¶ The Ground of Arts: teaching the perfect work and practise of Arithmetic, both in whole numbers and fractions, after a more easy and exact sort, than hitherto hath been set forth. Made by M. ROBERT RECORD, D. in Physic, and afterwards augmented by M. JOHN Dee. And now lately diligently corrected, & beautified with some new Rules and necessary Additions: And further endowed with a third part, of Rules of practise, abridged into a briefer method than hitherto hath been published: with diverse such necessary Rules, as are incident to the trade of Merchandise. Whereunto are also added divers Tables & instructions that will bring great profit and delight unto Merchants, Gentlemen, and others, as by the contents of this treatise shall appear. By john Mellis of Southwark, Schoolmaster. CERVA CHARISSIMA ET GRATISSIMUS PRO 〈◊〉 Imprinted by I. Harison, and H. Bynneman. ANNO DOM. 1582. To the Reader. THat which my friend hath well begun For very love to common weal, Of good will hath been over run directing each part every deal. And beautified for thy behoof With breifer ways for practise lore, As by the trial and the proof Was never yet in print before. Of numbers use the endless might No wit nor language can express, Apply, and try, both day and night And then this truth thou wilt confess. The Books Verdict. To please or displease sure I am, But not of one sort to every man. To please the best sort would I feign, The froward displease shall I certain. Yet wish I will, though not with hope, All ears and mouths to please or stop. To the Right worshipful M. ROBERT FORTH Doctor of Law, and one of the Masters of the Queen's majesties high Court of Chancery. BIAS the wise Philosopher of Greece, when his Country was spoiled, and the people carried their goods away, being demanded why he did not the like? answered, that he carried all that he had with him, meaning his virtue and learning: So I (my Right worshipful & singular good Master) having nothing of value in substance to bestow, do here present you with such treasure as the Lord hath vouchsafed upon me, which according to Bias opinion and mine own likewise, is the greatest jewel I have, not doubting but you will accept the givers mind more than the gift, which I confess is small. But being thoroughly acquainted with the great favour your Worship beareth to such as delight in any good exercise, it hath emboldened me to put forth this simple Addition under your Worship's defence. The entire love & exercise of this excellent Art, with drawing of proportions, Maps, Cards, Buildings, plats, etc. were the only studies whereunto I evermore have been inclined. Touching Drawing it was only Dei beneficio, naturally given me from my youth, without instruction of any man, more than Love thereof, delectation, desire, and practise. In this Art also having great delight, I had no other instruction at my first beginning but only this good Author's Book, but afterwards I greatly increased the same during the time I served your Worship in Cambridge, in going to the Arithmetic Lecture at the common School: And more furthered since the time that I left your Worship's service, which is about 18. years past, by continual exercise therein (the mother and nurse of Science,) during which time my only vocation hath been (thinking it a meet exercise for a common wealth) in training up of youth to write and draw, with teaching of them the infallible principles and brief practices of this worthy Science, having (I praise God for it) brought up a number to become faithful and serviceable to their masters in great affairs, and many of them good members of a common wealth, which is no small comfort to me in Christ. Amongst which number, a countryman of mine hath oftentimes been very inportunate with me, to do a deed of Charity upon the ground of Arts, uz. to peruse and amend it of the imperfections and faults, that have crept into it through negligence of often printing. Which earnest request of his bred two strifes in me: The one was, I was loath to do it, knowing myself inferior to a great number that might better do it than I. And yet considering it is a Book hath done many a thousand good, which when a young beginner cometh to a confused or mistaken figure, it bringeth him into a wonderful discouragement and maze: which thing considered, for mere love to a common wealth, and to the Book, being my first Author, I willingly granted to do my goodwill. And passing under the file of correction, I here and there increased it with such necessary Additions as I knew might encourage a young learner, and more would have done, but for fear the Book would rise too thick or grow too dear. And being thus entered into the vain thereof, and knowing that this Author was the only light and the chiefest loadstone unto the vulgar sort of English men in this worthy science, that ever writ in our natural tongue, I have (according to my simple knowledge) yielded again some part of my received talon with advantage: and endowed him to the further increase of his memory, with Rules of brevity and practise, abridged into a briefer method than hitherto hath been published in our English tongue, with other right necessary Additions, Rules and Tables, which I trust will do my country good, and be right commodious to all sorts of men: all which I commit to the favourable censure of your worship, and all such as love knowledge, desiring their favourable correction herein if ought be amiss. The which with greater affection than I am able to utter, I Dedicate to your Worship, as a meet Patron, both for learning godliness, and love of the same, which coming from your worship into the hands of many (shall I doubt not) do many good, as heretofore it hath already done. So shall you (as the best benefactor of these labours) be partaker of all their prayers that shall reap profit or knowledge by this worthy Art, in Commendation whereof if I should write, I should rather blemish than adorn it. For the Author's Epistle unto that famous Prince of worthy memory K. Edward the sixth, and his Preface to the Reader, are sufficient. Thus craving favourable acceptation of this my homely and dutiful present, I humbly leave you to the conduction of the Almighty, whom I beseech long to preserve you in continual health, with daily increase of worship, to the glory of his name, and to the joy of all such as love you. At my house in Southwark this 12. of june, Your Worship's most bounden to command, JOHN MELLIS. TO THE MOST mighty Prince Edward the sixth, by the grace of God King of England, France, and Ireland. etc. THe excellency of man's nature, being such, as it is by God's divine favour (most mighty Prince) not only created in highness of degree far above all other corporal things, but by perfection of reason and search of wit, much approaching toward the image of God, as not only the holy Scriptures do testify, but also those natural Philosophers, which exactly did consider the nature of man, and namely the far reach and infinite compass of the works of the mind, were enforced to confess, that man scarcely was able to know himself. And if he would duly ponder the nature of himself, he would find it so strange, that it might seem unto him a very miracle. And thereof sprang that saying: magnum miraculum est homo, maximum mariculum sapiens homo. For undoubtedly as man is one of the greatest miracles that ever God wrought, so a wise man is plainly the greatest. And therefore was it that some did account the head of a man the greatest miracle in the world, because not only of the strange workmanship that is in it, but much more of the efficacy of reason, wit, memory, imagination, and such other powers and works of the mind, which can more easily conceive any thing in a manner, than understand itself. And amongst all the creatures of God it findeth none more difficult to be perceived than these same powers of itself, whereby it doth conceive and judge as it may be well conjectured by the diversity of opinions, that the wisest Philosophers did utter touching the spirit of man and the substance of it: whereof at this present I intend to make no rehearsal, but who so sisteth to read thereof, may find it largely set forth not only in Aristotle his books De anima; but also in Galene his book called Historia philosophica, & again in Plutarch his work, De philosophorum placitis, whose words are also repeared of Eusebius in the xv. book, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, unto whom I remi● them that have desired to understand the intricate difficulty of knowing our own selves, as touching our best part, and that part whereby we deserve to bear the name of men. This matter seemed so obscure and difficult, in knowledge, that Galen, who for his excellent wisdom and judgement in natural works, is called of many men a Miracle in nature, yet in searching the nature and substance of the spirit of man, he not only confesseth himself ignorant, but counteth it plain temerity to attempt to find it: so far above the hope of man's knowledge is that part, whereby man doth know and judge of things. And although the ignorant sort (which hate all things that they know not) do little esteem the profoundness of man's spirit and of reason, the chief power and faculty of it, yet as there is a kind of fear and obedience of unreasonable beasts unto man by the working power of God, so is there in those small reasoned persons a certain kind of reverence toward wisdom and reasons, which they do show oftentimes, and by power of persuasion are enforced to obey reason, will they nil they. And hereby came it to pass, that the rudeness of the first age of man was brought unto some more civil trade, as