which of the two Rules is reverse, and which direct; this done, (as reason will dictate) it is not material whether of them be solved first, as 4 H. to 3 D. so 3 H. in a reversed proportion unto 4 D. Secondly, as 12 P. to 4 D. so 24 P. in a direct proportion to 8 D. as it was before: so in either Rule, the terms on which the demand is made, must always be the third number, as these were, viz: 3 Horses & 24 Pecks: so I will say no more of this double Rule performed at two operations, which by one Rule will be solved with more facility in less time by these following Examples.   

Question 3. If a man's 3 Sons spent 57 L. in 4 months, what would his 5 sons expenses have been at that rate in the space of 12 months?  
The first question I have  S. L. S. 1 Rule. 3 57 5  M.  M. 2 Rule. 4  12 Prod. 12 57 60 Or as 1 57 5 Facit  285 L.  hear stated again, to satisfy the Reader, in the operation of this double Rule at one work, by a single Rule of Three at most, if not reduced unto a Rule of Practice, as in this Example; where in the first Rule stands 3 S. 57 L. & 5 S. In the second Rule is placed for the two extremes the time, viz: 4 M. & 12 M. the demand is of the 5 S. and the 12 M. The other 3 numbers were proposed, viz: 3 S. 57 L. 4 M. if 57 L. were multiplied by 5 and divided by 3, the Quotient would be 95 for the fourth number, and a mean proportional in the second Rule, and that multiplied by 12 and divided by 4 will solve the Question: then for brevity, since the number or sum here required is contained in 57 L. twice multiplied, and as often divided, I say if the two Multiplyers, multiplied into one Multiplier, and the two Dividers into one Divider, the proportions must be the same; therefore by the 14 Axiom this Example with two Rules will be reduced to one, the product of the Multiplyers is 60, and the Divisors 12, so the proportion is as 12 to 57 L. so 60 to 285 L. or reduced by the Rule of Practice, as 1 to 57 L. so 12 will be in proportion unto 285 L. as before, the Question solved.   

Question 4. If 4 Horses did eat 12 Pecks of corn in 3 days, in what time will 3 horses eat 24 pecks of the same corn and daily allowance.  
This differs nothing from the second Question, but in the operation it being reduced unto a single Rule of 3, & in this manner: Here you may observe two Rules, whereof one is direct, the other reversed; all which rightly comprehended, the reason is the same as in the 3 Quest: of of this Parag: the form only differing: for here note, the first Rule is direct, the second reversed, and not material whether of the extremes in either Rule be multiplied crosse-wise, viz: 12 by 3, and 24 by 4, their products are 36 & 96, and 3 the medium: so the proportion is now direct, as 36 unto 3, so will 96 be to 8 a fourth proportional, and the number of days that 6 bushels or 24 pecks will serve 3 horses, with the same allowance that 3 bushels did serve 4 horses. Here note that 12, and 3 in the second Rule were both Divisors; and likewise 24 & 4 were in both Rules Multiplyers. And farther observe that if the first Rule had been reversed, and the second direct, their products set underneath their Multiplicands, it is always reduced to a Rule of 3, and of the same species with the first Rule, and the products placed under the Multiplyers, is ever the contrary, and of the same species with the second Rule.    

Paragraph XI. Declareth the Rules of Society or Companies both single and compound, in relation to time and quantity, both in whole numbers, or fractions, either in gain or loss.  

Definitions.  
THis Rule is either single, or compounded, and those consisting of multiplicity of parts, or manifold proportions either in gain or loss.  
Those are said to be single, when as the terms have a simple relation, either to Number, Weight, or Measure only.  
They are compounded, when each term or part consists of several species, or hath relation both unto time and quantity.  
This Rule is of most use in matter of accounts betwixt sundry men, employing several stocks within unequal times either of gain or loss in proportioning each man's charge, profit or misfortune, according to every particular adventure; this Rule consists of 4 terms at least, and is performed by the single Rule of Proportion recited so often as there be diversity of stocks, or adventures, according to the 15 Axiom, from whence this proportion ariseth, which by Examples shall be made evident, in gain or loss, or in equation of payments.   

A Rule of Society.  
As the total of any stock or adventure  
Shall be to the general profit or disprofit  
So shall each share or particular adventure  
Be in proportion to each respective gain or loss.   

Question 1. Two men, viz: A and B, joined their stocks together: A had 18 L. and B had 12 L. with which they bought wares, and quickly sold them again for 2 L. 10 s. profit: how much did either of them gain?  
According to Adventure Gain. 18 A 30 As 30 L. unto 50 s. so L. ● to s.   12 B 20 A gained 30 s. B 20 s. the total 50 the Rule before specified 30 L. the adventure stands in the first place of the Rule of Proportion: the middle number or mean is the whole gains, and in the third place stands the adventures, as the share of A 18 L. and B 12 L. these parts must ever be multiplied by the gain or loss, and divided by the whole adventure: as 18 L. increased by 50 s. produceth 900 which divided by 30, gives 30 s. for the gains of A. So 12 L. multiplied by 50 s. and divided by 30, showeth 20 s. for the gains which B hath made clear profit: to prove this Rule work them all backwards, and you will find the first adventure by every one of them; or for more brevity observe the particular shares, with their gains, the sum of those particulars, must be respectively equal to the total adventure with the gain or loss.   

Question 2. Five Merchants put their stocks together, as A, B, C, D, & E: whereof A did adventure 175 L. B. 140 L. C 70 L. D 35 L. E 30 L. this stock returned them 744 L. all charges defrayed: How much was each man's gain or loss?  
In the first place   stocks L. s d Stock Advent: 175 A 289 6 8 450 L. 744 L 140 B 231 9 4   70 C 115 14 8   35 D 57 17 4   30 E 49 12 0 The total sums 450  744 0 0 collect all the particular stocks, or adventures together, which is here 450 L. and being it is less than the stock returned, it is evident they were all gainers by the adventure, and in proportion according to each stock adventured, as in the last example; but here note, that all the three terms, are of one denomination, viz: pounds sterling, yet differing in their qualities, the first number being the total stock, and the third number, each particular stock, the mean proportion, viz: 744 is the total return of the adventure, to which a fourth number is required, to every particular stock; which here will be found with the profit together, because the middle number is of the same quality, viz: the whole stock as 450 L. and the profit of the adventure, as 294 L. so in these two examples there is no material difference, for in either way the increase of their stocks will be discovered by Addition, or Subtraction: for having multiplied each particular stock by 744, and the products divided by 450, the quotient will be each man's gain, and his stock adventured too; as the share of A comes to 289 L 6 s 8 d. so the clear profit he made of 175 was 114 1/● L. and so for all the particular adventures, as against each letter in the example will appear.   

Question 3. Three men unhappily employed their stocks together, as A, B, & C: whereof A deposited 20 L. B 30 L. C 60 L. all which they laid out; and afterwards sold those goods for 84 L. 6 s. 8 d. how much was each man's gain or loss?  
As in the former  Returned L.  No. Adven: 253 No 20 A 46 110 L. 84 L. 6 s. 8 d. 30 B 69  84 ⅓ L. 60 C 138 The total sums 110  253 Example, set all shares down in the third place, the sum of them in the first place, the money returned in this adventure must be the middle number as before, which is here 84 ⅓ L. which is less than the adventure or total stock, from whence it is evident that the money was employed ill, and returned with loss, and consequently the fourth terms found must be less, than was each particular; share which to find, differs nothing from the former, only the middle term being a compound number, must be reduced into one denomination, as into Groats, which will be 5060 Gr. or made an improper fraction, as 2●●/● L. and work it according to the Rules of Broken numbers, or in any other parts, that are readiest for use, as in this I will omit the fractions denominator, and so the sum, or stock returned, is 253 Nobles, which multiplied by each particular share, and divided by the total adventure, will produce these parts as in the Example, viz: A must have 46 N. B 69 N. C 138 N. so A had 15 L. 6 s. 8 d. and lost 4 ⅔ lb. B lost 7 L. and C 14 L. the Adventure being double to B 30 lb.   

Question 4. Three men held a pasture in common (as A, B, & C) for which they paid 44 L. per annum: A kept 10 steers upon the ground 20 weeks, B fed 15 steers 16 weeks, and C kept 20 there 8 weeks: what rent must each man pay?  
In all questions of this     L s. d  L. 5 A 14 13 4 15 44 6 B 17 12 0   4 C 11 14 8 The tot. 15  44 0 0 kind, observe double terms, as in respect of each particular stock & time, upon which all questions of this kind depend; and to effect this, or the like: multiply each man's terms together, as here the stock and the time, viz: for A 10 steers by 20 weeks, the product is 200; for B 240; and the product for the stock of C and his time, as 20 by 8, will be 160: the sum of these 600 for the first term, the rent 44 L. is the middle term, and the three last numbers are A 200, B 240, C 160, these multiplyers, with the divider 600 you may reduce by 40 unto 15 the first term: the other three will be 5, 6, 4; the mean proportional might have been reduced with the Divisor, but not so conveniently, the three multiplyers being made simple figures, the readier much for use, yet as true without reduction, as by the 13 Axiom, Parag: 7. the parts or shares here to be paid for rent are these, viz: A 14 L. 13 s. 4 d. for B 17 L. 12 s. and for C 11 L. 14 s. 8 d. all which sums added together do make 44 L. the whole rent, which shows the work is true: as by the Example does appear.   

Question 5. Four men joined their stocks together, as A, B, C, & D, whereof A ventured 40 L. B 160 L. C 100 L. D 280 L. by misfortune they were all losers: upon which they fell at difference, at several times, and broke off from this society, when D had lost 20 L. C 10 L. & B 30 L. A continued the trade 12 months, and lost 8 L. how long were all their stocks continued at that rate or proportion?  
In all questions of   L.  M D L  30 B 11 7 8 480 10 C 6 0   20 D 4 8 this nature, the double proportion must be made one, as here in this it is A, whose stock was 40 L. the time 12 months, the product 480; his loss was 8 L. for the first term, so the proportion will be, as 8 L. loss is unto the product of the time and principle, that is here 480; so shall each particular loss be proportionable unto the product of his time and principle, and being it contains them both, and the adventure known, divide that fourth proportional found by his principle or stock, and the quotient will discover the true time; as in this last Example, 480 multiplied by 30 L. (the loss which B sustained) the product will be 14400, which divided by 8 L. the first term, the quotient will be 1800 the true product of time and principle; therefore 1800 divided by 160 L. the adventure of B, the second quotient will be 11 ¼ months, that is, 7 days in all 45 weeks; and so long time did B continue his stock in the same company; in this manner, C that lost 10 L. will be discovered 6 months, and D that lost 20 L. kept in this society but 4 mon: and 8 days, as in the precedent Example is evident, where 68 L. was lost in all.  
Any question of this kind, must be tried by a contrary way; as with each man's adventure, his time, and the loss of one man's stock, to find the others: and so likewise in any other question wherein gain is made: and no more questions will I show here in this Paragraph, lest that I should lose more time, than the Reader shall gain benefit.    

Paragraph XII. Questions resolved by the rules of Alligation, or mixture of divers simples; by which are found a common price, and quantity for to be mixed, or taken, of any particular sort, and according unto any rate, price, or proportion required.  

A Definition.  
ALligation consists of two Species, the first admits but of one common rate, price, or proportion either in respect to the quantity, or quality of the compositions; the second Species is manifold.   

The rule.  
Alligation simply in itself is to find a common Medium in the mixture of divers things together, which is performed, by taking the sum of all the given quantities; and for the second number, the total value of all the simples to be mixed, and then the proportion will be, by the common rule of three.  
As the sum of the quantities to be mixed,  
Is unto the price, or value of the simples;  
So shall any quantity, or part of the mixture,  
Be in proportion to the price of that part.   

Question 1. A man mixed for provender 3 quarters of beans at 3 s. the bushel: 3 ½ quarters of pease at 2 s. 6 d. the bushel; with 5 ½ quarters of oats, at 1 s. 6 d. the bushel: what will a bushel of this mistling be worth, when all these are mixed together.  
Reduce all   Bushels, L. s. d The value of Beans 24 3 12 0 Pease 28 3 10 0  Oates 44 3 6 0  The tot. 96 & 10 8 0 A the measures into one denomination that is least, & against every quantity set down the price, or value of them, as in the margin; then take the total sum of both, as in this Example, the quantity mixed is 96 bushels, and the total of their several prices, is 10 L. 8 s. or 208 shillings: these place, as at A, according to the Rule, as the quantity ninety six bushels shall be to the price, 208 shillings; so will 1 bushel be unto the price thereof, viz: 2 ⅙ s. that is 2 s. 2 d. the bushel, the common price required: to try any such question, whether it be true or no, the quantity multiplied by the common price (if true) will be equal unto the sum of the particular prices; as here 96 times 2 s. 2 d. will be 10 L. 8 s. equal unto all the particular prices.   

Question 2. A Vintner put into a Wine-casck 15 gallons of Canary-sack which cost 4 L. 10 s. 30 gallons of Mallage 6 L. 5 s. & 35 gallons of Sherry sack 7 L. with 40 gallons of Greek wine 8 L. a pipe thus filled, the price of one gallon is demanded?  
The quantities,  Gall. L. S.  Canary s. 15 4 10 The value of Mallago s. 30 6 5 Sherry s. 35 7 0  Greek w. 40 8 0  The tot. 120 25 15 A and price of the several sorts of wine, being set down as in the Table here, take the totals of them as 120 gallons which cost 25 L. 15 s. or 515 shill: these place in the common Rule of 3 as at A. viz: if 120 gall: cost 515 shill: what shall 1 gallon cost? the fourth proportional number, or common price will be discovered 4 s. 3 ½ d. and will be proved, as was the last example, for 4 s. 3 ½ d. multiplied by 120, (the number of Gallons) and the product will be reduced unto 515 s. or 25 L. 15 s. as before: to find what the particular sorts cost a Gallon, or what the common price of this mixture is a quart, or in any other measure: were a question only fit for young beginners: so I will say no more in Questions that only concerns a common Medium, but proceed.   

The second Rule of Alligation.  
As the total difference of the prices given, and the price propounded.  
Is in proportion unto the total sum or quantity to be mixed:  
So shall each respective difference of the price given and the price set  
Be in proportion unto the quantity of each particular sort to be mixed.   