it is well declared by Cicero in the beginning of his first book De inuentione Rethorica, where he saith thus: Name suit quoddam tempus quum in agris homines passion bestiarum more vagahantur, & sibi victu ferino vitam propagabant, nec ratione animi quicquam, sed pleraque viribus corporis administrabant. N●ndum divine religionis, non humani officij ratio colebatur: Nemo legitimas viderat nuptias, non cer●os quisquam inspexerat liberos, n●n ius aequabile quid unlitatis haberet, acceperat: ita propter errorem atque inscitiam. caeca ac ●emeraria dominatrix animi cupiditas, ad se exple●dam viribus corporis abutebatur, perniciosissimis satellitibus. Quo tempore quidam, magnus videlicet vir & sapiens, cognovit quae ma●●ria esset, & quanta ad maximas res oportunitas in animis iness● th●ninum, si quis eam posset elicere, & praecipiendo me●●●r●n reddere: Qui dispersos homines in agris, & in tectis syluestribus abditos ratione quadam compulit in unum locum, & congregavit: & eos in unamquamque rem inducens v●ilen atque honestam primo propter insolentiam reclamantes, deinde propter rationem atque orationem studiosius audientes, ex feris & immanibus mites reddidit, & mansuetos. This long repetition of Tully's words will seem tedious to them which love but little, and care much less for the knowledge of reason, but unto your Majesty (I dare say) it is a delectable remembrance, and unto me it seemed so pleasant, that I could scarce stay my pen from writing all that mine eyes did so greedily read. This sentence of Cicero am I loath to translate into English, partly for that unto your Majesty it needeth no translation, but especially knowing how far that grace of Tully's eloquence doth excel any English man's tongue, and much more exceedeth the baseness of my barbarous style, yet for the fruit of the sentence, I had rather unto my mere English Countrymen utter the rudeness of my translation, than to defraud them of the benefit of so good a lesson, trusting they will also gladly and greedily embrace all good sciences that may help to the just furniture of the same, when they consider that informed reason was the only Instrument, or are least the chief mean to bring men unto Civil regiment, from barbarous manners and beastly conditions. For the time was (saith Tully) that men wandered abroad in the fields up and down like beasts, and used no better order in feeding than they, so that by reason's rule they wrought nothing, but most of their doing did they achieve by force of strength. At this time there was no just regard of religion toward God, nor of duty toward man. No man had seen right use of marriage, neither did any man know their own children from other, nor no man had felt the commodity of just laws: so that through error and ignorance, wilful lust, like a blind and heady ruler, abused bodily strength as a most mortal minister for the satisfying of his desire. At that time was there one, which not only in power, but also in wisdom was great: and he considered, how that in the minds of men was both apt instruments, and great occasion to the due accomplishment of most weighty affairs, if a man could apply them to use, and by teaching of rules frame them to better trade. This man with persuasion of reason, gathered into one place the people that were wandering about the fields, and lay lurking in wild cottages and woods: And bringing them into one common society, did trade them to all such things, as either were profitable or honest, although not without repining at the first, by reason that they had not been so accustomed before. Yet at length through reason and persuasion of words they obeyed him more diligently, and so of a wild and cruel people, he made them courteous and gentle. Thus hath Tully set forth the efficacy of reason and persuasion, how it was able to convert wild people to a mildness, and to change their furious cruelness into gentle cur●●e. Were it not now a great reproach in this our time (when knowledge reigneth so large) that men should show themselves less obsequious to reason? unless it may be thought, that now every man having sufficient knowledge of himself, needeth not to hearken to the persuasion of others. Indeed he that thinketh himself wise, will not esteem the reason of any other, be he never so wise, so that of such one it may well be said, He that thinketh himself wiser than he is, may justly be counted a double fool: wherefore such men are not to be permitted in open audience to talk, but must be put to silence, and made to give ear to reason, which reason consisteth not in a multitude of words, heaped rashly together, and applied for one purpose, but reason is the expressing of a just matter with witty persuasions, furnished with learned knowledge. Such knowledge had Moses being expert in all learning of the Egyptians, as the Scriptures declare, and therefore was able to persuade the stubborn people of the jews, although not without great pain. Druis was son to K. Sarron, & succeeded him in his kingdom. Such knowledge and such reasons did Druys show, which was the first law maker of all the West parts of Europe. Like reason and wisdom did Xamolxis use amongst the Goths: Lycurgus unto the Lacedæmonians, Zeleucus to the Locrians, Solon to the Athenienses, and Donwallo Molmitius two thousand years past, amongst the old Britanes of this realm. And thereby ti came to pass, that their laws continued long, till more perfit reason altered many of them, and wilful power oppressed most of them. At the beginning when these wise men perceived how hard it was to bring the rude people to understand reason, they judged the best means to attain this honest purpose to depend of learning in every kind, for by learning, as Ovid saith: Pectora moiles●unt, asperitasque fugit: Stout stomachs do wax mild, and sharp fierceness is exiled. Therefore as Berosus doth testify, Sarron▪ that was the third King over all this West part of Europe, for to bring the people from beastly rage to manly reason, did erect Schools of liberal Arts which took so good success, that his name continued in that sort famous above two thousand years after: for Diodorus Siculus, which was in the time of julius Caesar maketh mention of the learned men of the Goths, and named them Sarronides, that is to say, Sarron his Scholars and follower's. Among these Arts that then were taught, some did in form the tongue, and make men able both to utter aptly their mind, and also to persuade, as Grammar, Logic, and rhetoric, although not so couriously as in this time: some other did appertain to the just order of partition of Lands, the true using of weights, measures, and reckonings in all sorts of bargains, and for order of building and sundry other uses, those were Arithmerike and Geometry. Again, to encourage men to the honour of God, they taught Astronomy, whereby the wonderful works of God were so manifestly set forth, that no man's tongue nor pen can in like sort express his infinite power, his unspeakable wisdom, and his exceeding goodness toward man, whereby he doth bauntifully provide for man all necessaries, not only to live, but also to live pleasantly. And so was their confidence in God's providence strongly stayed, knowing his goodness to be such, that he would help man as he could, and his power to be so great, that he could do what he would: and thirdly his wisdom to be so pure, that he would do nothing, but that was best. Beside these Sciences they taught also Music, which most commonly they did apply partly to religious Sciences, to draw men to delight therein, & partly to songs made of the manners of men in praise of Virtue, and discommendation of Vice, whereby it came to pass, that no man would displease them, nor do any thing evil that might come to their hearing: This Bardus Druidius the v. king of the Celtes, reigned 69. years, and died 1832. years before Christ. for their Songs did make evil men more abhorred in that time, than any excommunication doth in this time. The posteritle of these musicans continue yet both in Wales and Ireland called Bards unto this day, by the ancient name of Bardus, their first founder. And as these Sciences did increase, so did virtue increase thereby. Again, as these sciences did decay, so virtue lost her estimation, and consequently was little in use: whereof to make a full declaration, were a thing meet for a Prince to hear, but it would require a peculiar treatise. Wherefore at this present I count it sufficient lightly to have touched this matter in general words, and to say no more of the particularity thereof, but only touching one of those sciences, that is, Arithmetic, by which not only just partition of lands was made, but also touching buying an selling, all assizes, weights, and measures were devised, and all reckonings and accounts driven: yea by proportion of it, were the true orders of justice limited, as (Aristotle in his ethics doth declare) & the degrees of estates in the common wealth established. Although that proportion be called Geometrical, and not Arithmetical, yet doth that proportion appertain to the Art of Arithmetic: & in Arithmetic is taught the Progression of such proportions, and all things thereto belonging. Wherefore I may well say, that seeing Arithmetic is so many ways needful unto the first planting of a common wealth, it must needs be as much required to the preservation of it also: for by the same means is any common wealth continued, by which it was erected and established. And if I shall in small matters in appearance, but in deed very weighty, put one example or two, what shall we say for the statutes of this realm, which be the only stay of good order in manner now? As touching the mesuring