Question 3. A Druggist had two sorts of Tobacco; the one was Spanish, at 10 s. the lb; the other Virginia, at 3 s. the pound; of these two sorts, he was to mix 112 lb weight, and so, as that it might be afforded for 8 s. the lb: how much must be taken of either sort?  
This second A C B D.— Lb. D Lb 7-112 5 80   2 32 rule differs much from the former, which, requires only a common price, from the total of several quantities mixed together; whereas this is confined unto a price and quantity in general, composed of particulars, from whence the mixture is to be made, and the parts taken, proportionally according to each price, quantity or quality; as in this Example, in the Table at A, where 8 s. is the price determined on; the best Tobacco was 10 s. the pound, the worst 3 s. the pound; which two prices I set one above another, and couple them together with an arch of a circle; the price set must be always lesser than the greatest price, and greater than the least of the particulars, as here 8 is lesser than 10, and greater than 3: this done the difference between the price set, and each particular must be found, and counter-changed with the number, to which it is coupled, as in the first Table, the difference betwixt 8 & 10 is 2, which is placed against 3. the difference between 8 & 3 is 5, which stands against 10. the sum of the differences is 7, which according to the Rule above, and the Table, at B, stands in the first place: 112 pound, the quantity to be mixed the second number; and each particular correspondent difference the third number: so it is now in the Rule of proportion, as 7 the total difference, to 112 pound the total quantity; so will 5 (the particular respective difference for the best price) be to 80 pound, the quantity to be taken of that sort. Again, as 7 to 112 pound, so 2 unto 32 pound the quantity of the worst sort; and thus repeating the Rule of 3, so often as there be particular differences, you will produce particular quantities to them, whose total must be equal to the sum, or quantity propounded to be mixed, if otherwise, the operation is false.  
Yet least in the mixture of these simples, you should remain ignorant of the reasons; this question, and all of this kind hath relation unto the 15 and 16 Axiom, parag. ●7. wherein the total difference of prices, hath proportion to the total sum or quantity to be mixed, as the particular difference or parts, have to their respective quantities, but here another Querie will be made, wherefore these differences between the price set, and each particular price is counterchanged with a greater and a lesser; by the difference of prices, will be discovered the several quantities in proportion to those differences; if the difference were equal, the quantities to be taken of all those sorts would prove alike: if the rate or proportion propounded, inclines unto the greater price given, it will have the lesser difference, by how much the nearer it comes to the greatest price, and yet the greater quantity of that sort must be in the mixture; and so consequently the least price (in this case) will have the greatest difference, in which the least quantity is required; and the contrary will be found by the rule of 3 direct; as by the last example, and the first table at A, where the price propounded is 8 s. the two prices given are 10 and 3, and the differences are 2 and 5: and being 8 is nearer 10 then 3, the greatest quantity of the best sort will be required; and the less of the worse sort; therefore as 7 to 112, so 5 unto 80, and as 7 to 112, so 2 unto 32.  
In the other Table at C admit a composition of the same commodity were to be made at 6 s, with any quantity assigned, 6 the price propounded is nearer 3 than 10, therefore 3 must have the greater difference, whereby to produce the greater quantity, and the best sort at 10 s. the least difference as 3. the worst sort 4. and so the proportion would have been, as 7 to 112, so 3 unto 48 lb. or as 7 to 112 lb so 4 unto 64 lb both of them making the just quantity of 112 lb and for a farther trial, the price stated, if multiplied into the quantity for to be mixed, the product will be equal, unto the products of the several prices, and their respective quantities: as 6 multiplied by 112 lb. produceth 672; so 48 by 10 is 480. and 64 by 3 maketh 192, which added unto 480 the sum is 672, or 33 L. 12 s. as before.   

Question 4. A Grocer had 3 sorts of Sugar, one was worth 6 d. the pound, another sort 10 d. a pound, and the best 15 d. a pound, out of these sorts, he made two parcels: either of them 56 pound weight, the one thus mixed to be afforded at 12 d. the pound: the other sort at 10 d. the pound. How many pound must be taken to be mixed of either sort?  
First to prepare the work, set down the price propounded, as 12 d. and likewise the prices given, viz: 6 d. 10 d. & 15 d. against these I prefix letters, as A, B, & C, and according to the last Question couple a greater with a less, whereof A is only greater, which must then be coupled with B 10, and C 6, and the differences transcribed and counterchanged; so often as there be numbers coupled to any one; as here the difference of prices betwixt C and 12, and also B and 12, be 6 & 2; which differences must be counterchanged with A. and the difference betwixt A & 12, which is 3; must be twice transcribed, as placed against B, & C. the total of differences is 14. thus the first question is prepared; and as for the second proposition, there are two numbers the one greater than the price propounded, the other less, viz: A 15. and C 6. with either of which you may couple the other two, and is indifferent in respect of the proposition, although the quantities will not be the same, in this I have again coupled them both with A; to avoid fractions, wherein I find the difference betwixt 10, the price propounded, and B & C to be 4 & 0. which must be placed against A, and the difference betwixt A & 10 the price propounded, for to be 5, which transcribed against B & C. the sum of these differences 14, as before.  
The differences  D lb.  D  lb.   C 3  12 1 As 14 to 56 so B 3 unto 12   A 8  32  The totals D 14  56  D lb 4 A  16 2 As 14 to 56 so 5 B unto 20   5 C  20  The totals 14   56 being found, and the rule prepared, as before; the proportion is as 14 (the total of the differences) to 56 pound (the sum or quantity for to be mixed) so will 3 the respective difference for C, be unto 12 pound, for its quantity, at 6 d. the pound; and likewise 12 pound of that sort for B, at 10 d. the pound. Then again, as 14 to 56 pound, so 8 to 32 pound: and so much must be had of that sort at A, of 15 d. the pound: all which 3 quantities makes 56 pound, as in the Table, which argues it is true, or may be thus proved: the common price propounded was 12 d. a pound, so 56 pound the quantity mixed comes unto 2 L. 16 s. now for the particulars, A must have 32 pound, at 15 d. the pound, that is, 2 L. 0 s. then B 12 pound, at 10 d. the pound, comes to 10 s. Thirdly, C must have also 12 pound, which at 6 d. the pound, comes to 6 s. the total sum 2 L. 16 s. as before.  
In the second composition or mixture, it will be as 14 to 56 pound, so 4 unto 16 pound, for the part to be taken of A. Again, as 14 to 56 pound, so 5 to 20 pound: for B, and so much for the composition must be taken of that sort at 10 d. the pound. then as 14 to 56 pound, so 5 unto 20 pound, the quantity of C. for being the Multiplicators are alike, C must have also 20 pound, at 6 d. the pound, the sum of these is 56 pound, in money 2 L. 6 s. 8 d. and so proves the particulars; for 16 pound of A, at 15 d. the pound, comes unto 1 L. 0 s. then 20 pound of B, at 10 d. the pound, is 16 s. 8 d. Lastly, 20 pound of C at 6 d. the pound comes unto 10 s. the total of the compound 56 pound, in money (according to their several rates and quantities) 2 L. 6 s. 8 d. and so much the 56 pound did come unto, at the set price, viz: 10 d. the pound: had A & B been coupled with C, the proportions had been changed in respect of their quantities, as by the following Example will be manifested.   

Question 5. A Grocer hath 4 sorts of currants; one at 4 d. another 6 d. a third sort 9 d. the best 11 d. the pound: the worst sort slighted as too mean, the best thought too dear, so two sorts would not sell: upon which the Grocer compounded two parcels out of the 4 sorts, each containing 120 pound: and so mixed, as that he might afford them at 8 d. the pound. How many pound had the man of every sort, in either parcel?  
Here I make two Tables again: in either of them set down the price propounded in this 8 d. next insert the several prices, viz: 4 d. 6 d. 9 d. & 11 d and so likewise in the second Tables: which prices you may note with letters against them, as A, B, C, & D. next couple a greater than the price set with a less, as in the first Table, 4 with 11, and 6 with 9; In the second Table 4 with 9, and 6 with 11: this done, take the difference between the price propounded, and the several prices, and transcribe. those differences, with counterchanging their places as in the former examples, and here are found to be in the first Table, A 3, B 1, C 2, D 4. In the second Table A 1, B 3, C 4, D 2: the total in either is 10. the quantity to be mixed 120 lb then say, As 10 the total difference is unto 120 lb. (the quantity to be compounded) so shall 3 the respective difference be to 36 lb, the quantity of the worst sort for the mixture, that is of A, at 4 d. the lb. Again, as 10 to 1●0 lb. so 1 unto 12 lb. the quantity of that sort for B at 6 d. the lb. As 10 unto 120 lb. so 2 unto 24 lb. for the quantity of that sort at C. As 10 unto 120 lb. so will 4 be to 48 lb. for the quantity of that sort at D, which is at 11 d. the lb. all which quantities inscribe against each peculiar price, or respective letter, as here in the Table, whose sum must be equal unto the quantity for to be compounded; and the whole quantity mixed, at the price set, to be equal unto the sum of each particular price and quantity, as the total of the particulars in this example is 120 lb. which at 8 d. the lb comes unto 4 L. now for the particulars, A 36 groats or 12 s. B 6 s. C 18 s. and D 2 L. 4 s. the total 4 L. as before.  
Now for the second composition, the proportion will be as 10 unto 120 lb. so 1 for A 12 lb. and so proceed as in the first mixture repeating the Rule of 3 so often as there be differences of several prices, & so these quantities will be found B 36 lb. C 48 lb. D 24 lb. the total 120 lb. and these particulars, at their own rates will be 4 L. as before.   

Question 6. A composition of several simples to be made in weight 1 ½ C gross; and to be afforded at the price of 5 s. the lb the particulars at these rates: viz: A 1 s. lb B 3 s. lb C 5 s. lb D 8 s. lb E 11 s. lb & F 14 s. lb How many pound must be taken of each sort, not exceeding either the price, or quantity propounded?  
This question differs not from the former, having here connext a greater with a less, excepting one, which is equal unto the price propounded, and might have been connexed unto a greater, but for one cause which shall be spoken of hereafter: so now to proceed in the operation of this Example, which is divided into several columns: In the first stands only the price propounded, viz: 5 s. In the second is placed each particular price: as 1 s. 3 s. 5 s. 8 s. 11 s. 14 s. with the least, as 1 s. I here connex 14 s. 11 s. & 5 s. & 3 s. is coupled with 8 s. this done, find their differences betwixt these particulars and the price set, which transcribe according unto the former Examples, so against 1 s. the difference will stand 9, 6, 0, against 3 will be 3. and against 5 stands 4 etc. find the total of these differences which is here 32. Then say by the Rule of proportion, as 32 is to 168 pound (the quantity to be mixed) so will 15 be to 78 ¾ pound. And thus repeating the Rule of Three so often as there be differences, the respective quantities to every one will be produced, as in the last column, whose total is 168 pound, the first quantity propounded.  
In most questions of this kind, having found any one number, the rest may be discovered without repetition of the Rule of Proportion: for by the Rule of Practice, Parag: 9 having found the difference and quantity for A or F, as admit F 21 pound, then must E & C have likewise the same quantity, their differences being equal, and D. 2 must be half so much, viz: 10 ½ lb, that is 10 lb 8 ℥, once that and ½ shall be the quantity for B 15 lb 12 ℥, and A being 5 times the difference of B, the quantity of that sort for A must be 78 lb 12 ℥, the totals as before, equal unto the quantity of the composition propounded: and 168 Crowns is equal to 42 L. equivalent to the particulars: for A 78 lb 12 ℥, at 1 s. the lb, comes unto 3 L 18 s 9 d. B 2 L. 7 s. 3 d. next C 5 L. 5 s. then D 4 L. 4 s. Fiftly, E 11 L. 11 s. Lastly, F comes to in money 14 L. 14 s. the total sum in money 42 L. as by the common price, and the quantity given.   

The same Question varied thus, with the former prices, and quantity propounded.  
In this Example there is the same common price with the particulars, and quantity to be mixed, which are thus alligated, viz: 1 s. & 14 s. next 11 s. & 5 s. also 8 s. & 3 s. these thus connext, and their differences transcribed (or counterchanged accordingly as this rule requires) the sum of those differences will be 24: so the proportion is as 24 to 168 lb, so 9 unto 63 lb, the quantity of that sort to be taken: and so proceed according to the last Example you will find the rest, as 21 lb 42 lb. 14 lb. 0 lb. & 28 lb. whose total is 168 lb. as before: against these stands the value of them, according to each particular price, and quantity of that simple, whose total is 42 L. as in the fi●st example; yet the state of the question is not performed according to demand, here being no part in the composition of that simple (represented by E) of 11 s the lb, as was in the last Example, which was the cause, it was connext as before; and a caution to you hereafter, in any question of this kind.  
These Rules prescribed I hope may satisfy the Ingenious: yet several Arts have various propositions, one of most note is in the commixing of metals, in which there is a loss in melting, except in Gold or Silver refined from the faces, besides an alloy is usually allowed, therefore I will render you one Example of that kind.   

Question 7. A Goldsmith hath several masses, or ingots of Gold, one purer than another; of 3 sorts he was to make a mixture, whereof one was 16 Carects: the second 18 Carects: the third was 20 Carects fine: this the Goldsmith was to mix with such an aloy, as that the whole mixture of 135 Ounces should be 14 Carects fine: How much must be taken of each sort?  
This last question differs from all the former in two things; the one is, the compositions or mixtures of them are made out of the simples, or particulars given, and so the common rate, or price, is always less than the greatest, and greater than the least propounded: Secondly, those may be varied in their allegations, whereas all questions of this nature admit of no connexion, but with the Allay; in the operation, and in other things, it differs not from the former, the total difference being the first number, the quantity given the second; and each particular correspondent difference the third number; and the sum of the fourth numbers found, must be equal unto the quantity propounded, if the operation be true, which is the trial of it.  
If Gold of several fineness were to be mixed together, without any Allay, but the quantity to be mixed known, and each particular difference, and the rate, or fineness of the whole composition to be better than the worst sort, yet not so fine as the best, the quantity of each particular will then be found by the former prescribed Rules: so here I will conclude this Paragraph, but not the subject. Having writ nothing as yet concerning the Apothecary, which shall be the next: but not to involve his Simples in conjuring words, to circumscribe Grammarians and charm the spirits of the Learned with common compositions, as Children disguised with visards do amaze the Wise and Valiant: but here I will only show how their Simples may be compounded according to their qualities, whether Hot, Cold, Dry, or Moist, in sympathy with the Elements, distinguished in 4 Degrees, and how to make a Medium with either of the two extremes.    

Paragraph XIII. Sheweth the solving of divers necessary questions by the Rules of Alligation, in the composition of Physical simples, according to their qualities, as Hot, Cold, Dry, Moist, with the quantities of those Simples augmented, or diminished according to any degree prescribed.  
ALL Simples are considered in their own natural qualities, as whether Hot, Cold, Dry, or Moist; betwixt these are contained two Mediums, proceeding from the two extremes; both these are called Temperate, viz: as the mean between the Hot and Cold, the others contained betwixt Drought and Moisture: from whence every Medicine is said to be either Temperate, or else Hot, Cold, Dry, or Moist, in some one of these 4 degrees, viz: in the first, second, third, or fourth.  

A preparative in the Alligation of Physical Simples, according to their several qualities or temperatures.  
In the first column The Degrees & Qualities. I— 9 4  H— 8 3  G— 7 2 Hot & Dry. F— 6 1  E— 5 0 Temperate. D— 4 1  C— 3 2 Cold & Moist. B— 2 3  A— 1 4  of this preparative, stands 9 letters of the Alphabet, beginning with A, and proceeding in order: against A is placed an unite, and so proceeding unto 9 in Arithmetical progression, where 5 must be the Medium, against which stands a cipher to denote a temperature, from thence ascending the degrees of Hot and Dry, viz: 1, 2, 3, 4, and from that temperate Mean, descending again the same degrees, into the other extreme of Cold & Moist; to both these, the first column here represents the the Index, and might have been in any other numbers (proceeding in this order) but the least are best; this done, the form and manner of operation differs not from the last prescribed Rules of Alligation, for in compounding of these Physical Simples, there is required to be connext a greater with a less, in respect of their temperatures, or natural qualities, for which the figures in the first column do stand, each as an Index to the quality of any simple, which numbers must be alligated, as were the prices, or quantities in the former questions, as shall be explained in sundry examples following.   

Question 1. Out of two Simples, to make a composition; which Medicine shall be of any mean temperature between the simples propounded, and of any quantity assigned.  
First you are to  Index. Differ: ℥ ʒ  5 1 2/7  6 6 6/7 1 As 7 to 12 ℥, so 3 unto 5 1 1/7 2 As 7 to 12 ℥, so 4 unto 6 6 6/7 2 The total of this is 12 0 understand that the weights used by the Apothecaries are these, & commonly thus noted: a pound lb, containing 12 ℥ Ounces: an ounce 8 ʒ Drams: a dram 3 ℈ Scruples: a scruple 20 Grains: this done, admit a composition to be made out of two Simples, one hot & dry in the fourth degree, the other cold and moist in the third degree, the composition to be temperate, and the weight of this mixture 1 lb or 12 ℥. Look in the Table for these degrees, where against 4 you will find the Index to be 9 I, and for that which is cold and moist in the third degree you will find at B in the next column the Index 2, and 5 the medium, or mean temperature for the composition; which first set down, and then either Index, viz: 9 & 2, then find the difference betwixt them, & the medium 5 which is here 3 & 4: these must be transcribed with each Index connext as in the former Examples: this done, the proportion will be as the sum of the difference, viz: 7, shall be in proportion unto the quantity to be mixed, that is 12 ℥, so shall each particular correspondent difference be unto their respective quantities to be taken in the Composition; which you may write (as they are found) against each Index, or proper letter; and these proportions must be repeated so often as there be particular differences: as in the bottom of the Table, noted the first and second Rule, whose total in the third row is 12 ℥, the quantity given, which shows the work is true. The Simple which was hot and dry, is least in the Composition, being in the greatest excess, and is here 5 ℥ 1 1/7 ʒ. the other B, cold and moist, but in the third degree 6 ℥ 6 6/7 ʒ, the sum 1 lb. This Rule (and likewise any in Alligation) may be also tried by the 16 Axiom, Parag: 7. by the former Rules in mixtures of Simples.   