of ground by length and breadth, there is a good and an ancient statute made by Art of Arithmetic, and now it shallbe to little use, if by the same Art it be not practised and tried. For the assize of bread and drink, the two most common and most necessary things for the sustentation of man, there was a goodly ordinance in the Law made, which by ignorance hath so grown out of knowledge and use, that few men do understand it, and therefore the statute books wonderfully corrupted, & the commons cruelly oppressed: notwithstanding some men have written, that it is too doubtful a matter to execute those Assizes by those statutes, by reason they depend of the standard of the coin, which is much changed from the state of that time when those statutes were made. Thus shall every man read that listeth in the Abridgement of Statutes in the title of Weights and Measures, in the seventh number of the English Book, where he should have translated a good ordinance, which is set forth in the French book: but no marvel if the Abridgement doth omit it, seeing the great book of Statutes doth omit the same Statute, as it hath done divers other very good laws. And this is the fruit of ignorance, to reject all that it understandeth not, although they use some cloaks for it: but such cloaks, as being allowed, might serve to repel all good laws, which God forbidden. Again there is an ancient order for assize of fire wood, and coals, which was renewed not many years past, and now how Avarice and Ignorance doth canvas the statute, it is too pitiful to talk of, and more miserable to feel. Furthermore, for the Statute of Coinage, and the standard thereof, if the people understood rightly the statute, they should not, nor would not (as they often do) gather an excuse for their folly thereby, but as I said, these statutes by wisdom and good knowledge of Arithmetic were made, & by the same must they be continued. And let Ignorance no more meddle with the use of them, than it did with the making of them. Oh in how miserable case is that Realm, where the Ministers and interpreters of the Laws are destitute of all good sciences, which be the keys of the laws? How can they either make good Laws, or maintain them, that lack that true knowledge whereby to judge them? And happy may that realm be accounted where the Prince himself is studious of learning, and desireth to understand equity in all laws. Therefore most happy are we the loving subjects of your Majesty which may see in your highness not only such towardness, but also such knowledge of divers Arts as seldom hath been seen in any Prince of such years, whereby we are enforced to conceive this hope: Certainly, that he which in those years seeketh knowledge when knowledge is least esteemed, and of such an age can discern them, to be enemies both to his royal person, and to his realm, which labour to withdraw him from knowledge to exeessive pastime, and from reasonable study to idle or noisome pleasures, he must needs, when he cometh to more mature years, be a most prudent Prince, a most just Governor, and a right judge, not only of his subjects commonly, but also of the ministers of his laws, yea, and of the Laws themselves. And to be able to conceive the true equity and exact understanding of all his Laws and Statutes, to the comfort of his good subjects, and the confusion and reproach of them which labour to obscure or pervert the equity of the same laws and statutes. How some of those statutes may be applied to use, as well in our time, as in any other time, I have particularly declared in this Book, & some other I have omitted for just considerations, till I may offer them first unto your Majesty, to weigh them, as to your Highness shall seem good: for many things in them are not to be published without your highness knowledge and approbation, namely because in them is declared all the rates of alloyes for all standards from one ounce upward, with other mysteries of mint matters, and also most part of the varieties of coins, that have been currant in this your majesties realm by the space of six hundred years last passed, and many of them that were currant in the time that the Romans ruled here. All which, with the ancient description of England and Ireland and my simple censure of the same, I have almost completed to be exhibited to your Highness. In the mean season most humbly beseeching your Majesty, to accept this simple treatise, not worthy to be presented to so high a Prince, but that my lowly request to your majesty is, that this amongst other of my books may pass under the protection of your highness, whom I beseech God most earnestly and daily, according to my duty, to advance in all honour and princely regality, and to increase in all knowledge, justice, and godly policy. Amen. Your majesties most obedient subject and servant, ROBERT RECORD. TO THE LOVING Reader. The Preface of Master Ro: RECORD. SORE oftentimes have I lamented with myself the infortunate condition of England, seeing so many great Clerks to arise in sundry other parts of the world, and so few to appear in this our nation: whereas for pregnancy of natural wit (I think) few nations do excel English men: But I cannot impute the cause to any other thing than to the contempt or misseregard of learning. For as Englishmen are inferior to no men in mother wit, so they pass all men in vain pleasures, to which they may attain with great pain or labour: and are as slack to any, never so great commodity, if there hang of it any painful study or travelsome labour. Howbeit, yet all men are not of that sort, though the most part be, the more pity it is: but of them that are so glad, not only with painful study and studious pain to attain learning, but also with as great study & pain to communicate their learning to others, and make all England, if it might be, partakers of the same, the most part are such, that uneath they can support their own necessary charges, so that they are not able to bear any charges in doing of that good, that else they desire to do. But a greater cause of lamentation is this: that when learned men have taken pains to do things for the aid of the unlearned, scarce they shall be allowed for their well doing, but derided and scorned, and so utterly discouraged to take in hand any like enterprise again. So that if any be found (as there are some) that do favour learning and learned wits, and can be content to further knowledge, yea only with their word, such persons, though they be rare, yet shall they encourage learned men to enterprise some things, at the least, that England may rejoice of. And I have good hope that England will (after she hath taken some sure taste of learning) not only bring forth more favourers of it, but also such learned men, that she shall be able to compare with any realm in the world. But in the mean season, where so few regarders of learning are, how greatly they are to be esteemed that do favour and further it, my pen will not suffice at full to declare. Therefore, gentle reader, where as I do upon most just occasion judge, yea and know assuredly, that there be some men in this realm, which both love, and also much desire to further good learning, and yet am not well able to write their condign praise for the same, I think it better with silence to overpass it, then either to say too little of it, or to provoke against them the malice of such other, which do nothing themselves that is praise worthy, and therefore can not abide to hear the praise of any other man's good deed. And considering their great favour unto learning, though I myself be not worthy to be reckoned in the number of great learned men, yet am I bold to put myself in press with such ability, as God hath lent me though not with so great cunning as many men, yet with as great affection as any man, to help my countrymen, and will not cease daily, (as much as my small ability will suffer me) to indite some such thing, that shall be to the instruction, though not of learned men. yet at the least of the vulgar sort, whose argument always shall be such, that it shall delight all learned wits, though they do not learn any great things out of it. But to speak of this present Book of Arithmetic, I dare not nor will not set it forth with any words, but remit it to the judgement of all gentle readers, & namely such as love good learning, beseeching them so to esteem it, as it doth seem worthy. And so either to accept the thing for itself, either at the least to allow my good endeavour. But I perceive I need not use any persuasions unto them, whose gentle nature and favourable mind is ready to receive thankfully, and interpret to the best, of all such enterprises attempted for lo good an end, though the thing do not always satisfy men's expectation. This considered, did bolden me to publish abroad this little Book of the Art of numbering, which if you shall receive favourably, you shall encourage me to gratify you hereafter with some greater thing. And as I judge some men of so loving a mind to their native country that they would much rejoice to see it to prosper in good learning and witty Arts, so I hope well of all the rest of Englishmen, that they will not be unmindful of his due praise, by whose means they are helped and furthered in any thing. Neither ought to esteem this thing of so little value, as many men of little discretion oftentimes do: For who so setteth small price by the witty devise and knowledge of numbering, he little considereth it to be the chief point (in manner) whereby men differ from all bruit beasts: for as in all other things, (almost) beasts are partakers with us, so in numbering we differ clean from them, and in