Question 2. A Composition is required of 3 Simples, whose qualities are known, and a temperature between one of those three is demanded.  
The temperature of this Composition is required in the first degree of Cold & Moist, whose Index is 4: the qualities of the Simples are these, the first, 4 degrees Cold and Moist, whose Index is an unite noted with the letter A. The second Simple is Hot and Dry in the third degree, whose Index is 8 H. The third Simple is Hot and Dry in the fourth degree, noted with I, and the Index 9 there being two qualities greater than the temperature assigned, connex them both with 1, and find the differences, as in former Examples, which proves here I 3, H 3, A 5 & 4, that is 9 the quantities of these Simples to be mixed, is 10 ℥ propounded. so the proportion will be (in the second row of this Table) as the sum of the differences 15, is to the quantity of the Composition, viz: 10 ℥. so 3 the difference is unto 2 ℥ for I. then H must have the same quantity, and A 6 ℥, as in the third row, the total of these is 10 ℥, equal to the quantity propounded, and may be also thus tried, by the 16 Axiom, Parag: 7. multiply each difference by its respective Index as it stands, as 9 by 3, and 8 by 3. And thirdly, 1 times 9, the sum of these is 60. so will the product be made of the temperature assigned, as in this 4, and the sum of all the differences 15, which produceth also 60 as before, either way is trial sufficient: yet convenient it is to know them both, our ways being doubtful, and Man prone to err.   

Question 3. A Composition of 4 Simples is required whose qualities are known, and the quantity of any one is to be mixed with such quantities of the rest, that the quantity of the Composition may be of any temperature required, greater than the least, and lesser than the greater given.  
Admit the temperature of this Confection were to be Hot and Dry in the first degree, the Index to it will be found in the former Table 6. the first Simple Hot and Dry in the fourth degree, whose Index is 9; of this there must be 6 ʒ added in the composition to the other three, whose qualities are these, G Hot and Dry in the second degree, whose Index is 7. D Cold and Moist in the first degree, the Index 4. Lastly, B Cold & Moist in the third degree whose Index is 2. the 4 Indices must be coupled, a greater with a less than 6, the Index unto the temperature required; in this, 9 is connext with 4, and 7 with 2. the differences are 2, 4, 3, 1, noted with these letters I, G, D, B, the quantity of I is here given 6 ʒ, whose Index is 9, and the difference belonging unto that (as counterchanged with another) is 2, which must be the Divisor, in questions of this nature, the quantity given the second number (in this 6 ʒ) and each respective difference the third number in the Rule of 3, excepting that which appertains to the quantity given, as in the three Rules of the Table, viz: As 2 unto 6 ʒ, so 4 to 12 ʒ; for the quantity of the Simple belonging to G; which writ in its proper place, and so proceed to the rest, where you will find under the title of ʒ the sum of 30. or 3 ℥ 6 ʒ: this question having no quantity assigned in the whole Composition, is proved as was the last: that is, each difference multiplied by its respective Index, whose total will be equal (if the operation be true) unto the sum of the differences, multiplied by the Index of the degree given: as here 6 by 10 produceth 60; and so is the sum of their several products, as in the trial of the Table appears, whose quantity proves 30 ʒ, or 3 ¾ ℥ in the temperature of Hot and Dry, and in the first degree as was desired.   

Question 4. A Composition being made of divers Simples, whose qualities and quantities are certainly known; and it is required, in what degree of temperature this Confection is in?  
In all questions  Index  ℥  Prod. ℥ 5  1  5 As 1 unto 6 so will 2 be to 12  2  3  6  3  4  12 The Index 3 ½ Totall 10  35 of this kind, there is no more to do than first to set down each Index, according to the temperature of the Medicine assigned: against each quality place the quantity of it, which multiplied by its respective Index, and the sum of those products divided by the total of the quantities, the quotient will be the Index unto the temperature of the whole Composition. As for Example, there is a Confection made of 4 Simples, whose qualities and quantities are as followeth: 1 ℥ temperate, whose Index is 5. 2 ℥ hot and dry in the first degree, whose Index is 6. Then 3 ℥ cold and moist in the third degree, whose Index is 2. Lastly, 4 ℥ in the second degree cold and moist, whose Index is 3: the products of these are, 5, 12, 6, 12, the sum of these is 35; the total of the quantities is 10 ℥, with which divide 35, the quotient is 3 ½ for the Index required, which in the former Table (of degrees and qualities) falls between C & D subtract the Index 3 ½ from 5 the medium, the remainder will be 1 ½ Cold & Moist, the question solved: if the Index had been greater than 5. subtract then 5 from it, the remainder will be your desire, in the degrees of Hot and Dry. This question depends upon the common Rule of Proportion, viz: As an unite is to the Index, so will the quantity be unto a fourth proportional number, as in this last Example does appear.   

Question 5. A Confection to be made of several Simples, whose particular quantities and qualities are known, as in respect of any degree in Heat, Drought, Cold, or Moisture, how to find the temperature of such a composition.  
In Compositions Ind. of hot & cold. Ind. ℥ Pro. Ind. to dry & moist Ind. ℥ Pro. 1 5 5 4 5 20 3 4 12 9 4 36 5 2 10 5 2 10 8 3 24 7 3 21 9 1 9 3 1 3  15 60  15 90 of this nature set down the quantity propounded twice, as in these Tables of Hot & Cold, with the other two extremes Dry and Moist: and according to the degree of the Simples temperature, writ down the Index, which vultiply by each particular respective quantity, the total of the products divided by the sum of the quantities, will show the Index for the temperature required: as in this Composition, whose qualities and quantities are these, one Simple in weight 5 ℥ could in 4 degrees, whose Index is 1, and moist in the first degree, the Index 4. then 4 ℥ could in the second degree, the Index 3. and dry in the fourth degree the Index 9 then was there taken 2 ℥ of a Simple in the qualities temperate, and mixed with 3 ℥ of a Simple hot in three degrees, and dry in two, the Indices 8 & 7. Lastly, 1 ℥ of a Simple hot in the fourth degree, and moist in the second degree, the Index to these is 9 & 3. these multiplied by their quantities produceth 5, 12, 10, 24, 9, and 20, 36, 10, 21. 3. their totals 60 & 90 these divided by 15, the quotients will be 4 & 6, each an Index to the quality of the Composition required, viz: hot in one degree, and in the first degree of moisture, the question solved.   

Question 6. To increase or diminish in quality any composition or medicine, according to any degree of temperature that shall be assigned.  
Admit the Composition or Medicine given which was hot in the first degree, and moist in the first, as in the last question; and suppose that 6 ʒ was the quantity of the Medicine, which by another confection is to be increased or diminished in either of the qualities, to any degree assigned, as here, the quality which was hot in the first degree, is to be increased unto the third degree of heat; or that in the first degree of moisture, to be made temperate. In this Example there are two Tables, the first is in one degree of heat, whose Index is 6. yet by the commixture of another Simple is to be increased unto the third degree of heat. Take any Simple or Composition, which is either equal in the temperature, or greater, in this I take one in the fourth degree of heat, whose Index is 9: place this under the Index, whose temperature is known, viz: 6. the difference of these two, and 8 propounded is counterchanged 1 & 2, then say, as the difference of that known, and that given, or the difference of the Index 1 propounded to the quantity of the Confection made 6 ʒ; so shall the difference of 2 (the Index) be in proportion unto the quantity of the Simple that is to be mixed: so it is, as 1 unto 2, so 6 ʒ to 12 ʒ; or as 1 to 6 ʒ, so 2 unto 12 ʒ. And so much of that which is hot in 4 degrees, commixed with another hot in the first degree, the temperature will be in three degrees of heat, as will appear in the Table of their qualities. And so likewise in the second Table of this Example, the Confection which was moist in the first degree is to be made temperate: D 4 is the Index unto the quality given, which must be annihilated, like the common people of these times, as neither hot nor cold. Take the Index of some required Simple, whose degree is either equal, or greater: as here admit one, in the second degree of heat, whose Index is 7 G. the difference of each Index transcribed according to the last, or first Question of this Parag: the difference D 2. & G 1. now the proportion will be, as 2 to 1 so 6 ʒ unto 3 ʒ, or 2 unto 6 ʒ: so 1 to 3 ʒ. and 3 Drams of a Simple, hot in the second degree, added to a Confection of 6 Drams, moist in the first degree, in respect of moisture it would have been in that quality, neither hot nor cold, nor yet dry or moist, if their qualities correspond.  
In the first of these Examples, the Composition 6 ʒ, hot in the first degree, being mixed with 12 ʒ hot in the fourth degree, the whole Confection was made hot in the third degree, but yet is held moist in the first degree, as it was before: although some affirm, not only heat does abate cold, drought & moisture, but also moisture and drought lessens cold; two of these qualities being but by accident. So to return, if the last Composition which was made hot in the third degree, and moist in the first, the whole Confection of 18 ʒ, may be increased, or diminished in the degree of moisture by one of the last Examples, to which I refer you, repetitions being unnecessary: if the quantities of any Simple, or Composition to be made, should happen in a fraction, or in several denominations, viz: lb ℥ ʒ ℈ or the like: reduce the given quantity into the least denomination, and then proceed as before: if there happen a fraction in the Index, subtract it from 5, or the Index of the mean temperature from that, and you will find your desire: as admit the Index were 3 ½ subtracted from 5, shows the degree 1 ½ cold and moist: if the Index were 8 ¾, the degree appertaining to it (by the subtraction of 5) will prove 3 ¾ hot and dry, and so for any other fraction in these degrees and qualities; of this subject I will write no more, lest the Apothecaries should take me for a simple, or this the worst of their drugs.   

A general Rule.  
In all Questions of Alligation wherein a price, without quantity, or quality is propounded: the difference or differences counterchanged gives a solution unto the demand; for the sum of all such differences will be both a Multiplier & a Divisor, as Parag: 7. Axiom 16. & Parag: 12. Quest: 3. In all Rules of Alligation: in this last I do not question the wise Children of Aesculapius, nor the learned Disciples of Galen or Hypocrates, or presume to teach their expert Apothecaries any Rules, but to give an insight of theirs to please some, and to assist others, who have more Practice than Theory, and less Art than Experience.    

Paragraph XIV. By false positions to discover the truth; this Rule teacheth how unto a number known to suppose others analogically, and from thence, four proportional numbers will arise, that shall answer the question, or in proportion to it.  

A Definition of false Positions.  
THe Rules of False, or false Positions, consists in supposing of numbers, representing the quantity or quality proportionable to the thing required, and is twofold, viz: Single, or Double.  
The single Rule of False, showeth by one position of proportional, yet supposed numbers, to resolve a question propounded.  
The double Rule requires two positions of numbers: but first for the single Rule, to be explained by Examples.   

Question 1. There was a man had two sons and one daughter, viz: A, B, & C, the old man (in his last Will & Testament) bequeathed to his children all his estate (in value unknown) in this manner: to his eldest son A, he gave a portion double to B, his second son; and his part, triple his daughters C: the estate after his death was praised at 745 L. What must the children's parts be?  
Most Rules of single position consist of more parts, or proportional numbers than one, and usuallly differs not  Suppos. L s d Suppos. L. 6 A 447 0 0 10-745 3 B 223 10 0  1 C 74 10 0 The totals 10  745 0 0 in form, from the Rules of Society, nor yet in the manner of operation: in these questions take such numbers as may be proportionable, or answer the state of the question, and yet generally avoiding all fractions: first set down the names, or the thing, upon which the demand is made, and those underneath one another, as here A, B, C, take commonly the least number, or part, and in the least denomination; as for C, I suppose 1. than B must be 3 times so much, against which I place 3. the portion of A must be twice that, for which I put 6, the sum of these is 10. then by the 15 Axiom, Parag: 7. the proportion will be, As the sum of the supposed numbers 10 shall be in proportion to the total sum of the Legacies given, so will each particular or proportional part be to his respective share or portion: and this Rule of proportion must be repeated so often as there be supposed parts for the required shares, that is, As 10 unto 745 L. so will 6 A be to 447 L and in the same manner proceeding you will find for B 223 L. 10 s. and for C the Daughter's portion 74 L. 10 s. the total 745 L. which shows the work is true. Many times in these questions you may ease yourself in the operation, as here, 1 being a Multiplier, and 10 the Divider; by divers former Rules there needs no other division than cutting off the first figure on the right hand, so it will be 74 5/10, that is, 74 L. 10 s. for C three times, that is, B 223 L. 10 s. twice, that is, 447 L. for A, as before.   

Question 2. Five Merchants, as A, B, C, D, E, entered into Society with several stocks, and according to their shares for to stand the pleasure of Fortune either in gain or loss: profit was their hopes, and proved the end: their stock unknown, the gain 675 L. their conditions were, that ½ the gains of A, should be ¼ part of B. and ¼ part of B, should be ⅕ part of C. three times A the gains of D. and twice what B did gain, was the gains of E, each particular share is required.  
The conditions  Suppos. L. Suppos: L. 2 A 54 As 25 to 675, so 4 B 108  5 C 135 Or as 1 to 27, so 6 D 162  8 E 216 The totals are 25 & 675 here being only of their gain or loss, you may take any proportional numbers in this case; but such as will solve the question without fractions, is the best, for which you are not confined to any particular number, as 2, 4, 5, etc. herein I do take the least, and suppose 2 for the gains of A; since half that is to be the fourth part of B, in this proportion, B must be 4, and C 5, being five times the fourth part that B did gain: then D must be 6, being three times A. and E will gain 8, being twice the adventure of B. these supposed numbers, being placed under one another, (against each particular profit) find the sum of them, as in the Table 25. which must be the first number, the whole gains, or loss the second, in this 675 L. so will 2, 4, 5, 6, 8, each particular supposed number (according to the 15 Axiom, Parag: 7.) be in proportion to their respective shares, so A must have 54 L. B 108 L. C 135 L. D 162 & E 216 L. the sum of these 675 L. the whole gains, which is proof sufficient, as by the Table appears. The common multiplier 675 L. with the total of the supposed parts; viz: 25, the Divider unto all may be reduced by 5 or 25 (according to the former Rules of Practice, Lib: 2. Parag: 9 Quest: 3.) unto this, as 1 unto 27, so each supposed number to its particular part; by which means a division is avoided in all.   

Question 3. The sum of 570 L. was to be distributed unto 3 men, A, B, C; but at several payments, and upon these conditions, that when A received 2 L then B should have 4 L. and so often as B took 3 L. was C to have 5 L. in this proportion was the money paid, and here their shares are required.  
The Table of 1 2 A 6— B 12-20 C 3 A 3— B 6-10 C 4 The total of 3, 6, 10, is 19 5 As 19 to 570 L. so 3 to 90. this Example is divided into 5 rows, as by the column in the head appears: the state of the question is, where A had 2 L. B must have 4 L. and in that proportion, as if B had 3 L. then C should have 5 L. in the first row they stand, as A 2. so B 4: and for B 3 then C must have 5. in all such cases proportional parts may be found in whole numbers thus: multiply A 2 by B 3, the product is 6 for A. then multiply B 3 by B 4, the result will be 12 for B. and 4 thus increased by 5, produceth 20; that is B by C, so these three numbers are A 6. secondly, B 12, & C 20, which are proportional to the state of the question, and the thing required: for B 12 is in a duplicate proportion to A 6. as A 2 was to B 4. and so 12 is to 20. as B 3 is unto C 5. so in the second row the three proportional numbers are A 6. B 12. & C 20. which you may reduce if you please, as in the third row unto 3, 6, 10, the sum of them, as in the fourth row is 19; this must be the first number; the quantity or sum, as 570 L. the second, the other the parts of A 3, B 6, & C 10, according to the 1 or 2 Quest: of this Parag: and should be placed in the same manner, the operation as in the fift row where the share for A is 90 L. and so find the rest as B 180 L. and the part for C 300 L. the total of them 570 L. as before; each share being in proportion, unto the state of the Question propounded.   