manner peculiarly, sith that in many things they excel us again. The Fox in crafty wit exceedeth most men, A dog in smelling hath no man his peer, To foresight of weather if you look then, Many beasts excel man, this is clear. The wittiness of Elephants doth letters attain, But what cunning doth there in the Beeremaine? The Emmet foreseeing the hardness of winter, Provideth vitailer in the time of Summer. The Nightingale, the Lines, the Thrush, the Lark, In Musical harmony pass many a Clerk. The Hedgehog of Astronomy seemeth to know And stoppeth his cave, where the wind doth blow The Spider in weaving such are doth show, No man can him mend, nor follow I trow. When a house will fall, the Mice right quick, Flee thence before, can man do the like? Many things else of the wittiness of beasts & birds might I here say, save that another time I intend to write wherein they excel in manner all men, as it is daily seen: but in number was there never beast found so cunning, that could know or discern one thing from many, as by daily experience you may well consider, when a Bitch hath many whelps, or a Hen many Chickens: and likewise of other whatsoever they be, take from them all their young, saving only one, and you shall perceive plainly, that they miss none, though they will resist you in taking them away, and will seek them again if they may know where they be, but else they will never miss 'em truly, but take away that one that is left, and then will they cry and complain: and restore to them that one, then are they pleased again: so that of number this may I justly say, It is the only thing (almost) that separateth man from beasts. He therefore that shall contemn number, he declareth himself as brutish as a beast, and unworthy to be counted in fellowship of men. But I trust there is no man so foul overseen, though many right smallie do it regard. Therefore will I now stay to write against such, and return again to this book, which I have written in the form of a Dialogue, because I judge that to be the easiest way of instruction, when the Scholar may ask every doubt orderly, and the master may answer to his question plainly. Howbeit I think not the contrary, but as it is easier to blame an other man's work than to make the like, so there will be some that will find fault, because I writ in a Dialogue: but as I conjecture, those shall be such, as do not, cannot, either will not perceive the reason of right teaching and therefore are unmeet to be answered unto, for such men with no reason will be satisfied. And if any man object that other books have been written of Arithmetic already so sufficiently, that I needed not now to put pen to the book, except I will condemn other men's writings: to them I answer. That as I condemn no man's diligence, so I know that no one man can satisfy every man, and therefore like as many do esteem greatly other Books, so I doubt not but some will like this my Book above any other English Arithmetic hitherto wrtiten, and namely such as shall lack instructors, for whose sake I have so plainly set forth the examples, as no Book (that I have seen) hath done hitherto: which thing shall be great ease to the rude readers. Therefore gentle reader, though this book can be small aid to the learned sort, yet unto the simple ignorant (which needeth most help) it may be a good furtherance and mean unto knowledge. And though unto the King his Majesty privately I do it dedicate, yet I doubt not (such is his clemency) but that he can be content, yea and much desirous, that all his loving subjects shall take the use of it, and employ the same to their most profit: Which thing if I perceive that they thankfully do, and receive with as good will as it was written, then will I shortly with no less kindness set forth such introductions in to Geometry and Cosmography, as I have at other times promised, and as hitherto in English hath not been enterprised, wherewith I dare say all honest hearts will be pleased, and all studious wits greatly delighted. I will say no more▪ but let every man judge as he shall see cause. And thus for this time I will stay my pen, committing you all to that true fountain of perfect number, which wrought the whole world by number and measure: he is Trinity in Unity, and Unity in Trinity: To whom be all praise, honour and glory. AMEN. Here followeth a Table of all the Contents of this Book. The Contents of the first Dialogue containeth the Declaration of the profit of Arithmetic. Numeration with an easy & large Table. Addition. Subtraction. Multiplication. Division. with divers Examples, and all their proofs, and some new forms of workings, etc. Reduction, with divers declarations of Coins Weights and Measures of sundry forms newly added, with a new Table, containing most part of the gold Coins throughout Christendom, with the true weight and valuation of them in currant money English, etc. Progression both Arithmetical and Geometrical, with divers sundry questions touching the same. The Golden Rule of three: and the Backer Rule of three: with divers questions thereunto belonging, newly added & augmented. The double Rule of Proportion. The Rule of three composed of 5 numbers. The Rule of Fellowship, both with time, and without time. Unto all these are added their proofs. The second Dialogue containeth The first 5 kinds of Arithmetic wrought by Counters. The common kinds of casting of accounts after the Merchant's fashion, & Auditors also. Numbering by the hand newly added. The Contents of the second part, touching Fractions. What a Fraction is. Numeration in Fractions. The order of working fractions. with divers familiar questions proponed for the perfect understanding, & proof of each of them. Multiplication. Division. Reduction of divers fractions into one denomination in 3 varieties. Fractions of Fractions. Improper Fractions. Fractions to the smallest denomination, with easy rules how to convert them thereunto. Fractions in other parts of things, with a Table demonstrative of their proportions. Fraction, and how it may be turned into any other Fraction, or into what Denomination you list. Again of Multiplication Duplation Division Mediation Addition Subtraction. The Golden Rule with divers questions, and their proofs. The Backer Rule. A question of Loane. The statute of Assize of Bread and Ale recognized and applied to this time, with new tables thereunto annexed. The Statute of Measuring of ground, with a table thereof faithfully calculated and corrected. Questions of Society, with the reason of the Rules and proofs of their works. To find three numbers in any proportion. The Rule of Alligation, with divers questions and the proofs of their works, with many varieties of such solutions. The rule of Falsehood, or false Position, with divers questions, and their proofs. The Contents of the third Addition to this Book. The first Chapter entreateth of Rules of Brevity and Practice, after a briefer Method than ever yet was published in the English tongue. The second Chapter treateth of the briefer Reduction of divers Measures, as , yards, Braces, etc. by Rules of Practice. The third Chapter entreateth of the Rule of three in Broken numbers after the trade of merchants, something differing from Master Records order, which is comprehended in 3 Rules. The fourth Chapter entreateth of Loss and Gain in the trade of Merchandise. The fifth Chapter entreateth of Loss & Gain in the trade of Merchandise upon time, etc. with necessary questions therein wrought by the double Rule of three, or the Rule of 3 composed. The sixth Chapter entreateth of Rules of payment, and of the necessariest Rules that appertaineth to buying and selling, etc. The seventh Chapter entreateth of Buying and Selling in the Trade of Merchandise, wherein is taken part ready money, and divers days of payments given for the rest, and what is won or lost in the 100 lb forbearance for 12 months, etc. The eight Chapter entreateth of tars and allowances in the trade of Merchandise sold by weight, and of their Losses and Gains therein, etc. The ninth Chapter entreateth of Lengths and Breadths of Arras, and other Clothes, with divers questions incident thereunto. The tenth Chapter entreateth of reducing of Pawns of Geanes into English yards. The eleventh Chapter entreateth of Rules of Loane and Interest with divers questions incident thereunto. The twelfth Chapter entreateth of the making of Factors. The thirteenth Chapter entreateth of Rules of Barter or Exchange of Merchandise, wherein is taken part ware, & part ready money with their proofs, and divers other necessary questions thereunto belonging. The fourteenth Chapter entreateth of exchanging of money from one place to an other, with divers necessary questions incident thereunto. The fiftéeeths Chapter entreateth of six sundry forms of practice for the Reduction of English, Flemish, and French money, and how each of them may easily be brought to others money sterling. The sixteenth Chapter containeth a brief note of the ordinary Coins of most places of Christendom for traffic, and the manner of their exchanging from one City or town to an other, which known the Italians call Pary: whereby they find the gain or loss upon the Exchange. The seventeenth Chapter containeth also a Declaration of the diversity of the weights and measures of most places of Christendom for traffic, at the end whereof are two Tables, the one for weight, and the other for measure, proportionated to an equality unto our English measure and weight, whereby the ingenious practitioner may easily reduce the weight and measure of each Country into other. The eighteenth Chapter entreateth of divers Sports and Pastimes, done by Number. FINIS. A Collection of such Tables as are contained in this Treatise. A large Table of Numeration. A Table of Multiplication. A Table of Division. A Table of the money currant in this Realm when the Author first published this book. A Table of all the usual silver Coins now currant in this Realm, newly added. A Table of all the gold Coins in this realm with all the most usual Gold Coins throughout Christendom, with their several weights of Pence and Grains, and what they are worth in currant money English. Certain Tables or Notes of the contents of Ale, Beer, Wine, Butter, Soap, Salmon, eels, etc. both what such vessels ought to contain by the Statute, and what those vessels empty ought to weigh. A Table of the quantity of dry measures, as Pecks, Bushels, Quarters, ways, etc. A Table of the proportion of measure, touching Lengths or breadths: to wit, from the inch to the foot, and so to the yard, the Ell with their parts: the perch, the rod- the furlong, the mile, etc. A Table made by Progression Arithmetical, which containeth a double table of Multiplication. A Table of the Art of Numbering by the hand. A Table or demonstration of a figure or measure for the perfect understanding of Fractions of Fractions. A Table of the contents of the Statute for the assize of the weight of bread. From 1 s the quarter to 20 s faithfully corrected, and amended. A necessary Table of the Statute of measuring of ground, upon the breadth given, what length it ought to contain: faithfully corrected according to the equity of the statute: wherein the Author declareth how necessary this worthy Art of Arithmetic is unto Gentlemen Students of the law, and such other as are desirous of infallible truth. Brief Tables of the ready reducing of English, French, and Flemish money, each into others common currant moneys. A brief Table or collection of the common & usual moneys of most places of Christendom for traffic, the manner of their payments or exchanging from one City or town, to an other: right necessary for Merchants, and other Occupiers, Traveluellers, etc. Tables of the Weights, Measures, and Customs of most places of Europe for traffic. 2 Tables, the one for weight, the other for measure, reduced to an equality, unto our Measures and weight here at London, with the help of which 2 Tables, and the aid of the rule of 3, the ingenious may easily reduce our measure to the perfect valuation of other Country's measure or weight, and likewise theirs to ours. Lastly, a Table demonstrating the true solution of three divers things hidden of 3 several persons in pastime. A dialogue between THE MASTER and the Scholar, teaching the Art and use of Arithmetic with pen. THE SCHOLAR SPEAKETH. SIR, SUCH is your authority in mine estimation, that I am content to consent to your saying, and to receive it as truth, though I see none other reason that doth lead me there unto: whereas else in mine own conceit it appeareth but vain, to bestow any time privately in learning of that thing, that every child may and doth learn at all times and hours, when he doth any thing himself alone, and much more when he talketh or reasoneth with other. Master. Lo this is the fashion and chance of all them that seek to defend their blind ignorance: that when they think they have made strong reason for themself, than have they proved quite contrary. For if numbering be so common (as you grant it to be) that no man can do any thing alone, and much less talk or bargain with other, but he shall still have to do with Number: this proveth not Number to be contemptible and vile, but rather right excellent and of high reputation, sith it is the ground of all men's affairs, so that without it no tale can be told, no communication without it can be long continued, no bargaining without it can duly be ended, or no business that man hath, justly completed. These commodities (if there were none other) are sufficient to approve the worthiness of Number. But there are other unnumerable far passing all these, which declare Number to exceed all praise. Wherefore, in all great works are Clerks so much desired? Wherefore are Auditors so richly fed? What causeth Geometrians so highly to be enhanced? Why are Astronomers so greatly advanced? Because that by Number such things they do find, which else should far excel man's mind. Scholar. verily Sir if it be so, that these men, by numbering their cunning do attain, at whose great works most men do wonder, than I see well I was much deceived, and numbering is a more cunning thing than I took it to be. Master. If Number were so vile a thing as you did esteem it, then need it not to be used so much in men's communication. Exclude Number and answer to this question. How many years old are you? Scholar. Mum. Master. How many days in a week? how many weeks in a year? What lands hath your father? How many men doth he keep? How long is it sith you came from him to me? Scholar. Mum. Master. So that if Number want, you answer all by Mummes: How many mile to London? School. A pooke full of Plums. Master. Why, thus you may see, what rule Number beareth, and that if Number be lacking, it maketh men dumb, so that to most questions, they must answer Mum. Scholar. This is the cause sir, that I judged it so vile, because it is so common in talking every while: For plenty is not dainty, as the common saying is. Master. No, nor Store is no sore: perceive you this? The more common that a thing is, being néedefully required, the better is the thing, and the more to be desired. But in Numbering as some of it is light and plain, so the most part is difficult, and not easy to attain. The easier part serveth all men in common, and the other part requireth some learning. Wherefore as without Numbering a man can do almost nothing, so with the help of it, you may attain to all things. Sc. Yea sir? Why? then it were best to learn the Art of Numbering first of all other learning, and then a man need learn no more, if all other come with it. Master. Nay not so: but if it be first learned, then shall a man be able (I mean) to learn, perceive, and attain to other sciences, which without it, he should never get. Scholar. I perceive by your former words, that Astronomy and Geometry depend much of the help of Numbering, but that other Sciences, as Music, Physic, Law, and Grammar and such like, have any help of Arithmetic, I perceive not. Master. I may perceive your great Clerkelinesse by the ordering of your Sciences: but I will let that pass now, because it toucheth not the matter that I intend, and I will show you how Arithmetic doth profit in all these, somewhat grossly, according to your small understanding, omitting other reasons more substantial. first (as you reckon them) Music hath not only great help of Arithmetic, but is made, and hath his perfectness of it: for all Music standeth by Number & Proportion. And in Physic, beside the calculation of Critical days, with other things which I omit, how can any man judge the Pulse rightly, that is ignorant of the proportion of Numbers? And as for the Law, it is plain, that the man that is ignorant of Arithmetic, is neither meet to be a judge, neither an Advocate, nor yet a Proctor. For how can he well understand another man's cause appertaining to distribution of goods, or other debts, or of sums of money, if he be ignorant of Arithmetic? This oftentimes causeth right to be hindered, when the judge either delighteth not to hear of a matter that he peceyveth not, or cannot judge it for lack of understanding: This cometh by the ignorance of Arithmetic. Now as for Grammar, me thinketh you should not doubt in what it needeth number, sith you have learned that Nouns of all sorts, Pronouns, Verbs, & Participles, are distinct diversly by Numbers: besides the variety of Nouns of Numbered, and Adverbs. And if you take away Number from Grammar, then is all the quantity of Syllables lost. And many other ways doth Number help Grammar. Whereby were all kinds of metres found and made? Was it not by Number? But how needful Arithmetic is to all parts of Philosophy, they may soon see, that do read either Aristotle, Plato, or any other Philosopher's writing. For all their examples almost, and their probations, depend of Arithmetic. It is the saying of Aristotle, that he that is ignorant of Arithmetic is meet for no science. And Plato his Master wrote a like sentence over his schoolhouse door. Let none enter in hither (quoth he) that is ignorant of Geometry. Seeing he would have all his Scholars expert in Geometry, much rather he woulds the same in Arithmetic, without which Geometry cannot stand. And how needful Arithmetic is to Divinity, it appeareth, seeing so many Doctors gather so great mysteries out of Number, and so much do write of it. And if I should go about to write all the commodities of Arithmetic in civil acts, as in governance of common weals in time of peace, and in due provision and order of armies in time of war: For numbering of the ho●̄e, summing of their wages, provisions of victuals, viewing of Artillery, with other armour: Beside the cunningest point of all, for casting of ground, for encamping of men, with such other like. And how many ways also Arithmetic is conducible for all private weals, of Lords and all possessioners, of merchants, and all other occupiers, and generally, for all estates of men, besides Auditors, treasurers, receivers, stewards, bailiffs, and such like, whose offices without Arithmetic is nothing. If I should (I say) particularly repeat all such commodities of this noble Science of Arithmetic, it were enough to make a very great book. Scholar. No, no sir, you shall not need: For I doubt not, but this that you have said, were enough