Question 4. There came unto a wedding 6 Lords, 8 Knights, 24 Esquires, and 48 Burgesses: these gave at an offering 120 L. in this manner, viz: 2 Lords gave as much as 4 Knights, 5 Knights equal unto 6 Esquires, and 5 Esquires gave so much as 10 Bur-Burgesses. How much did each distinct degree offer?  
In all questions of this nature you must first find out proportional numbers answering the state of the question propounded, which to effect, according unto the last Example, set down the proportions The products 2 & 5 by 5 is 50 2 L— 4 K— 5 & 5 by 4 is 100 5 K— 6 E— 5 & 4 by 6 is 120  4 & 6 by 10 is 240 5 E— 10 B— The totals are 50. 100 120. 240 Reduced are 5-10-12-24 given, as in the Table to this Question, where the offering of 2 L, 5 K, & 5 E are multiplied together, whose product is 50. then 5 K, 5 E, & 4 K, is 100 next 5 E, 4 K, & 6 E 120. Lastly, 10 B, 6 E, & 4 K, which multiplied produceth 240. the sums of these are 50, 100, 120, 240, and reduced unto their least denominations, 5, 10, 12, 24, which are proportional numbers, and manifesteth that 5 Lords offered equally unto 10 Knights, 12 Esquires, or 24 Burgesses.  
These proportions    L s d 6 Lords 5 6 0 0 8 Knights 10 4 0 0 24 Esquires 12 10 0 0 48 Burgesses 24 10 0 0 The total — 30 0 0 being found, place them in order, as here you see: then suppose for 1 Lords offering any sum you please, admit 20 s. then 6 Lords offered 6 L. which set down; the proportional numbers found were 5, 10, 12, 24, and according to the supposition, 5 L. answers unto any of the proportional numbers, by which find the rest, as 10 to 5, so 8 unto 4 L. the supposed part for 8 Knights. and in this manner find the rest, viz: as 12 to 5 L. so 24 unto 10 L. and as 24 unto 5 L. so 48 unto 10 L. the sum of all the offerings (according unto supposition) amounts unto the sum of 30 L.  
   L s Suppos. L. 6 Lords 24 0 As 30 is to 120, so 4 Knig. unto 16 0  10 Esqu. 40 0  10 Burg. 40 0 The total sum— 30  120 0  
This sum discovered by the supposition is but 30 L. and the offering was certainly known 120 L. therefore institute the Rule of Proportion again, viz: as 30 to 120 L. so 6 unto 24 L. and so much did the 6 Lords offer, and so repeating the Rule you will find the 8 Knights offer 16 L. the 24 Esquires 40 L. and likewise the 48 Burgesses offered 40 L. the total is 120 L. the sum propounded, and the particular parts required.    

Paragraph XV. Sheweth the double Rule of False positions, from whence two errors will arise, and by those errors, a truth will be discovered, as by sundry examples shall be demonstrated.  

A Definition of the double Rule of False positions.  
THe double Rule of False positions, is when the particular quantities, or qualities of things are unknown, which are supplied by supposing of false numbers, from whose errors, a truth will be discovered, and the question solved.  
From two false supposed numbers, two errors will arise, these errors must be examined whether they are greater, or lesser, than is the thing required, and must be noted with a sign usually with these Algebraicall characters, viz: ✚ more: but if either error be defective thus, viz:— less: if equal thus = an equation.  
The first Rule of proportion in False positions, when the signs of the errors are alike, viz: both ✚ or both— 

I. As the difference between the two errors,  
II. Shall be in proportion unto the first error:  
III. So will the difference of the two supposed (numbers  
FOUR Be unto a fourth proportional number, and that added to the first supposition (if the signs be—) or subtracted from it (if the signs be ✚) the sum or remainder will be the number required.     

The second Rule of Proportion in False positions, when the errors have unlike signs, as one ✚ the other—.  

I. As the total sum made of both the errors  
II. Shall be in proportion unto the first error,  
III. So will the difference of the 2 supposed numbers  
iv Be unto a fourth proportional number, which added unto the first supposition (if it be less than the second) but if greater, subtract it from thence: the sum or remainder will be the number desired.    

A Directory unto this double Rule of False positions.  
In the first place, the former  8 10 16 Addit. 11 13 19 Subtract. 4 6 12 Multipl. 40 50 80 Divis. 4 5 8 Differenc. ⅓ in all ⅓ Rules require the same proportion between the errors, as there is betwixt the differences of the supposed numbers, and the true number that is required, and though unknown, it is always comprehended within these differences, which in this Rule are either added, or subtracted, multiplied or divided interchangeably by one another, and so one thing involved is in all their several operations, that is the number sought: for the Errors in this Rule, are nothing else but the differences between the true and supposed numbers, by the operation of some common number, from whence they are in the same proportion, as are the differences of the supposed numbers, and that required: and according to the fourth Paragraph the differences betwixt any numbers increased or diminished by a common number will continue in the same proportion, as in this Example, where 8, 10, 16, are three numbers given, whereof one is a number required, the other two are supposed numbers; the differences between 8 & 10, and also 10 & 16 are 2 & 6, that is, as 1 to 3 or ⅓: and so will these differences prove, if increased or diminished by any one common number, as in the Example does appear.  
By these former grounds are all questions solved in this kind, and not otherwise, unless some Geometrical progressions be employed in the operation, as Squares, Cubes, Squared Squares, etc. which divers Authors have introduced, and amplified this Rule of False positions very much, and to effect that, which before was not thought circumscriptable, or within any bounds but Cossick numbers; yet all their Rules involved with such obscurities, that those ways are more difficult to find, and less certain to continue in, then is the illustrious Algebra itself, which I have reserved for my next work, if these my labours and directions be acceptable, and so I will proceed.   

The common way, and directions, in the double Rule of False positions.  
It is a common Adagy, that there are more ways to the wood than one, the plainest road, or readiest tract in most things ought to be followed, which in this I shall endeavour evidently to show, & this way: first draw two lines (like a St. Andrews Cross) as in this figure: then make two suppositions for the thing required, but both of one denomination, as in respect of quantity or quality: the first false position, place at the upper end of the Cross, upon your left hand; the other on the right, as in this 8 & 12: underneath them directly, place the errors respectively from which false position it did arise: as under 8 set 6, and beneath 12 insert 14 note them with the signs, as whether more, or less: in this both are noted with ✚ than multiply the errors by their contrary supposed numbers, that is crosswise, the first false position by the second error: and the second position by the first error: as 6 by 12 is 72, and 14 by 8 is 112. thus fare 'tis general. Next observe whether the signs are alike, or no: if both in excess, or both defective; subtract the lesser from the greater product, and the remainder shall be the dividend; and the difference of Errors the Divisor.  
In this Example 40 is to be divided by 8, the quotient will be 5 the number required, which place upon the right hand of the Cross, betwixt the second supposition, and its error: and if the whole number or quantity known were 20, the other must be 15, which place against it, upon the left hand of the Cross, as in the figure; but if the errors be of different kinds, as the one ✚ in excess; and the other— being defective: the sum of all such products, must be the Dividend, and the sum of the errors must be Divisor; the Quotient will be the number sought; and by either way, it will be always in the same denomination, with the false supposed numbers, as shall be illustrated by examples.   

Question 1. There was an Excise upon all goods, sold by wholesale men: and by an Edict, if any man rendered not a true account, the penalty (if discovered) was double the commodity: yet all tradesmen allowed to be as obscure as they could, the truth being affirmed: it was demanded of one man, what he had sold: Who replied, 2 pieces of Canvas, 2 of Fustian, and 7 of Holland, every piece a crown more than other; and one piece of the best, was 3 times the price of the worst: what was the value of the 11 pieces of cloth?  
In this Proposition I take the last Example, in which I choose the meanest price, and make two suppositions thereof, and supposing the price of that piece to be 8 crowns, and being there were 10 p eces more, every one dearer by 5 shillings, the best than must consequently be 18 crowns, which according to the supposition, should contain the worst three times, whereas here 3 times 8 is 24, from whence take 18, the remainder is 6 for the first error, and too much: for the second supposition you may take a less, but here I will again suppose a greater, as 12 crowns, and being it is of the same denomination, the best piece should be worth 22 crowns, and that equal unto three times the supposition; but 3 times 12 is 36, from whence take 22, the remainder will be 14 for the second error, which place under 12 the supposition with the sign of more: this second error, multiplied by 8 the first supposition, produceth 112: and 6 the first error, multiplied by 12 the second supposition; the product will be 72: the difference betwixt these products is 40, for the dividend, the difference of errors 8, the divisor; the quotient 5, for the true number required, and then the best piece must be 15 crowns; being 3 times the value of the other, the prices of all the rest will be easily discovered, as thus:  
Two pieces of Canvas, the first 1 L. 5 s. the other 1 L. 10 s. then two pieces of Fustian, the first 5 s. more than the last piece of Canvas, viz: 1 L. 15 s. the other 2 L. the piece, then for the 7 pieces of Holland, the first 2 L. 5 s. the second 2 L. 10 s. the third 2 L. 15 s. the fourth 3 L. the fift 3 L. 5 s. the sixth 3 L. 10. the last 3 L. 15. that is 15 crowns, and three times the first piece in value, according unto the state of the question, the total sum is 27 L. 10 s. the thing required: this is a sufficient trial; and yet for to please all, according unto the former Rule, as 8 is to 6, so will 4 be to 3, which fourth proportional number, taken from 8 (the first supposition) the remainder will be 5. the number required: the errors being both in excess.   

Question 2. To divide any number propounded into any parts that shall be required; and those parts for to be in any proportion one unto another, that shall be assigned.  
The given number here is 45 for to be divided into 2 parts, and those to be in a triple proportion one unto the other: the Cross made as before, let 8 be the first supposition, and the least part, then 24 should have been the greatest, the sum of them is 32, but the number given is 45, therefore this supposition is 13 defective, as by the error does appear: then suppose 10 for the number required, the triple of it is 30, the sum of them both is 40, which shows the second error for to be 5 defective: these errors multiplied crosswise, into their supposed numbers, will produce 40 & 130, their difference 90 for the Dividend: the difference of Errors 8 the Divisor, the quotient 11 ¼ for the lesser number required, three times that is 33 ¾, their total 45, the question solved. By the first Rule, the proportion is as 8 to 13, so 2 unto 3 2/8 or ¼, which 3 ¼ (according to the first Rule) must be added unto 8 the first supposition, the sum will be 11 ¼ as before, answering the state of the question, for the lesser number which being discovered, the other is easily found; this question will be performed by the Rules of single position, if you take proportional numbers, answering the state of the proposition.   

Question 3. To divide a given number into any two parts, and those in any quantity assigned, as to part 10 in two; and so, as that the greater divided by the less, the quotient shall be 20.  
In all questions of this kind, suppose any one number, the other is the remainder; as here I suppose 2: the other must be 8, both numbers being 10: and according to the state of the question, the greater should contain the less 20 times, then consequently 20 times the less would be equal unto the greater number, so 2 multiplied by 20 produceth 40, and should have been equal to 8, the error is 32 too much, then take a second supposition, as admit 1, the other part must be 9, and the second error 11: and both too much, which note, and then multiply them into their contrary suppositions, the products will be 22 & 32, the difference 10 for the Dividend, the difference of errors 21 the Divisor, the Quotient 10/21 for the lesser part, which subtracted from 10 the remainder will be 9 11/21, which is 20 times the other, and consequently 9 11/21 or ●00/21 divided by 10/21 the Quotient will be 20, the thing required: and for trial, by the first Rule of False positions, as 21 to 32, so 1 unto 1 11/21; which (according to the same Rule) subtracted from 2, the first supposition, the remainder will be 10/21 for the true number as before.   

Question 4. A vessel of 63 gallons was filled with French wine of two sorts; the one was at 2 s. the gallon, the other at 2 s. 6 d. the gallon; the wine in the hogshead thus filled, did come unto in money 7 L. 4 s. and it is here demanded how much there was of either sort.  
The quality here, and the number of gallons in either supposition, will solve this question, the denominations being the same in both, as in respect of the quality, that is, supposing the best, or worst sort, or price in either; yet to avoid any great number, the quantity of gallons being odd as 63, and the meanest price or quality being even, I take that in both, and presuppose 23 gallons of the meanest sort, than there was 40 gallons of the best, which at 2 s. 6 d. the gallon comes unto 5 L. and the other sort unto 2 L 6 s. in all 7 L 6 s. the price was 7 L. 4 s. from whence it is apparent, that the first error was ● s. too much● then suppose 33 gallons (or what you please) which granted, there must be 30 gallons of the best, which comes unto 3 L. 15 s. and 33 gallons of the worst (at 2 s. the gallon) amounts unto 3 L. 6 s. the sum 7 L. 1 s. the error 3 s. defective; these errors multiplied crosswise by the suppositions, will produce 69 & 66: and according to my former directions (being the signs are unlike) their sum is 135 for the Dividend, the sum of the errors 5 for the Divisor, the Quotient 27, the number of gallons, of the worst sort of Wine, then must there be 36 gallons of the best, the total quantity 63 gallons; the price of the worst is 2 L. 14 s. and the best comes unto 4 L. 10 s. the just sum of 7 L. 4 s. according unto the proposition. And by the second Rule, the proportion is, as 5 unto 2, so 10 to 4, which 4 if it be added unto 23 (the first supposition) the sum will be 27 gallons, as by the former, operation in the figure does appear.   

Question 5. Hiero King of Sicylia, caused a Crown of Gold for to be made in weight 10 lb. and it was conceived, that the Workman had put a great Alloy of Silver unto it, which abuse of the Artificer Archimedes detected; and by False positions may be thus discovered.  
lb lb [1] lb lb As 10 Gold to 2 water so 6 Gold to 6/5 water As 10 Silver to 3 water so 4 Silver to 6/5 water The total of these fourth proportionals 2 ⅖ lb lb [2] lb lb As 10 Gold to 2 water so 7 Gold to 7/5 water As 10 Silver to 3 water so 3 Silver to 9/10 water The total of these proportionals is 2 3/10  
Admit by putting the Crown into the cistern of water, the quantity run out was 2 ⅕ lb. the mass of pure gold avoided but 2 lb water, and that of silver 3 lb. which in the first place shown the difference of metals, for they being of equal weight, that most compacted and heaviest of nature will have the lesser body, and consequently possess the lesser room: by this a great allay appears, and will be explicitly known, as thus, suppose there was 4 lb of silver, then was there 6 lb of gold: here institute the Rule of proportion twice for either mettle, as in the first Table, viz: if 10 lb of gold voided 2 lb of water, how much will 6 lb of gold void, which will prove 1 ⅕ lb of water; and so according to the supposition find how much water the silver will avoid, which is here also 1 ⅕ lb, the sum is 2 ⅕ whereas the water which the Crown forced out was 2 ⅗ lb. the difference only ⅕ lb for the first error. then suppose there might be 3 lb of silver in the Crown, there must be of gold 7 lb. and according to the second Table the fourth proportional number will be 14/10, or 7/5. then again, as 10 s. to 3 W. so 3 lb silver to 9/10 lb of water, the sum of these is 2 3/10 lb water, the difference of this 2 3/●0 and 2 ⅕ is 1/10 too much for the second error. These multiplied by their contrary suppositions, will produce ⅖ & ⅗, and according to my former directions, the difference is ⅕ for the Dividend, the difference of errors 1/10, the Quotient 2 lb of silver, the quantity of the allay, and 8 lb the weight of gold, that was in the Crown, the thing required.   