to persuade any man to think this Art to be right excellent and good, and so necessary for man, that (as I think now) so much as a man lacketh of it, so much he lacketh of his sense and wit. Master. What? are you so far changed since, by hearing the few commodities in general? By likelihood you would be far changed, if you knew all the commodities particular. Scholar. I beseech you sir, reserve those commodities that rest yet behind, unto their place more convenient. And if ye will be so good as to utter at this time this excellent treasure, so that I may be somewhat enriched thereby, and if ever I shall be able, I will requite your pain. Master, I am very glad of your request, and I will do it speedily, sith that to learn it you be so ready. Scholar. And I to your authority my wit do subdue, whatsoever you say, I take it for true. Master. That is too much, and meet for no man to be believed in all things, without showing of reason. Though I might of my Scholar some credence require, yet except I show reason, I do it not desire. But now sith you are so earnestly set this Art to attain, best it is to omit no time, lest some other passion, cool this great heat, and then you leave off before you see the end. Scholar. Though many there be so unconstant of mind, that flitter and turn with every wind, which often begin, and never come to the end, I am none of their sort, as I trust you partly know. For by my good will what I on's begin till I have it fully ended I would never blind Master. So have I Found you hitherto in deed and I trust you will increase rather than go back. For better it were never to assay, than to shrink and flee in the middle way. But I trust you will not so do, therefore tell me briefly. What call you the science that you desiere so greatly? Scholar Why sir? you knowe. Master. That maketh no matter. I would hear whether you know, and therefore I ask you. For, great rebuke it were to have studied a Science, and yet cannot tell how it is named, Scholar. Some call it Arsmetrik, and some Augrime. Master. And what doth those names betoken? Scholar. That if it please you, of you would I learn. Master. Both names are coruptly written, Arsmetrike for Arithmetic, as the Greeks call it, & Augrim for algorithm, as the Arabians sound it, which both betoken the science of Numbering. For Arithmos in Greek, is called number: and of it cometh Arithmetic, the Art of Numbering. So that Arithmetic is a science or art teaching the manner and use of Numbering. This Art may be wrought diversly, with Pen or with Counters. But I will first show you the working with the Pen, and then the other in order. Scholar. This I will remember. But how many things are to be learned, to attain this Art fully? Master. There are reckoned commonly seven parts or works of it. Numeration, Addition, Subtraction, Multiplication Division, Progression, and extraction of roots: to these some men add Duplation, Triplation, and Mediation. But as for these last three, they are contained under the other seven. For Duplation and Triplation, are contained under Multiplication, as it shall appear in their place. And Mediation is contained under Division, as I will declare in his place also. Scholar. Yet then there remain the first seven kinds of Numbering. Master. So there doth: Howbeit, if I shall speak exactelye of parts of Numbering I must make but five of them: For Progression is a compound operation of Addition, Multiplication and Division. And so is the extraction of roots. But it is no harm to name them as kinds several, seeing they appear to have some several working. For it forceth not so much to contend for the number of them, as for the due knowledge and practising of them. Scholar. Then you will, that I shall name them as seven kinds distinct. But now I desire you to instruct me in the use of each of them. Master. So will I, but it must be done in order: for you may not learn the last as soon as the first, but you must learn them in that order, as I did rehearse them, if you will learn them speedily and well. Scholar, Even as you please. Then to begin, Numeration is the first in order: what shall I do with it? Master. first you must know what the thing is, and then after learn the use of the same. NUMERATION. NUMERATION is that Arithmetical skill, whereby we may duly value, express and read any number or sum propounded: or else in apt figures and places, set down any number known or named. Scholar. Why? then me thinketh you put a difference vetwéene the value and the Figures? Master. Yea so do I: For the value is one thing, and the figures are an other thing: and that cometh partly by the diversity of figures, but chiefly of the places wherein they be set. Scholar. Then I must know there three things: the Value, the Figure, and the Place. Master. Even so: but yet add Order to them as the fourth. And first mark, that there are but ten figures, that are used in Arithmetic: and of those ten, one doth signify nothing, which is made like an 0, and is called privately a Cipher, though all the other sometime he likewise named. The other nine are called Signifying figures, and be thus figured. 1 2 3 4 5 6 7 8 9 And this is their value. j ij.iij.iiij.u.uj.vij.viij.ix. But here must you mark, that every Figure hath two values: One always certain that it signifieth properly, which it hath of his form: and the other uncertain, which he taketh of his place. A Place is called the seat or room that a Figure standeth in. And look how many Figures are written in one sum, so many places hath that whole number. And the first place must be called that that is next to the right hand, and so reckoning by order towards the left hand, so that that place is last, that is next to the left hand. As for example: If there stood before you six men in a row, side by side, and you should tell them as they stand in order, beginning with the man that were next to your right hand: then he that were next him should be called the second, and so forth to the farthest from your right hand, which is the sixth and the last. Scholar. Sir, I perceive you well: so might I reckon letters or any other thing. As if I should write eight letters after this order, a, b, c, d, e, f, g, h, now must I say h, is the first g the ij, f the iij, e the iiij, d the v, c the uj, b the seven, and a the viii. Master. That is well done. And after the same sort use hereafter, that what I declare by one example, do you express by an other, and so I shall perceive whether you understand it or no. And so pass over nothing, till you perceive it well, and be expert therein. Scholar. Sir, I pray you how many of these places be there in all? Master. There is no certain number of them, but they are sometimes more and sometimes fewer, according to the sum that is expressed. For so many as the figures are, so many are the places: and the last place is so called, not because it is last of all other, but it is the last of that present sum, and it may be the middle place in an other sum. Scholar. Me seemeth I perceive this very well, as touching the order of reckoning of the places: But as for the number of them, you say there is no certainty. Now there resteth to declare the value of the figures by diversity of places, which you called, the Value uncertain. Master. But first let me hear whether you know perfectly the certain value. Scholar. Yes sir, as you wrote them, so I marked them. Master. How writ you then five? Scholar. By this figure 5. Master. And how six? Scholar. Thus, 6. Master. Writ these three numbers each by itself as I speak them. vij.iiij.iij. Scholar. 7.4.3. Master. How writ you these four other, ij, j, ix, viii? Scholar. Thus (I trow,) 2, 1, 6, 8. Master. Nay, there you miss: Look on mine example again. Scholar. Sir, truth it is, I was too blame, I took 6 for 9, but I will be warer héreafter. Master. Now then take heed, these certain values every figure representeth, when it is alone written without other Figures joined to him. And also when it is in the first place, though many other do follow: as for example: This figure 9 is ix. standing now alone. Scholar. How? is he alone and standeth in the middle of so many letters? Master. The letters are none of his fellows. For if you were in France in the middle of a M. French men, if there were no English man with you, you would reckon yourself to be alone. Scholar, So it is. Then 9 without more figures of Arithmetic, betokeneth ix, whatsoever other letters be about it. Master. Even so, and so doth it, if it be in the first place joined with other, how many soever do follow, as in this example, 3679. you see 9 in the first place, and doth betoken nine, as if he were alone. Scholar. I perceive that. And doth not 7 that standeth in the second place, betook seven? and 6 in the third place, betoken uj? And so 3 in the fourth place, betoken three? Master. Their places be as you have said, but their values are not so. For as in the first place, every figure betokeneth his own value certain only, so in the second place every figure betokeneth his own value certain ten times: as in the example, 7 in the second place is seven times x, that is, lxx. And in the third place, every figure betokeneth his own value a hundredth times, so that 6 in that place betokeneth vj. C. And in the fourth place, every figure betokeneth his own value a M. times, as in the foresaid number 3. in the fourth place, standeth for●. M. And in the fi●th place, every figure standeth for his own value x. M. times. And in the uj place a C.M. times. And in the v●. place a M.M. times: And in the viii. place x. M.M. so that every place exceedeth the former x. times. Scholar. As thus: if I make th●s Number at all adventures, 9●359●84, here are eight places. In the first place is 4, and betokeneth but four: in the second place is 8, and betokeneth x. times 8, that is, 80: In the third place is 6, and betokeneth six hundredth: In the 4. fourth place 9 is nine thousand. And ● in the fifth place is x. M. times 5, that is, fifty M. So ● in the sixth place, is a C.M. times ●, that is CCC.M Them in the seventh place, a M.M. And 9 in the eight place ten thousand thousand times 9, that is xc. M.M. But now I can not easily nor quickly read it in order. Master, That shall you practise by this means. First put a prick over the fourth figure, and so over the seventh. And (if you have so many) over the tenth, thirteenth, sixteenth, and so forth, still leaving two figures between each two pricks. And those two rooms between the pricks, are called ternaries. Then begin at the last prick, and see how many figures are between him and the end, which can not pass three, reckoning himself for one: then pronounce them as if they were written alone from the rest, and add at the end of their value so many times thousand as your number hath pricks. After that come to the next three figures, and sound them as if they were apart from the rest, and add to their value so many tunes thousands, and there are pricks béetwene them, and the first place of your whole number. And so do by every other three Figures following, if you have more. As in example, 91359684. this was your number. Put a prick over in the fourth place, and over in the seventh place, and then no more, (for your places come not to ten) as thus: 91359684. Now go to the last prick over 1, and take it and the figure 9 that followeth it, and value them alone. Scholar. 91 that is xcj Master. So is it: but then add for the number of your pricks twice M. Scholar. That is xcj, thousand thousand. Master. So is it. Then take the three other figures from one to the next prick, and value them. Scholar. 359. that is CCC.lix. Master. Now add for the one prick▪ that is between them and the first place, M. Scholar. CCC.lix. thousand. Master. Then come to the other three f●gures that remain. Scholar. 684. that is, vj. C.lxxxiiii. Master. Now have you valued all. And at the end of the last number you shall add nothing, because there remaineth no prick nor number after it: yet prove in an other number, as thus, 2 3 0 8 6 4 0 8 9 10 5 3 4 0. Scholar. 2 3 0 8 6 4 0 8 9 1 0 5 3 4 0. I have pricked them as you taught me: but I am in doubt, whether I have done well or no, because of the Cyphers: for I remember, you told me that they do signify nothing, and therefore I doubt whether I should reckon them for a figure in setting of the pricks: and again, I know not wherefore they serve. Master. That will I tell you now. Indeed they are of no value themselves, but they serve to make up number of places, and so maketh the figure following them to be in a further place, and therefore to signify the more value: as in this example 90, the Ciphre is of no value, but yet he occupieth the first place, and causeth 9 to be in the second place, and so to signify ten times 9, that is, xc, so that two Ciphres thrusteth the figure following them, into the third place, and so forth. Scholar. Then I perceive in the example above I have pricked well enough: for though that Ciphre that is pricked signify nothing, yet must he have the prick, because he came in the xiii place. Then will I prove to number that sum. First there is 230 M.M.M.M. and then followeth 864 M.M.M. And what shall I now do? There is a Cipher in the third place, and no figure after him, but they that have reckoned. Master. He did serve for them that you have already reckoned, to make them in a place further than they should be if he were away: and therefore now you shall let him go. And so do always when he occupieth that place next before any prick, which is the last of that ternary, and a Cipher in the last place doth nothing. Scholar. Then shall I say but 89 M.M. Master. So, but go forth. Scholar. 105 thousand. Now are all my pricks spent, and yet remain 340, so that I must value them CCC.xl. only. Master. Now can you reckon after this sort: and remember, that every such room so parted, is called a ternary or Trinity. Some do part such great numbers with letters, after this manner. 2 c 3 b 0 a 8 b 6 c 4 a 0 b 8 c 9 a 1 b 0 c 5 a 3 b 4 c 0 a. In which example ye may see, that a supplieth the room of your prick. And some do part the numbers with lines after this form. . where you see as many lines as you made pricks, and is one intent, save that the lines do more plainly part every three figures, according as they should be valued under one Denomination. Scholar. Yea sir, but if you should show me a number so parted, I should take it for many numbers, and not for one. Master So might you do, not knowing my meaning. But what if I did set forth the number without lines, and yourself (for the ease of reckoning) did so part it with lines, would you forget wherefore ye did it, and then take them for many numbers? Scholar. No I trow not, but yet I doubt. Master. Thou use that that you like best, for all the three ways are to one intent, save (as I said) that the lines us more plainly distinct the denominations. Scholar. What call you Denominations? Master. It is the last value or name added to any sum. As when I say: CC.xxii. pounds: pounds is the Denomination. And likewise in saying: as men, men is the Denomination, and so of other. But in this place (that I spoke of before) the last number of every ternary, is the Denomination of it. As of the first ternary, the denomination is unites, and of the second. ternary, the Denomination is thousands: and of the third ternary, thousand thousands, or millions: of the iiij. thousand thousand thousands, or thousand Millions: and soforth. Scholar. And what shall I call the value of the iij. figures that may be pronounced before the Denominators? as in saying: 203 000000, that is CCiij. millions. I perceive by your words, that millions is the Denomination: but what shall I call the CCiij. joined before the Millions? Master. That is called the Numerator or valewer, and the whole sum that resulteth of them both, is called the Sum, value or number. Scholar. Now is there any thing else to be learned in Numeration? or else have I learned it sully? Master. I might here show you who were the first inventors of this Art, and the reasons of all these things that I have taught you, but that will I reserve till ye have learned over all the practice of this Art, least I should trouble your wit, with over many things at the first. But yet this must you mark, that there are three kinds of number: one called Digits, an other articles, and the third mixed numbers. A Digit is any number under 10, as this: 1, 2, 3, 4, 5, 6, 7, 8, 9 And 10 with all other that may be divided into ten parts just, and nothing remain, are called Articles: such are 10, 20, 30, 40, 50, etc. 100, 200, etc. 1000 etc. And that number is called mixed, that containeth Articles, or at the least one article, and a digit: as 12, 16, 19, 21, 38, 107, 1005. and so forth. And for the more ease of understanding and remembrance mark this: The diget number is never written with more than one figure, but the article and the mixed number are ever written with more than one figure. And thus they differ, that the article hath evermore this Cipher c, in the first place: and the mixed number hath ever there some Diget. Scholar. By these last words, I perceive it much better than I did before, and now (I think) I will never miss to know those three asunder. Master. If you remember now all that I have said, you have learned sufficiently this first kind of Arithmetic, called Numeration, Howbeit, I will yet exhort you now, to remember both this that I have said, and all that I shall say, and to exercise yourself in the practice of it: For Rules without practice, are but a light knowledge: and practise it is, that maketh men perfect and prompt in all things. And as you have learned to gather and express the value of a sum propounded, and set down before you: so must you practise to mark note, or writ down, with apt figures, and in due places, any number, only named or recited to you, or of yourself imagined: as for a proof: How note you, or writ down this sum, five thousand, two hundredth, fifty and seven. Scholar. This troubleth me now, whether I should begin at the first figure or at the last. For reason (me thinketh) should cause me to beginnne at the first: and yet if I writ it as you speak it, I must begin at the last. Master. When you know your places perfectly, you may begin where you list. But the more ease for your hand is to begin with the last, that is to say, as I did speak them. Yet for the more surety, a while you may begin with the first, repeating my words backward thus: Seven, Fifty, two hundredth, five thousand: or else sounding them all by their diget or valewer, as thus: seven, five, two, five: for that way is easiest. But then must you look well, whether there be any Ciphre in your sum that he may be set in his place. As if your last valewer of your sum (as you speak it) be above 9, then is there a Cipher in the first place. And if it be a hundred or above, then is there two Cyphers one in the first place, & an other in the second, and so forth. But because this thing is such that cannot be set forth without many words, I think best here now at the end of Numeration to add a table easy and ready for the first exercise of it. Lo, this is the Table. The left side or hand. The names of the digits, values certain, or valewers. The denominatours of the place or value uncertain Nine. Eight. Seven. Six. Five. Four. Three. Two. One. Ciphre. The order of the places. Unites. 