The examen, or trial.  
For the proof of this or the like, take the quantities of both metals found, and likewise the quantity of the thing or mass propounded; then institute twice the Rule of proportion; if 10 lb of gold forced out 2 water, how much will 8 of gold put forth, facit 1 ⅗ lb water; then by the second Rule, if 10 lb of silver expelled 3 lb of water, than 2 lb of silver will force out of the same cistern ⅗ lb of water, the sum of these is 2 ⅕ lb of water, and so much did the Crown itself put forth, or by the first canon to this double Rule of False positions, as 1/10 to ⅕ so will 1 be to 2, which according unto the same Rule (if subtracted from the first supposition) the remainder will be 2 lb the quantity required as before: this question may be solved by any less quantity of mettle in either sort: for by finding how much one ounce, or any other quantity shall force out of a cistern, by the common Rule of proportion, you will easily find what quantity of water shall be expulsed by any greater or lesser mass of the same mettle, therefore I will write no more of this.  
Truth by these two last Paragraphs is extracted from False positions, and gross errors multiplied and divided by errors more and less, obsurd in themselves, yet in these the thing lies involved, which is inquired after, though benighted in obscurity, and by correcting the errors will be brought to light, the Aenigmas solved, the Objections cleared, and I discovered in my intentions a friend to the Truth, and really wishing this for the common good, reflecting upon honest ingenious men, to whose candid and merciful censure I refer myself, and for instructions in the Rules of false, because it is so generally beloved and daily put in practice, I will recommend this Breviate to their memories, as a Directory, whereby to avoid some errors of this kind in future, and thus conclude the second Book.   

The Rule of False Positions to discover the Truth: or by erroneous suppositions to find things really true, a paradox, and no hyperbole.  

AXIOMS in Rules of False, are briefly these:  
Take two convenient numbers as you please,  
Two errors will arise, from what you guess,  
Which note with signs, as whether more or less:  
Then multiply those errors you disclose,  
Crosse-wise, by both the numbers you suppose:  
If in your work, two signs alike you make,  
Difference of errors than all numbers take:  
If signs unlike (as ✚ &—) this do,  
Add both the errors, and the numbers too;  
The sum or difference th' errors must divide  
Which quotient then, the riddle will decide,  
Unless both numbers sought it does contain,  
Or both products alike, th' Aenigma's vain.  
Where naught is to divide, you may descry  
The Question's false, or but a fallacy.  
If more than two positions you must need,  
The Rule of false will then be false indeed.  
Some other notes, and queries might be shown,  
Whose use is best to Cossick-numbers known.   

The Rules prescribed here will guide you true,  
Yet take these caveats too along with you,  
Though many say (and will my counsel shun)  
What need we Rules, when as this Work is done?  
To which I answer: All I here impart  
Are but the grounds, and principles of Art.  
Some Rules, search tracts, that various ways do wind,  
Yet leave not inquisition till they find:  
Errors, Maeanders are; in which we're led,  
That none can well return without a thread.  
Make no positions, errors to maintain,  
As blind sects do, who seek for truth in vain,  
With miscalled Lights, pretend to guide men right,  
They're Faux his lanterns, & a snuffs their light,  
'Tis not to such my labours I intent,  
But to the good, or those that would amend:  
The new, but false inspired Saints suppose,  
All things of truth are now revealed to those:  
He that from them, can any truth descry  
By false positions; hath more art than I:  
These, I as errors eat; and do implore  
To be Christ's servant, and I wish no more:  
The well-disposed amend will what's amiss,  
And find their errors, as I do in this,  
Which found, correct; they'll vanish then away,  
And Truth will rise from thence, like break of day:  
Errors, are mists, that do benight our Spheres,  
Withdraw those vapours, and the day appears:  
Man did by errors fall, whence Art in vain  
Labours, in part, for to restore again.  
By false positions, you may errors find;  
Who cannot see their faults, are very blind,  
By what is false, I hope you'll find what's true.  
I wish your Errors small; and so Adieu.   

All that I know, I know was given to me  
For others good, so this I give to thee:  
And for my labours, give (if you be eased)  
All glory unto God; and I am pleased,   
To be the Servant Unto his Servants, THOMAS WILSFORD.     

Artificial Arithmetic: OR, NUMBERS, Divided into SECTIONS, And these in CHAPTERS.  
Containing decimal Arithmetic, with the Definition, Reduction, Annotation, Numeration, and Construction of these fractions; with their several rules in Addition, Subtraction, Multiplication, and Division; with decimal Tables of the Coins, Weights, and Measures, commonly used in England. Also one of Minutes and Seconds.  
THE THIRD BOOK  
LONDON, Printed by J. G. for Nath: Brook at the Angel in Cornhill, 1656.   

THE THIRD BOOK: Containing decimal Arithmetic.  

SECT. I.  

CHAP. I. The Definition of Artificial Arithmetic, with the Reduction of the decimal fractions, and the art of framing those numbers.  

The Definition.  
Artificial Arithmetic by decimal fractions, doth Add, Subtract, Multiply, and Divide with whole numbers and fractions commixed together in one sum; and their totals, Remainders, Products, and Quotients, shall produce mixed numbers; as integers, and fractions, in one total sum; these decimal fractions have always for their Denominators, an Unite with Ciphers annexed unto that Unite towards the right hand; as 1/10, 2/190, 3/1000, or 1/1000, etc. but if any Fraction propounded shall not have such a Denominator, it must be reduced unto it by art, from whence this kind of Arithmetic derives it exordium, or name originally.   

Rule 1. The reduction of common or vulgar fractions unto decimals, with the first grounds thereof.  
Any vulgar fraction may be reduced unto a decimal, by division, or very near the same quantity, without any sensible error, as thus: unto the Numerator of the fraction given annex cyphers, as in extracting the Quadrat root, Lib: 2. Parag: 1. Example 5. but in all these cases at pleasure, as 1, 2, or 3 cyphers, etc. this done, divide the whole by the Denominator of the fraction propounded, the Quotient will be the Numerator of the fraction, whose Denominator shall be an Unite, with so many Ciphers, as the Numerator hath places: of these there are two kinds, viz: Rational and Irrational: those are called Rational, whose Numerators are just quantities, without having any remainder, as all the others have, and yet those decimals retaining a proportion so near their vulgar fractions, as that humane works can require no more exactness, as shall be instanced in some following examples.  

Example 1. Of some vulgar fractions reduced to decimals, retaining true proportions.  
The Denominator of every fraction is in proportion unto the Numerator, as are the Integers to their parts, according to the Rule of Fractions: Lib: 1. Sect: 2. Parag: 1. Paradig: 9 then by the first of these 4 Examples, ½ is a fraction propounded, for to be made a decimal, of which suppose 10 to be the integer: then say by the Rule of Three, As 2 is to 1, so 10 unto 5, which are proportional numbers, Lib: 2. Parag: 7 Axe: 11. that is, as the Denominator of the given fraction is unto its Numerator, so shall 10, 100, 1000, etc. (the decimal Denominator) be in proportion unto the decimal Numerator required, which is 5/10 equal to ½. Thus the second Example is ⅗, which is to be converted unto a decimal, and thus as 5 to 3, so 10 will be to 6; so the fraction is 6/10. The third Example is of ¼ which is made 25/100 equal to ¼. The fourth Example is ⅝ reduced to a decimal of 625/1000 equal to ⅝, being in the same proportion with the vulgar fraction, from whence it was derived: in composing of these decimal fractions, there needs no multiplication, as Lib: 1. Sect: 1. Parag: 4 Example 4. but annex cyphers unto the Numerator of the vulgar fraction given, as occasion requires, but my second way of Division is best, as Lib: 1. Sect: 1. Parag: 5. Example 6. by which means you may continue annexing cyphers at pleasure, and so take no more than you need; for cyphers only after a significant figure, are quite unnecessary in this kind of Arithmetic, as you will see hereafter in the following Rules; besides the Denominator of every decimal being an unite with cyphers, they must be superfluous in the Numerator.   

Example 2. Of some vulgar fractions that are irrational, reduced unto their nearest decimals.  
All fractions in this are such numbers as cannot be reduced unto perfect decimals, but something will remain; in all such cases annex cyphers to the Numerator, as occasion, and the state of the question shall require, let 3 or 4 cyphers be the least, whereby the error may not be an unite in 100 or 1000 parts, then proceed according to the last Example, and find these in this, as ⅓ whose Numerator converted to a decimal will be 3333 that is 3333/10000 which is very near equal to ⅓, as by the first Proposition of this Example. In the second 2/7 is propounded, whose Numerator 2 multiplied by 100000, or which is all one, annex 5 cyphers to it, and then it will be 200000, which divided by 7 (the vulgar fractions Denominator) the quotient will be 28570 for the new Numerator, and will stand thus 28570/100000, or thus 2857/10000 equal to the former, as Lib: 1. Sect: 2. Parag: 1. Parad: 4. of this last decimal fraction; if the Numerator had been required but of 3 places, viz: 285/1000 add 1 unto the unite place, and so make it 286/1000, and the reason is evident ●86/1000 being 3/10 too much, whereas 285/1000 is 7/10 defective, and of two errors the least is to be chosen; and so for any other fraction in decimals.    

Rule 2. The annotation of decimal fractions, and how denominated.  
decimal fractions are 
A Table of annotations.  - Unites - 3 3 L 0 Primes 1 1 9 Seconds 2 5 3 S 8 Thirds 3 6 7 Fourths 4 5 1 D 6 Fifts 5 1 5 Sixts 6 0 4 Sevenths 7 4 2 Q 3 Eights 8 1 2 Ninths 9 6 1 Tenths 10 7 ¼ I degrees TWO III IV thus denominated, descending from an unite: the first place below an integer is called a Prime; the next unto that are Seconds; the place beneath them are Thirds, etc. denoted according to the degrees, series or order of the Arithmetical figures, descending by Ten to what number of places you please, or the state of the question shall require: this Table is made unto ten places, although 5 or 6 will be abundantly sufficient for most questions: in the uppermost row of this Table noted I stands o Prime, 9 Seconds, 8 Thirds, 7 Fourths, etc. and so any other numbers reckoned according to their degrees, descending from an unite, as in the second row TWO, whose first place must be always noted with this mark, or a point only prefixed before them, as thus: 25 denoting 2 Primes & 5 Seconds: the third row shows a number compounded of integers and fractions together, continued unto ten places; and according to the quality of this Artificial Arithmetic, comprehends (as in the fourth row) 3 L. 3 S. 1 D. 2 Q & ¼ all in one sum, as shall be made evident hereafter.  
According to this One Unite makes 10 Primes Prime Seconds Second Thirds Third Fourth's Fourth Fifts Fift Sixts sixth Sevenths Seventh Eights Eight Ninths Ninth Ten last prescribed Rule in all Fractions of Artificial or decimal Arithmetic, the degrees proceeds by 10; so that one Unite contains 10 Primes: one Prime 10 Seconds, one Second 10 Thirds, one Third makes 10 Fourths &c. So an unite of any one denomination makes 10 in the succeeding degree descending: and 5 is the half of any precedent decimal, as 5 Second; is half a Prime; and 5 Primes half an integer or unite whether it be of Number, Weight, or Measure, that is, of any divisible thing.   

Rule 3. The numeration of decimal fractions without Denominators.  
 
Numerators of all decimal fractions are numbered from an unite, descending towards the right hand, contrary to all numbers in Natural Arithmetic; as by the front of this Pyramid will appear: which figure is divided into two parts, containing upon the right hand the Numerators of fractions, descending in 8 degrees of decimals, as by the numeral letters on the right side of the Pyramid appears, from 1/10 of an unite 1/10000000: Upon the left hand (in this figure) do stand all the respective Denominators contained between them: At the basis of these stands letters of the Alphabet: over the capital letters the 9 significant figures are inscribed, ascending by integers according unto Natural Arithmetic; viz: over A the unite place; over B the place of ten: above C the degree of hundreds etc. ascending ten every degree or place towards the left hand: against these upon the right hand stands as many small Roman letters, with Arithmetical figures over them, to show the order of decimals, from an integer or unite descending towards the right hand, viz: a denoting the unite place over which stands 1, and that under the point, representing the place of integers: b notes the Primes, being ten times less than the former: c stands under Seconds the third degree, or place of hundreds descending, being ten times inferior to the last, and thus these decrease from an unite by ten, as integers increase: and as from an Unite one ascends, the other descends, both in the same order, and without end, or limitation, as in Numeration continued; the Denominators in this kind of Artificial Arithmetic, are totally to be omitted, observing to make good the places of the Numerator, unto Primes inclusive, by prefixing cyphers, as thus, 1 for 1/10 and − 01 for 1/100 and − 001 for 1/1000: see Lib: 1. pag: 3. by which means the Denominator of any decimal is always known being an unite with so many cyphers, as the Numerator hath places, and for finding all decimal fractions, the following Rule declares and illustrates with Examples.   

Rule 4. A general method for reduction of vulgar fractions unto decimals without any sensible error, as thus:  
This differs nothing really from the 1 Rule, & yet depends upon the two last for here you are notwithstanding farther to observe the Unite place of the Denominator (which in all questions must be made Divisor) the first figure of the quotient shall be of the same denomination, as is the figure over it in the Dividend; and if the Divisor extends itself so, as that the unite place shall stand underneath the first cipher of those annexed unto the Dividend (which is the place of Primes) the first figure of the Quotient must be of the same denomination, viz: Primes, according to the first Example: but if in case the unite place of the Divisor extends itself, to stand under the second cipher annexed unto the Dividend, the first figure of the Quotient will be also Seconds, and so must have a cipher prefixed before it: if unto the third place 2 cyphers: if to the fourth place 3 cyphers etc. because the first figure in the Quotient will require the same denomination, as hath the figure of the Dividend, over the unite place of the Divisor, and all the other places must be made good by prefixing of cyphers to Primes inclusive, as shall be illustrated by these Examples.  
First, 15/●6 of an ounce weight is to be made a decimal; to 15 the Divisor annex 4 cyphers, the sum is 150000, and that divided by 16 the Denominator, the Quotient will be − 9375 the true decimal required: in the second Example 3/40 will be reduced unto − 075, that is − 0●5/1000, the true decimal. Thirdly, 1/365 parts of a common year (that is, one day) will be made − 00274: but this is irrational, yet without any sensible error, for it is not an unite too much in 100000, the Denominator of the decimal: for if it were continued on to another place it would have been − 002739 & the next − 0027397, so this 00274/100000 is sufficient, and exact enough for any use.   

A Proviso.  
In case the Denominator of any vulgar fraction, shall have a cipher, or cyphers, after the significant figure or figures, you must note what degree of the Dividend the unite place of it will stand under; and so accordingly prefix cyphers if any be required, as in the second Example of the last Rule, and that done you may omit the cyphers in the Divisor: and if the Numerator of a vulgar fraction shall have cyphers after any significant figure, as 10/17, or 100/1655 etc. you must note the figure or cipher of the Dividend that stands over the unite place of the Divisor, as before, for the first cipher on the left hand of those annexed is always the place of Primes, and according to that denomination shall the first figure of the Quotient be: as for Example 10/17 the decimal will be 588: and the decimal of 100/1655 shall be − 0604. but when the vulgar fraction may be abreviated, reduce it unto the least denomination, as in Lib: 1. Sect: 2. Parag: 1. Parad: 4. and then make them decimals: if fractions of fractions are required to be made decimals, reduce them first into a single fraction, as Lib: 1. Sect: 2. Parag: 1. Parad: 1. which done, proceed according to my former directions: in all vulgar fractions of what kind sooner (that are to be made decimals) consider what their integers are, and so accordingly make their decimals, these are but fractions of fractions; as for example, a Farthing is but ¼, if a Penny be the integer; but of a Shilling, it is ¼ of 1/12 that is 1/48, and if a Pound sterling were the integer it will be ¼ of 1/12 of 1/20 that is 1/960, and so likewise of all other things in Number, Weight, Measure, Time, etc. but lest I spend time without measure, this Chapter shall be here concluded, although it be necessary in this kind of Arithmetic to have Tables ready calculated; which by the former Rules may be easily performed, and that in part you shall have at the end of this Section which treats of decimals only: this Chapter was made the longer, whereby to make all the others short and easy.    

Sect. I. Chap. II. Addition of decimals with integers, fractions, and compounded numbers.  
THis Table of Addition Money Weight Measure 1 2 3 − 65 − 83333 13 Gall. − 15 − 16667 26 Gall. − 1 − 075 − 5 − 01667 − 025 − 25 − 04167 − 05 − 125 − 03750 − 00087 − 09375 − 00313 − 00009 − 03125 − 99897 1 − 15096 the total the total the total is divided into 3 parts or columns: the first consists in the number of several Coins or pieces of money, 1 pound sterling being the integer. The second column is of Troyweight, whereof 12 Ounces makes 1 lb. The third is of liquid Measures consisting of whole & broken numbers together, whose integer is a Gallon, the other Quarts, Pints, and their parts. In the first column is propounded these several sums of money, to be added together according unto decimal Arithmetic, viz: 13 s. 4 d. secondly 3 s. 10 d. thirdly 2 s. 9 d. & ¾. the decimals of these, take in what order you think best, but place them right under one another according to their degrees whether they be integers, primes, seconds, or thirds etc. and where there are no such inferior degrees, make points to keep the fractions orderly in their places under one another: in this I seek the decimals of the greatest denominations first, as 13 s. that is 13/20, whose artificial number by the last Chapter will be found − 65, then for 3 s. or 3/20 the Decimal will be— 15 the next 1 for 2 s. or 1/10. the shillings thus entered make points towards the right hand, as in this unto fifts, and so proceed unto the Pence, where first I find 4 d. whose vulgar fraction in respect of 20 shillings is ⅓ of 1/20 that is 1/60 and the Decimal will be found − 01667 and to 5 places, next 10 d. whose fraction is ⅚ of 1/20 that is 5/20 or 1/24 the decimal − 04167. and the decimal for 9 d. or 3/80 is − 0375. Lastly, 3 farthings whose compound fraction is ¾ of 1/12 of 1/20 that is 3/960 or 1/320 whose decimal is − 00313, the total of these is 99897. the true decimal of these particular sums, which total might have been inscribed a shorter way, as thus, the total sum propounded is 19 s. 11 d. ¾ whose 3 decimals are these; 95 secondly − 045833. thirdly − 003125- the total of them is − 998958, which differs little from the former, being an unite less, yet both defective, which is caused by the irrational numbers.  
The second example is of Troy-weight, whereof 24 Grains makes a Pennyweight, 20 of them an Ounce, and ●2 Ounces 1 Pound: the particulars here given are these, viz: 10 Ounces: 18 P. secondly, 2 O. 6 P. weight: thirdly, 12 P. weight: 5 G. ½, the sum of these is 1 lb. 1 Ou: 16 P. 5 G. ½, herein is contained one integer; and as for the fractions, you may find their decimals thus: for 1 Ou: take 1/12 the decimal − 083333— for 16 P. the fraction is reduced to ⅘ of 1/12 that is 1/1● the decimal − 066667. next for 5 G. the vulgar fraction 1/1152 the decimal is − 000168, the ½ G. is ½ of 1/24 of 1/20 of 1/12. that is 1/●000087 the total is 1 lb − 150955, in the same manner you may take the particulars, as in the second Table to 5 places, whose sum will be 1 − 15096 not differing an unite from the former in the fift place: herein you are to note that although these were fractions given, yet their totals do make one integer, parted from the place of Primes with a point. The third Table is of liquid measures, viz: Gallons, and their inferior parts to be added together, as 13 G. 2 Q. 1 P ¾. to be added unto 26 G. 1 Q. 0 P ¼. the sum of these is 40 Gall●ns: but to add them according to their particulars begin first with 2 Q that is ½ whose Decimal is always 5. Next 1 Q. is ¼ the decimal − 25, than 1 P. is ½ of ¼, that is ⅛ the decimal. 125. then ¾ of ½ of ¼ is 3/32 the decimal is − 09375. Lastly, ¼ of ½ of ¼ is 1/32, and for the decimal − 03125, these numbers are all rational, and so no fraction remains. Here observe that all Ten in the place of Primes, are integers, as was said before, and must be added as Unites (if there be any whole numbers) or inscribed beyond the point, or Prime line, as in the second and third Example. Addition of these Artificial numbers differs not from the common or vulgar way, but in this, and placing them according to their degrees. And so much for this Chapter, which is proved by Subtraction as in Natural Arithmetic.   

Sect. I. Chap. III. Subtraction of decimals with integers and fractions commixed together.  
If several     L.  1 − 9260417 2 20 275  − 3989583  8 2875 Re. − 5270834 Re. 11 9875  L.   L.  3 40 1666667 4 100 —  − 325  − 008333 Re. 39 8416667 Re. 99 991667 sums be given to be subtracted, reduce them unto totals, and place them according to their degrees, as in the last Chapter, and then subtract the lesser number from the greater, and if the decimals are unequal, that defect may be supplied with points, or suppose cyphers to be annexed unto them, in all things else (except their degrees) it differs nothing from the subtraction of whole numbers, as by these 4 Examples shall be made evident: in the first is propounded 18 s. 6 d. 1 q. from whence 7 s. 11 d. 3 q. is to be taken in decimals, which to find I have shown already, yet for your ease one more shall be inscribed, and first for 18 s. the vulgar fraction is 9/10 the decimal − 9 then 6 d. is ½ of 1/20 that is 1/40 the artificial number − 025, than 1 q. is ¼ of 1/12 of 1/20 that is 1/960 the decimal − 0010417. the total − 9260417: and in this manner you will find the second number to be − 3989583 which taken from the former, there will remain − 5270834. which is the decimal of 10 s 6 d. ½. the true remainder of the vulgar numbers if subtracted. In the second Table 8 L. 5 s. 9 d is given to be subtracted from 20 L 5 s. 6 d. the decimal of 5 s. is − 25 of 6 d. − 025 the sum is 20 L 275. the total of 5 s. 9 d. is − 2875, so the lesser sum is 8 L. − 2875 the difference is 11 L. − 9875 according to the Table, which is 19 s. 9 d. The third Table is 6 s. 6 d. to be subtracted from 40 L 3 s. 4 d. the sum of the decimals answering 3 s. 4 d. is − 1666667 − the decimals belonging to 6 s. 6 d. is − 325 to which annex cyphers or make points as in the Table, and subtract them according to vulgar or Natural Arithmetic, the difference is 39 L. and this decimal − 8416667 equal to 16 s. 10 d. The fourth and last Example is of 100 L. from whence 2 d. is to be deducted, whose decimal is − 008333: suppose cyphers or points representing their places, and subtract the decimal, as in the Table, where you will find the remainder 99 L. − 991667 the decimal of 19 s. 10 d. answering the truth and your expectation too. This Chap: is tried by the last, and that proved by this, as by subtraction in Natural Arithmetic, to which I refer you for the form and reason, and for the practic part to this Chapter, in 4 Examples.   

Sect. I. Chap. iv Multiplication of decimals commixed with whole, and broken numbers.  
THe increasing of Decimals 1 2 L 456 − 91875 − 15 − 25 2280 459375 456 183750 6840 − 2296875 by integers or mixed numbers differs nothing from multiplication in Natural Arithmetic; yet 4 things in this are particularly to be observed: first for conveniency, make that number (which consists of most places) the Multiplicand, although it be oftentimes the least in quantity. Secondly, you must distinguish the decimals from the integers (if there be any) with a point, or Prime line. Thirdly, cut off with a Prime line, so many places of the product (numbered from the right hand towards the left) as there were fractions both in the Multiplier and Multiplicand: all the figures from thence to the left hand are integers, and those to the right are decimal fractions; or this may be done with a point, to note the place of Primes. Fourthly, if the Product shall have fewer places than the terms given had decimal fractions, those places must be made good by prefixing cyphers on the left hand, this will often happen when the places of Primes, Seconds, or Thirds shall be cyphers: all which by Examples will be illustrated.  
In the first of these two Tables there is propounded 456 L. to be multiplied by 3 s. whose decimal is 15, and according unto Multiplication in whole numbers the product will be , that is 68 L. 4/10 or 8 s. and so will 456/1 multiplied by 3/20, which is the vulgar fraction of 3 s. the Multiplier propounded: and in the second Example 18 s. 4 d. ½ is to be multiplied by 5 s. the decimal fraction of the Multiplicand is − 91875. and 5 s. is − 25 as a decimal, these multiplied will produce − 2296875; and being there were 7 places in both terms, the whole Product is but a decimal fraction, and must have a point prefixed, as in the example; the fractions of several denominations may be reduced unto a single fraction, as 1323/1440 or 147/160, and this unto a decimal as before is shown in reduction of Artificial numbers.  

Example 2.  
In the first of these 1 2 Tables, there is given a whole number with a fraction to be multiplied by a whole number only, viz: 16 & 55 to to be increased by 132, which is the greater number, but the fewer places, and therefore is made Multiplier, whose product is , and being there are two decimals in the Multiplicand 2 figures or places are cut off from the Product, as 60: so the true product is 2184 & 6 the thing required, as in the Example, for 60/100 & − 6/10 is all one, as was said before in the first Chap: in the same manner, the second Example is multiplied.   

Example 3.  
The first of these two Tables is 1 2 an integer with a fraction, multiplied by a fraction, viz: 4 & − 125 multiplied by − 0039 whose product is 160875, which contains but 6 places, whereas the decimals of both terms have 7 places, therefore ● prefix a cipher, and then a point, as in the Example. In the second Table is propounded − 004 for to be multiplied by − 25, the product is 100, and being there are 5 places in both terms, I prefix 2 cyphers, and cut off the other two, as unnecessary and independants in this k nde of Arithmetic, so the true product will be − 001, as in the Table: so − 25 multiplied by − 2 will be − 05, that is 05/100 equal to 1/20 in vulgar fractions; and 25/100 & 2/10 the terms propounded were equal to ¼ & ⅕ whose product in Natural Arithmetic is also 1/20 the proposition evidently proved; and so in any other, if the decimals be not irrational, howsoever they may be tried without sensible error; and Multiplication by Division, as in Natural Arithmetic.    

Sect. I. Chap. V Division by decimals with integers, and compounded numbers divided into 6 cases or rules.  

Case 1.  
IN all cases of Division by decimals, if the Dividend be greater than the Divisor, the Quotient will be an integer or mixed number, but if less, the Quotient will prove a decimal fraction.   

Case 2.  
Ciphers may be annexed unto the Dividend at pleasure, or as occasion shall require, whereby the Quotient may be continued to so many places as are necessary, according to the state or condition of the question propounded, and so likewise in whole numbers: as if 45 were to be divided by 7, the Quotient will be 6 3/7: but if cyphers be annexed, the Quotient may be 6 4/10 or 6 42/100, or 6 428/1000, or 6 4285/10000, or 6 42857/100000; but according to the Rules of Decimals, the Denominator must be omitted, as was said before, and then the artificial number stands thus, 6 − 42857 − and so may any other whole, or broken number be divided, and the remainder (if there be any) made a decimal fraction.   

Case 3.  
All decimals or mixed numbers, must be divided as integers are; and if the Divisor be an integer, or a compounded number, the first figure in the Quotient, will be of that denomination, as is the figure of the Dividend over the unite place of the Divisor: the Quotient found, (according unto Division in Natural Arithmetic) separate the integers from the fractions (if there be any) and find the quality and quantity of those broken numbers, as by the following Examples.  

Example 1.  
a unite or integer here is propounded for to be divided by 95 or 9 − 5 this whole number for the Divisor, placed as the Rule requires, annex so many cyphers unto the Unite given as you please, and so proceed unto Division, where you will find the first point, or the unite place of the Divisor, under the denomination of Seconds in the Dividend, and the first figure to be 1, which according to his degree must have a cipher prefixed, or set before it to denote its place; so the decimal of 1 divided by 95 is − 01, or by continuing the Division unto more places you will find this decimal, viz: − 0105263: but if in case the Divisor were 9 − 5 the decimal will be − 105263, because in this mixed number − 5 be decimals, and 9 is the unite place, which stands under D●cimalls in the Dividend, from whence it takes the Denomination the first figure being a Prime— and 1 thus divided by 36 − 5 will be − 0274— or 1 by 123 − 45 the quotient will be − 008 or − 0081— for the figure 3, being the unite place of the integers will stand under the place of Thi●ds in the Dividend, and consequently the first figure in the Quotient must be of the same denomination, and therefore hath two cyphers prefixed; and so for any other this must be observed, otherwise the decimal degrees will be confounded.   

Example 2.  
Here are two compounded numbers given, viz: 1655 − 21 for to be divided by 45 − 2 having set down the Dividend 1655 − 25 − & the Divisor in his place, I find the integers of it will be twice had in the whole numbers of the Dividend, and consequently two integers in the Quotient: annex cyphers unto the Dividend, and then divide as if they were whole numbers, the Quotient will be 36 − 62. the mixed number required, as in the Table does appear: so if 16·845 were to be divided by 14, the Quotient will be 1 − 203. for 4 the unite place of the Divisor will stand under 6 the integer, and upon the second remove under 8, the place of Primes, so the second figure in the Quotient must be a decimal fraction: if 168·45 were to be divided by 24, annex cyphers to the Dividend and make it 168·45000, and then divide it by 24 the Quotient will be 7 − 01875. an exact decimal: but if 16 − 845 should be divided by 24 as before, the Quotient will be − 701875 a decimal only, because the unite place of the Divisor, viz: 4 will at the first demand stand under · 8 in this question the place of Primes.    

Case 4.  
If the Divisor he a decimal fraction only, and the Dividend either a whole or compounded number, the Quotient (in all such cases) will be Integers unto that place inclusive where the degrees of the Divisor and Dividend are equal: and if the Dividend shall want decimal fractions, annex cyphers, to make their places equal in degrees, as whether Primes, Seconds, Thirds, &c and having found what Integers will arise in the Quotient, you may annex more cyphers to the Dividend at pleasure, as by examples shall be made apparent.  

Example 1.  
The Dividend in this is a mixed number, viz: 58 − 05 for to be divided by which consists of 3 places, therefore annex a cipher unto the decimal of the Dividend which will stand thus, 58 having a decimal fraction equal in places unto the Divisor, viz: in Thirds; and the least of these degrees being equal, there must be so many Integers in the Quotient, as the Divisor can be contained in the Dividend, and in these cases to avoid any error from the cipher or cyphers annexed, continue points to supply the defective place of cyphers, as in the Example; which divide by 96, the cipher being omitted, standing for nothing but to supply a place, and show the decimal degrees, so here the Quotient will be found 604 − 6875, which is apparently caused, the Divisor not being a tenth part of 1 unite.   

Example 2.  
In this example the Dividend is 14886, and the Divisor is − 75 which consisting of decimals only, and the Dividend a whole number, viz: 14886, there must be 2 cyphers annexed unto it, for to make the places of the Dividend equal in degrees unto those of the Divisor, which are Primes & Seconds; this done, divide it by − 75 the Quotient will be 19848 Integers or whole numbers, as in the example does appear.    

Case 5.  
When both terms given are decimals, yet the Dividend the greater number; the Quotient will be Integers, while both terms are equal in their degrees, and all the other decimals, as by the Examples following.  

Example 1.  
In this Table there are three Examples propounded, the first is − 18 for to be divided by − 0045 both decimals, the Dividend − 18 is the greater number consisting of Primes & Seconds, whereas the degrees of the Divisors significant figures, are Thirds and Fourths, therefore annex two cyphers to the Dividend, and then it will be − 1800 to be divided by − 45 − the Quotient will be 40 whole numbers. In the second Example − 24102 is given to be divided by − 16: their degrees being equal in the foremost places, the first figure in the Quotient will be an Integer, the other decimals, viz: 1 − 506 − in the third Example − 1872 is given to be divided by − 24 these are both decimal fractions: but the Seconds in the Divisor will at first demand stand under the Thirds of the Dividend which is a degree lower, and therefore the first figure in the Quotient must be a decimal, which here is 78. and so for all others in this case.    

Case 6.  
In all Fractions of this kind, if the terms be decimals, and consisting of equal places, and the Divisor the greater number, superscribe the Dividend as the Numerator of a fraction; but if the places of the terms be unequal, supply the defect of the Numerator, by annexing cyphers unto it, and so making it a proper vulgar fraction, which reduce by the first Chap: unto a decimal, if required.  

Example.  
In this Table are contained 3 I III III Examples: the first is − 4 for to be divided by − 12 and that reduced to ⅓. The second Table is 48 for to be divided by 64 which is reduced to ¾. The third Table is 5 to be divided by 16●5, and reduced to ∷ 1/●31, unto the Numerator of it annex 2 cyphers, and then it will stand as a proper vulgar fraction thus ●00/3●1, these 3 Examples are thus reduced to decimals, viz: ⅓ unto 3333 − and ¾ to − 75 and 1/331 unto − 003. which you may continue unto more places by the first Chap: if you please: and as for the trial of Division in Artificial numbers, it will be proved by Multiplication, and Multiplication by Division, as in Natural Arithmetic, only observing the prescribed Rules of decimals; and here I will put a period to this subject, having in a breviate laid the foundation, and described a little model of a great structure in decimal Arithmetic, as to the theory; and for the practic part it is convenient to have Tables ready calculated, which shall be the subject (God willing) of the next Chap: for your present practice of those past, and your ease in future.   

A general Rule. Any decimal, or compounded number being given, to find the quantity, or parts of the fraction.  
By the Tables following L 4 − 990625 S 19 812500 D 9 750000 Q 3 000000 Totall 4 L. 19 s. 9 d ¾. this question may be answered: but in case you have no such Tables, the f actionall parts may be as exactly found, & but with little trouble by this general Rule: the Integers of any compound number being known, the quantity of any decimal fraction will be discovered, as in vulgar fractions by Natural Arithmetic, as Lib: 1. Sect: 2. Parag: 1. Paradig: 10. and so here, multiply the decimal by the known parts of the Integer, and from the product out off so many places as there be cyphers in the Denominator, for the Unite divides nothing, as in this Example where there is propounded 4 L. and this decimal, viz: − 990625 (which are parts of a pound sterl:) these multiplied by 20 the product is 19 − 812500, that is 19 S. the decimal fraction consisting of 6 places, whose Denominator is 1000000; then descend unto the next denomination which are Pence, 12 the Integer, so 812500 multiplied by 12 produceth 9 D. remaining − 750000, which multiplied by 4 will produce − 3 Q. and nothing remaining, the true sum required: and if any thing shall remain after the least denomination, you may cast it by, or make a decimal fraction of it, as you please, and be contented with this; a little to the Ingenious being sufficient.     

Sect. I. Chap. VI Tables of the chief Coins and Weights, with the long dry and liquid concave measures commonly used in England.  

Of English Coins.  
THE least 1 Farthing makes 1 Farthing 2 Farthings 1 Halfpenny 4 Farthings 1 Penny 4 Pence 1 Groat 3 Groats 1 Shilling 20 Shillings 1 Pound sterling Fraction, or part of our English money is a Farthing, from whence these Coins in scribed do proceed (as by this Table) unto 1 Pound sterling the Integer; these are subdivided into divers other parts, necessary to be known, but unnecessary for the present purpose, some of these being superabundant: for 1 Shilling being 1/20 of the Integer, the fraction of one Farthing will be written as thus, ¼ of 1/12 of 1/20 which are easily reduced to a single fraction as 2/960. Lib: 1. Sect: 2. Parag: 1. Parad: 1. and if a compound fraction be made of all these parts it will be thus, viz: ½ of ½ of ¼ of ⅓ of 1/20, which if reduced will be 1/960 as before: the shortest way is best, yet being there may be use of these parts, they were purposely inserted: thus one Halfpenny is 1/24 of 1/20, that is 1/480 and one Groat, ⅓ of 1/20 that is 1/60: and so for any other Coin, whose parts of the Integer are known.   

Of Troy-weight.  
1 Grain of wh: makes in weight 1 Grain. 24 Grains 1 Penny weight 20 Penny weight 1 Ounce 12 Ounces 1 Pound-Troy 14 Ounces 12 penny Troy 1 Pound-Averdupois  
These are in use for the weighing of Bread, Electuaries, Gold and Silver; whereof a Grain is the least, and in former times 32 Grains of Wheat was accounted a Pennyweight, Vid: Stat: de compositione ponderum, 51 Hen: 3.31 Edw: 1. & 12 Hen: 7. But times are altered so well as we, and now 24 Grains is reduced unto a Pennyweight, whereof 20 such Grains did make an Ounce Troy, as now they do, but termed Pence until Queen Elizabeth's reign, who changed the value of those Pence unto 3 Pence the piece, as now they stand: so a Grain of Wheat as a fraction of a fraction to 1 Pound-Troy the Integer, will stand thus, viz: 1/24 of 1/29 of 1/12● that is by reduction 1/5760 parts.   

Averdupois weight.  
24 Grains of wheat 1 Scruple. 3 Scruples make 1 dram. 8 Dragmes make 1 Ounce, Averdup: 16 Ounces make 1 Pound Averdup: 14 Pound Averdup: 1 Stone. 2 Stone, or 28 lb ¼ Of an Hundred. 4 Stone, or 56 lb ½ a Hundred. 8 Stone 112 lb 1 Hundred weight. 5 Hundred lb 1 Hogshead. 10 Hundred lb 1 Butt or Pipe. 20 Hundred lb 1 Tun or Load.  
This is called Civil or Merchant's weight, with which is weighed all gross commodites and Merchandizes, (Malynes lex Mercat: pag: 49. & 252.) of these there are two kinds, viz: the lesser and the greater, these proceed originally from a Grain of Wheat, (Georgius Agricola de pond: & mens:) and so in several parts and denominations they increase to a pound the lesser weight, by which are sold commodities by retail, as Butter, Cheese, Flesh, Tallow, Wax, and what hath the name of garbel, and whence issueth Waste or Refuse; of this a Pound is the Integer, and the least of the greater weight, whose Integer is 1 C, that is 112 lb, and as fractions they may be thus expressed, the lesser weight proceeding from a grain of Wheat, viz: 1/24 of ⅓ of ⅛ of 1/16 which if reduced is 1/9216. and the greater weight proceeding from a pound thus 1/14 of ½ of ½ of ½, that is, if reduced 1/1●2: or thus ●/14 of ⅛ which is the same, 1 stone being ⅛ part of a hundred.   

Long or radical measures.  
4 Barley corns 1 Inch or finger. 4 Fingers or Inches 1 Palm or hand. 12 Inches or 3 Palms 1 Foot. 18 Inches or 1 ½ Feet 1 Cube. 3 Feet or 2 Cubes 1 Yard. 3 Feet and 9 Inches 1 Elle. 5 Feet 1 Pace Geometrical. 6 Feet or 2 yards 1 Fathom. 5 ½ Yards or 16 ½ Feet 1 Perch or Pole. 40 Perches 132 Paces 1 Furlong. 8 Furlongs 320 Pole 1 Mile English. 3 Miles 1 League.  
These are named long or radical, by reason the superficies of divers things are measured by the Squares composed of their sides, commonly called Roots (vide 33 Edw: 1. & 25 Eliz:) in Geometry: the least of these is a Barley corn in breadth, being ¼ of an Inch, from whence all the other measures are derived, as in the Table: the Integers of these are Feet, Yards, Paces, Poles, etc. the fractions (as to the greatest denomination) may be thus expressed, ¼ of ¼ of ⅓ of ⅓ of 2/11 of 3/40 of ⅛ of ●/3; these fractions reduced into a single fraction will be 1/760320 which may be made out of fewer compositions, or more: for these proceed from a Barley corn, and so to a Palm, a Foot, a Yard, a Pole, a Furlong, a Mile, and a League, the greatest denomination here.   

Of concave dry measures.  
2 Pints or pounds 1 Quart. 2 Quarts 1 Pottle. 2 Pottles 1 Gallon. 2 Gallons 1 Peck. 4 Pecks 1 Bushel Land measure. 5 Pecks 1 Bushel Wat. measure. 4 Bushels 1 Coombe. 2 Coombes 1 Quarter. 4 Quarters 1 Chalder. 5 Quarters 1 Tun or Wey.  
These measures are derived from a Pint, which of Wheat is supposed to weigh 1 pound Troy, from hence proceeding unto Gallons, 8 of them making 1 Bushel, usually called Land measure: and 5 pecks do make 1 bushel of Water measure; 5 quarters is the greatest denomination, containing 1 Tun, Wey, or sized Load: the Measures here proceeding from a Pint, may be expressed in broken numbers or fractions of fractions thus, ½ of ½ of ½ of ½ of ¼ of ¼ of ½ of ⅕, and these fractions by reduction will be made a single fraction, as 1/512 if 1 quarter were the Integer: but if a Tun, it must be ⅕ more, and then it will be 1/2560, the fraction required proceeding from a Pint unto a Wey: by these are measured dry commodities, viz: all kinds of Grain, Salt, Lime, Sea-coal, etc.   

Of concave liquid measures.  
2 Pints 1 Quart. 2 Quartes 1 Pottle. 2 Pottles 1 Gallon 8 Gallons 1 Firkin, of Ale, Soap or Hearing. 9 Gallons 1 Firkin of Beer. 2 Firkins 1 Kilderkin. 2 Kilderkins 1 Barrel, 36 Gallons. 42 Gallons 1 Tierce. 63 Gallons 1 Hogshead. 2 Hogsheads 1 Pipe or Butt. 2 Butts, 252 Gallons 1 Tun.  
By these all liquid substances are measured, proceeding from a Pint unto a Tun: the Integer containg 252 Gallons or 2016 Pints, which will be expressed by fractions of fractions thus ascending by these particulars, excepting the Ale Firkin, Kilderkin, and Pipe, viz: ½ of ½ of ½ of 1/9 of ¼ of ●/7 of ⅔ of ¼ which is 12/24192, or reduced 1/2016, the fraction made of a Pint, and a Tun the Integer.   

A Table of Time.  
60 Seconds makes 1 Minute. 60 Minutes 1 Hour. 24 Hours 1 Day natural. 7 Days 1 Week. 4 Weeks 1 Month. 13 Months, 1 Day, & 6 Hours 1 vulgar Year.  Or 365 D. 5 H. 48 M. & 55 S. The magnitude of a common Year.  
This is a Table of Time (but not of these) from hence the World's infancy derives a pedigree, with a continual succession of Days, Months, & Years unto this declining Age, proceeding here from a Second, and terminated with a Year, wherein I will conclude, being there is a time for all things. I could have derived these from Thirds and Fourths, etc. but do conceive Seconds are sufficient for common use, 60 making a Minute, as in the Table, not perfectly true, errors increasing as the times; an Hour unlimited in humane understanding, and is only known to GOD, the sole Creator of all: as I will instance in a Natural day, generally conceived for to consist of 24 Hours just, which opinion is reprehensible in humane sense, and found to contain 57 Seconds more, yet one Day not equal unto another, some being greater, and others less, for which I have inserted (at the bottom of the Table) the magnitude of a common Intercalary year, according to the opinion of divers learned men: the fractions are not here subscribed, being written as the last, and so pronounced, their terms respectively considered: besides the year is also divided into 12 Solar months, each containing 30 equal parts, and by some into 30 days 5/12 or 10 hours, which is 4 seconds, and more to little; so fearing I should write too much of this, and consequently lose time, I will here conclude this Chapter and proceed to the next.    

Sect. I. Chap. VII. Of decimal Tables calculated to 7 places, according to the fractions before, in Number, Weight, and Measure, with Time made apt for use in this kind of Artificial Arithmetic.  

I. The decimal Tables of reduction of English Coins unto sevenths are these.  
An explanation.  
THis decimal Table (of English Coins) is divided into 4 columns, or denominations, & each of those in two: the first of them contains a row of Shillings, descending from 19 s. unto 1 s. and from 11 d. unto 1 d. and from 3 farthings unto 1 q. Lastly, from 3 Mites unto one M inclusive, against every one of these stands their respective 
ENGLISH COINS.  Shillings. Pence. 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 0041667 8 4 Farthings. 7 35 3 003125 6 3 2 0020833 5 25 1 0010417 4 2 Mites. 3 15 3 0007812 2 1 2 0005208 1 05 1 0002604 decimals: as for Farthings, they were never lawful money of England, yet each of those is subdivided into 4 Mites or lesser parts, for more exactness in some questions, one of these being a fraction to the Integer, thus expressed, ¼ of ¼ of 1/12 of 1/20 parts of a Pound sterling, which if reduced is 1/3840 whose decimal will be 0002604, according to the 1 Chap. and 4 Rule of this Book, and all the rest may be discovered in the same manner: or more compendiously by Multiplication, into all the other parts ascending; but then calculate the first decimal in more places than you intent for the Table, otherwise you may commit errors, and those beget many more: As for example, the decimal of 1 Mite is 0002604, which multiplied by 4 will produce 0010416, which is too little: 1 Farthing being 0010417, so Reduction may be sometimes necessary to regulate your Table, where the decimals are long continued. The Tables thus made, the decimal for 18 S. will be expressed with a Prime only, viz: 9 and 19 S. with a Prime and a Second, as & 6 S. 8 D. thus: for the 6 S. with & for 8 D. 0333333 the sum 3333333— and 19 S. 11 D. 3 Q. 3 M. for 19 S. writ for 11 D. for 3 Q. 003125. and for 3 M. 0007812, the total is 9997395, which is true unto an unite in 10,000,000, the Integer of a pound sterling, according unto these Tables, and the Rules of decimals: one Mite being 0002604, which 9997395 wants of the Integer, and an unite more in the seventh place. I dilated this, to abreviate the rest: which observe diligently with the next.   

II. The decimal Tables of reduction of Troy weights, unto 7 places are these.  

The construction and use of this Table.  
This Table of Troy weight is framed as was the former, according to the 4 Rule, & 1 Chap: of this Book: or having discovered the least decimal number, you may find the others by Addition or Multiplication: or having found the greatest number, you may find any of the other numbers by Subtraction or Division, yet care must be had in it, especially from those which are irrational: As for example, in the decimals of 14 Penny w. wherein 0041667 and Troy-weight. Ounces. 11 9166667 10 8333333 9 75 8 6666667 7 5833333 6 5 5 4166667 4 3333333 3 25 2 1666667 1 0833333 Pennyweight. 19 0791667 18 075 17 0708333 16 0666667 15 0625 14 0583333 13 0541667 12 05 11 0458333 10 0416667 9 0375 8 0333333 7 0291667 6 025 5 0208333 4 0166667 3 0125 2 0083333 1 0041667 Grains. 23 00●9931 22 0038194 21 0036458 20 0034722 19 0032986 18 003125- 17 0029514 16 0027778 15 0026042 14 0024306 13 0022569 12 0020833 11 0019097 10 0017361 9 0015625 8 0012889 7 0012153 6 0010417 5 0008681 4 0006244 3 0005108 2 0003472 1 0001736 0541667, that is the decimal for 1 Penny and 13 Pennyweight, whose sum is 0583334 which should be but 0583333- or the Decimal of 7 Pennyweight which is 0291667, this doubled is 0583334 as before: and the like will be in Subtraction or Division; as for the use in setting down a sum, as admit these, viz: 11 O. 19 P. 23 G. thus 9166667 & 0791667 & 0039931: the total is 9998265, a decimal expressing the parts required; this wants but 1 Grain of an Ounce Troy, which in the Table is 0001736, and added to the former sum it will be an Integer, and an Unite more in 10000000 as 10000001, and so for any other in this Table.    

III. The decimal Tables of reduction of Averdupois little weight unto sevenths, are these.  
Averdupois-weight. Ounces. 15 9375 14 875 13 8125 12 75 11 6875 10 625 9 5625 8 5 7 4375 6 375 5 3125 4 2● 3 1875 2 125 1 0625 Drachmas. 7 0546875 6 046875 5 0390625 4 03125 3 0234375 2 015625 1 0078125 Scruples. 2 0052083 1 0026042 Grains. 23 0024957 22 0023872 21 0022786 20 0021701 19 0026616 18 0019531 17 0018446 16 0017361 15 0016276 14 0015191 13 0014106 12 0013021 11 0011936 10 0010851 9 0009766 8 0008681 7 0007595 6 000651 5 0005425 4 000434 3 0003255 2 000217 1 0001085  

The use of this Table of Averdupois little weight.  
The construction of these I have explained in the two former Tables, and the use of them little differing from the others, only in the fractional parts, whereof this consists in Ounces, Drachmas, Scruples, and Grains, and a weight by these will be thus expressed: As for example, 8 lb 15 O. 7 D. 2 S. 23 G. the Decimal in the Table for 15 O. is 9375 for 7 Dr: for 2 S. 0052083, and for 23 Gr: 0024957 the total is 9998915, and the whole number thus inscribed 8 lb 9998915, which fraction wants but 1 Gr: to make it 9 lb, to the former fraction, add the Decimal of 1 Gr: which is 0001085, the total will be 10,000,000 the unite or Integer unto these decimals of 7 places.    

iv The decimal Tables of reduction of Averdupois great weight, unto 7 places are these.  
The great weight. The Stone or 14 lb 7 875 6 75 5 625 4 5 3 375 2 25 1 125 lb Pound w. 13 1160714 12 1071429 11 0982143 10 0892857 9 0803571 8 0714286 7 0625 6 0535714 5 0446429 4 0357143 3 0267857 2 0178572 1 0089286  

This Table of Averdupois great weight explained.  
As for the framing of these numbers I refer you unto the first or second Table, and as for the use of this it will not differ from any of the former, only observing 8 Stone, and 112 lb, to be the Integer, 14 lb makes one Stone: and 1 lb the least decimal fraction of this gross weight: and by these artificial numbers 3 C. 7 St. 13 lb. is thus expressed: the 7 St. being 875. and the decimal for 13 lb is 1160714, the sum of these is 9910714, or in all 3 C. 9910714, the number required; this wants but one pound of 4 C weight, then add the decimal of 1 lb to it (that is, 0089286) the sum is 10000000 which is 112 lb. to this add 3 C. the total will be 4 C. the thing required, viz: 40,000,000.    

V decimal Tables of reduction of long or radical measures from a Foot unto 1/10 part of an Inch, and the artificial numbers to 7 places.  
The use of this Long measures. 1 Foot or 12 Inches. Tenths of inches. 11 9166667 9/10 075 10 8333333 8/10 0666667 · 9 75 7/10 0583333 8 6666667 6/10 05 7 5833333 5/10 0416667 6 5 4/10 0333333 5 4166667 3/10 025 4 3333333 2/10 0166667 3 25 1/10 0083333 2 8666667   1 0833333   Table.  
This Table of long Measures consists of Inches, subdivided into ten parts, the Integer to these is 1 Foot, or 12 Inches, their decimals are thus to be subscribed, as in this example, 2 Feet, 11 Inches & 1●/20, the decimal of 11 Inches is 9166667, under which place the artificial number of 9/10 which is 075, the sum of these (according unto the second Chapter in addition of decimals) will be 9916667 the true number of those fractions; which wants 1/10 of a foot, then add the Decimal of 1/10 unto it, that is 0083333, the sum is 10,000,000 an Integer, so the total is 3 Feet, and the sum of 9/10 & 1/10 will be 0833333, the decimal number of an Inch, as in the Table appears.   

VI decimal Tables of reduction, of long Measures, in Yards, els, and their parts unto 7 places.  
In this Table is Long measures in Yards & Ells.  Nailes. 3 1875 2 125  1 0625 Quarters. Quarters. 3 75 ¾ 046875 2 5 ●/2 03125 1 25 ¼ 015625 expressed long Measures, as Yards and the parts, the greatest is ¾ or 27 Inches, each Quarter is commonly subdivided into four lesser parts, usually termed Nails, each containing inches 2 ¼, and these Nails are again divided into 4 parts, with artificial numbers appropriated to them, and are thus expressed, viz: for ¾ Y. 75 for 2 N. 125— for ¼ N. or Quarter 015625— and thus are expressed (by the same decimals) all the parts of an Elle: As for example, 3 E. 3 Q. 3 N. & ¾. the decimals to them are these, viz: 75— & 1875— & 046875 the total 984375— in all 3  984375. but here note that one Naile is understood 1/16 part, being ¼ of ¼ both of Yard & Elle.   

VII. decimal Tables of reduction of long measures in Statute perches down to 6 inches in 7 places.  
By this little Table Long measures ½ foot or 6 inches. 32 9696969 16 4848485 31 9393939 15 4545454 30 9090909 14 4242424 29 8787879 13 3939393 28 8484848 12 3636363 27 8181818 11 3333333 26 7878788 10 3030303 25 7575757 9 2727273 24 7272727 8 2424242 23 6969697 7 2121212 22 6666667 6 1818182 21 6363636 5 1515151 20 6060606 4 1212121 19 5757576 3 0909091 18 5454545 2 0606061 17 5151515 1 030303- are found the decimal fractions of a Statute Pole or Perch, containing 16 ½ feet, whereof 33 half feet makes the Integer; so the decimal of 1 is 030303— and that of 8 feet, or 16 halfs is 4848485; and of 32 is 9696969, which if added with an Unite unto the decimal of 1, v z: 030303— will be 1000000 the Integer, if ¼ or ½ or ¾ were required, their decimals will be expressed generally, as in all other fractions of this nature, viz: 5 or 25 or 75: their use will be the same; if less parts were required, divide the last by the parts according to the Rules prescribed: as admit to an inch, the decimal will be 0050505 being but ⅙ of the last decimal in the Table.   

VIII. decimal Tables of reduction of dry concave measures in Bushels, Pecks, & Quarts, to 7 places.  
Dry measures.  Bushels. 3 325 2 0625 5 0195313 7 875 2 25 1 03125 4 015625 6 75 1 125  Quarts. 3 0117188 5 625  Pecks. 7 0273438 2 0078125 4 5 3 0937 6 0234375 1 0039063  

An explanation of this Table of dry concave Measures.  
Here you have the decimal fractions of Bushels, Pecks, and Quarts; whereof 8 Bushels or 1 Quarter is the Integer; the greatest decimal (which is 7 Bushels) is 875; and that of a Quart (which is the least denomination here) is 0039063, the half of it will be 0019531, the Decimal of 1 Pint, if it be required, and may easily descend lower if you please.    

IX. decimal Tables of reduction of liquid concave measures in Quarts, Pints, and Quarterns to 7 places.  
The use of this Table Liquid measures. Quarts. Quarterns. 3 75 3 09375 2 5 2 0625 1 25 1 03125 Pint.   1 125   differs nothing from the former, either in construction or use; the Integers of this is a Gallon, the greatest denomination here is 3 Quarts, whose decimal is 75— the least is a quartern, or ¼ part of 1 Pint its decimal 03125, and from these a greater or a lesser may be calculated.   

X. decimal Tables of reduction of hours or degrees, in minutes and seconds unto 7 places.  
A Table of Minutes unto Hours or Degrees. A Table of Seconds continued to one Minute. M.  M.  S.  S.  59 9●33333 29 4833333 59 0163889 29 0080556 58 9666667 28 4666667 58 0161111 28 0077778 57 95 27 45 57 0158333 27 0075 56 9333333 26 4333333 56 0155556 26 0072222 55 9166667 25 4166667 55 0152778 25 0069444 54 9 24 4 54 015 24 0066667 53 8833333 23 3833333 53 0147222 23 0063889 52 8666667 22 3666667 52 0144444 22 0061111 51 85 21 35 51 0141667 21 0058333 50 8333333 20 3333333 50 0138889 20 0055556 49 8166667 19 3166667 49 0136111 19 0052778 48 8 18 3 48 0133333 18 005 47 7833333 17 2833333 47 0130556 17 0047222 46 7666667 16 2666667 46 0127778 16 0044444 45 75 15 25 45 0125 15 0041667 44 7333333 14 2333333 44 0122222 14 0038889 43 7166667 13 2166667 43 0119444 13 0036111 42 7 12 2 42 0116667 12 0033333 41 6833333 11 1833333 41 0113889 11 0330556 40 6666667 10 1666667 40 0111111 10 0027778 39 65 9 15 39 0108333 9 0025 38 6333333 8 1333333 38 0105556 8 0022222 37 6166667 7 1166667 37 0102778 7 0019444 36 6 6 1 36 01 6 0016667 35 5833333 5 0833333 35 0097222 5 0013889 34 4666667 4 0666667 34 0094444 4 0011111 33 55 3 05 33 0091667 3 0008333 32 5333333 2 0333333 32 0088889 2 0005555 31 5166667 1 0166667 31 0086111 1 0002778 30 5   30 0083333    

An explanation of this Table in time.  
This Table consists of equal parts of an hour or a degree, each being divided into 60 minutes, and each of them again into 60 seconds, so the least fraction here is 1/60 of 1/60, that is 1/3600 whose decimal is 0002778, the artificial number representing 1 second of an hour or of one degree; the rest being discovered, as by the 1 Chap: of this Book, and the decimals of 59″ S. is given to be added unto 59′ M. whose decimals are these 0163889 & 9833333 the sum is 9997222 the decimal for 59′ M. & 59″ S. to which add 000277 S. (representing 1 second) the total will be 10,000,000 denoting the Integer, which is either an hour or a degree, by which I have here come unto an end of this Section, and so in time will put a period to my labours, and your trouble.      

A Conclusion.  
Benevolent Supervisors,  
BY the persuasion of some friends I have presented you here with 3 Books in the Art of Numbers, containing variety of difficult Questions, and doubtful Rules, confirmed and made facile by ocular demonstrations; how well I know not, that's referred to your better judgements, unto whom I do appeal for justice, against all capricious heads, and sinister calumniatore, enwalled upon the times: This little Volume now attends your pleasures, not for an applaudie, but approbation, hoping you will rather err in the clemency of mercy, than in the rigour of justice, in which we are all lost: the greatest faults in the Book you will find at the end corrected, and inserted in a Table by themselves, not written on the front in capital characters, but, as men carry their crimes behind them, in a small print: I hope you will not add to the errata, this being in the nonage, and if born with so happy a fate as to live unto maturity, by a second impression the errors of the infancy shall be expunged, whereof some of them will be fathered upon me 'tis like, although I like them not: but since errors are originally incident to all humane race, I hope you will with humanity cover or excuse them so fare as you can, without blemish to your reputations.  
The chiefest, and most useful Rules (I hope) you see explicitly delivered, and so scientifically as my Genius could direct, or dictate to my judgement, and those delineated by Geometrical demonstrations, extracted from the original and principles of Art, derived from precedent Ages; yet some (perhaps) will censure it in particulars, as the Cobbler who questioned Zeuxis about a picture exposed to the public view, in which Table the figure of a man was pourtraicted so artificially to life, as there wanted only motion to deceive the Spectators; the Cobbler found fault with his shoes, and according to the skill in his own trade made it apparent, which satisfied many of the beholders, insomuch that the man presumed from thence to give his judgement upon the whole figure: at which, Zeuxis reprehended him, saying, Suitor, ne ultra crepidam.  
Some will make queries, wherefore I used Paragraphs and Paradigmas, for Chapters and Examples, which I have done for variety and distinction only, having composed many Books of Mathematical Sciences: others will ask wherefore I treated of this subject, when the Stationer's shops seem oppressed with them already: to this the Cobbler might reply, another man's shoe may not fit me so well; as for my Arithmetic, I intent not to teach them how to cast figures, or to throw others by, since some men's works I cannot with reason object against, and as for others, I will not, out of humanity, and the principles of morality, my condition being the same, subject to err as much as they, or more.  
As for this Arithmetic I confess it is old, and so is all what these later Ages have produced in this kind, yet in respect of method 'tis new to the World a● day (so fare as I know) when the model was first cast by me, portrayed by Geometry, in a sympathetic union betwixt Number and Magnitude, by Art founded upon Reason, supported by Axioms like Pillars in Architecture: the cause I writ it, was that diversity of capacities and understandings, will require diversity of ways to approach their apprehensions; I have seen ingenious men, and good Arithmeticians, who knew well the practic part, and almost quite ignorant of the speculative, or any reason what they did, nor satisfied by the writings or dictates of others, which if I have explicated to their understandings, and my endeavours prove acceptable, then are my labours recompensed, myself pleased, my Book graced, I obliged and encouraged to expose my private Manuscripts unto a public view, according to my first intentions & sole scope, which is to the glory of God and benefit of my Country: but if the candid Lectors shall think this too much, I shall be disanimated, and conceive so of my labours too, and rest.  
Being wearied with attending each proof from the Press, and some printed off before I could peruse them, from hence perplexed with faults, solicitous and doubtful of a civil entertainment, coming forth in a blustering distempered Age, I was easily persuaded by the Stationer not to hazard any more at sea in this bottom, which made me put a period here abruptly, leaving out many Rules both in civil and rural affairs although finished, which I intent quickly to adventure forth, if this makes a prosperous voyage, and the public voice prove auspicious gales to fill my sails, I shall be then encouraged to weigh Anchor again, fraught with customary Rules in Commerce and Trade, both for Sea and Land, viz: Society in equation of payments; of Barter, Tear, Neat, Cloffe, Trett, Reductions of Coins, Weights and Measures, with Exchanges, Cambi●-Maritimo, Factorage, Interest and Discount of money: divers Questions erected upon Geometrical foundations, as the dimension of each Superficies and solid Body: as Board, Wainscott, Land, Circles, Timber, Stone, Cylinders, Pyramids, Cones, Segments, Gauging of Vessels, Spheres and Globes, to find the weight of Bullets, with sundry Military propositions.  
I have also finished divers Manuscripts, as Algebraicall & Logarithmecall Arithmetic with their applications in there books, viz: Geometry, Altimetrie, Geographie, cosmography, Astronomy, Navigation, both by right Lined and Spherical Triangles, with exact solutions to their propositions by addition only, and all questions in the Spherical Triangles by two operations at most, without Rules or Theorems: besides all these I have composed sundry Manuscripts of several subjects, expecting to see what hospitality this finds in the mean time.  
Farewell.   
FINIS.   

Errata Typographica emendata: Lib. I.  
PRoëme pag. 2. line 13. read, with the dictates & p. 9 l. 4. commonly known to  
p. 5. l. 19 r. they are p. 6. l 18. r. following p. 12, l. 12. for a or. nothing p. 25. l. 19 r. 1 Acre p. 28. l. 12. for 91 r. 81 p. 32. line 17. r. then p. 33. in the Table 730 1460 r. 730 1460 p. 36. l, 23. for 800 lb r. 8 C. in the table 5376 896 r. 5376 896 p. 44 in the table 2000 40 r. 2000 40 p 45. in the table r. p. 55. l. 21. r. found 10 p. 73 l. 7. r. fraction p. 74. l. 19 r 〈◊〉 p. 77. l. 20. r. so 12 p. 78. table 3. r. 22 p. 86. l. 14. r. and p 90 l. 14. r. Sect: 2. & l. 23. r. 3 ⅓ p. 95. in the table r. 56/320 p. 97. l. 15. r. as p. 104. l. 23: ¾ or ⅔ r. ¾ & ⅔ p. 108. l. 16. ⅛ r. 4/5  
Lib. II. Page 144. l. 4. r. prefix: in the Table r. 1728 p. 181. l. 17 & 18, r 26. 18 & l. 20. r. 2 & 18 p. 188. l. 22. r. extreme p. 194 l. 1 &. 3. r. Multiplyers for Dividers. p. 196. l. 7. r. 445, or 450 p. 203. for 9 squares r. two p. 209. r. Rule of 3 p. 211. l. 19 r. 63/8 p. 213. l. 24. r. of 45 p. 262. l 4. four deal p. 268. r. Knights offered p. 315. l. 7. such grains deal