9 8 7 6 5 4 3 2 1 0 First. tennis. 9 8 7 6 5 4 3 2 1 0 Second. Hundreds. 9 8 7 6 5 4 3 2 1 0 Third. thousands. 9 8 7 6 5 4 3 2 1 0 Fourth. x. thousands. 9 8 7 6 5 4 3 2 1 0 fift. C. of thousands. 9 8 7 6 5 4 3 2 1 0 Sixte. Millions. 9 8 7 6 5 4 3 2 1 0 Seventh. x. of millions. 9 8 7 6 5 4 3 2 1 0 Eighth. C. of millions. 9 8 7 6 5 4 3 2 1 0 Ninth. M. of millions. 9 8 7 6 5 4 3 2 1 0 Tenth x. M. of millinos. 9 8 7 6 5 4 3 2 1 0 Eleventh. The 〈…〉 This Table (as you may see) hath eleven places, and in each of them are set all the digites, whose certain value is written in the right hand of the Table, & the value uncertain on the left hand. So that by this table you may learn both how to express any number that you list, (if that it exceed not eleven places) that is to say, lxxxx. thousand Millions, and so may you by the help of it, value all sums proposed under the said number. For example: take the sum that I proposed before, which was five thousand, two hundred, fifty and seven. And if you will express it, take the first number (as I speak it) which is five M. whose valuer or certain value is v. and his uncertain value or denomination is M. First you shall seek at the right hand of the valuer v. Then seek along under the title of Denomination toward the left hand, till you find thousands, and under it right at the soot of the Table, is the number of the place, that is the fourth, wherein you must write your diget or valuer five. afterward come to the second part of the number, two hundred, whose valuer is 2, and his denomination C. Seek two at the right hand of the Table, and go along under the denominations toward the left hand, till you come under C: then look to the table, and there shall you see the number of the place, that is to say, three, wherein you must set your diget 2. Then do so by your other two numbers that remain, and you shall find five in the second place for your fifty, and 7 in the first place for your seven. And thus may you do with other numbers. Scholar. Master I thank you heartily. I perceive you seek to instruct me most plainly and briefly, and not to hide your knowledge with subtle words as many do. For this rule is so plain, that I can desire it no plainer. And though it seem somewhat long, yet I perceive it to be a sure way. Master. So is it, and though it be long, yet it is neither too long, neither too plain for young learners that lack practise: for this table is in stead of a teacher, to them that lack one. But now I trust I have said enough of Numeration: which after you have well practised, then may you learn forth. Scholar. Yet I pray you in one thing to tell me your judgement. Why do men reckon the order of the places backward, from the right hand to the left? Master. In that thing all men do agree, that the Chaldeyes, ☜ which first invented this Art did set these figures as they set all their letters: for they write backward as you term it, and so do they read. And that may appear in all Hebrew, Chaldeye, and Arabike books, for they be not only written from the right hand to the left, and so must be read, but also the right end of the book is the beginning of it: whereas the Greeks latins, and all nations of Europe, do write & read from the left hand toward the right: And all their books begin at the left side. Scholar. That reason doth satisfy me. Master. It neither sact●fieth me, neither liketh me well, because I see that the Chaldeys and hebrews do not so use their own numbers, as at another time I will declare. But this plain reason may best satisfy you presently: That seeing in pronouncing of numbers we keep the order of our own reading, from the left hand to the right: And again, we do ever name the greater numbers before the smaller: it was reason, that the lesser places containing the lesser numbers, should be set on the right hand, and the greater places containing the greater numbers, to proceed toward the left hand. Scholar. This reason is to me so plain, that it seemeth now against reason to make a doubt of that order. So that now for Numeration I am satisfied: so that only practice shall make me fully ready and expert in it. And in the mean season, I desire to learn the other kinds of Arithmetic. Master. That is well said: but what should you next learn can you tell? Scholar. I remember you said that Addition was next. Master. Even so, and what that is must you first know, ADDITION. Addition is the gathering together and bringing of two numbers or more into one total sum: as if I have 160 Books in the Latin tongue, and 136 in the Greek tongue, and would know how many they be in all, I must write these two numbers one over an other, writing the greatest number highest, so that the first figure of the one, be under the first figure of the other. And the second under the second, and so forth in order. When you have so done, draw under them a right line, then will they stand thus. Now begin at the first places, toward the right hand always, and put together the two first figures of those two numbers, and look what cometh of them, writ under them, right under the line. As in saying, 6 and 0, is 6. Writ 6 under 6: as thus. And then go to the second figures, and do likewise: as in saying, 3 and 6 is 9: writ 9 under 6 and 3, as here you see. And like wise do you with the figures that be in the third place, saying: 1 and 1 be 2: write ● under them, and then will your whole sum appear thus. So that now you see, that 160, and 136 do make in all, 296. Scholar. What? this is very easy to do, me thinketh I can do it even sith. There came through Cheapside two droves of cattle: in the first was 848 sheep, and in the second was 186 other beasts. Those two sums I must write as you taught me, thus. Then if I put the two first figures together, saying: 6 and 8 they make 14. That must I write under 6 and 8, thus. Master. Not so, and here are you twice deceived. First, in going about to add together two sums of sundry things, which you ought not to do, except you seek onelis the number of them, & care not for the things. For the sum that should result of that addition, should be a sum neither of sheep, nor other beasts, but a confused sum of both. Howbeit sometimes ye shall have sums of divers denominations to be added, of which I will tell you anon: but first I will show you, where you were deceived in an other point, and that was in writing 14, (which came of 6 and 8) under 6 and 8, which is unpossible. For, how can two figures of two places be written under one figure, and one place? Scholar. Truth it is: but yet I did so understand you. Master. I said indeed, that you should write that under them, that did result of them both together: which saying is always true, if that sum do not exceed a Digit. But if it be a mixed number, then must you write the Digit of it under your figures, as I have said before: but and if it be an Article, then writ 0 under them, and in both sorts you shall keep the article in your mind. And therefore when you have added your second figures, which occupy the place of tens, you shall put that 1 thereto, which you kept in your mind: for though it were ten indeed, yet in that place it is but as one, because, that every 1 of that place, is ten, for it is the place of tens. And in like manner: if you have in the second place so great a number, that it amounteth above 9, then writ the digite, and reserve the article in your mind, ever adding it to the next place following: and so of all other places, how many so ever you have. And if you have a mixed number, when you have added your last figures, then writ the digit under the last figures, and the article in the next place beyond them: so shall your number resulting of Addition, have one place more than the numbers which you should add together. Scholar. Now do I perceive you, and the reason of this is, (as I understand) because that no one place can contain above 9, which is the greatest figure that is, and then all tens or articles must be put to the next place following: for every place (as I may see) exceedeth the other place next before him, by 10. Now (if it please you) I will return to my example of cattle. But I remember you said, I might not add sums of sundry things together, and that might I see by reason. Master. Truth it is, if you seek the due sum of any things, but if you only seek a bare sum, & have no respect to the thing, than were it better to name the sum only without any thing, as in saying 848, without naming sheep, or any thing else. And likewise 186, naming nothing. Now let me see: how can you add those two sums? Scholar. I must first set them so, that the two first figures stand one over an other, and the other each one over his fellow of the same place: then shall I draw a line under them both. And so likewise of other figures, setting always the greatest number highest, thus, as followeth: Then must I add 6 to 8, which maketh 14, that is mixed number: therefore must I take the diget which is 4, and write it under 6 & 8 keeping the article 1 in my mind thus. next that do I come to the second figures, adding them together, saying, 8 and 4, make 12, to which I put the 1 reserved in my mind, and that maketh 13, of which number I writ the diget 3 under 8, and 4, & keep the article in my mind thus: Then come I to the third figures, saying: 1 and 8, ● make 9, and 1 in my mind maketh 10. Sir, shall I write the cipher under 1 and 8? Master Yea. Scholar. Then of 10 I writ the cipher under 1 and 8, and keep the article in my mind. Master. What needeth that, seeing there followeth no more figures? Scholar. Sir, I had forgotten, but I will remember better hereafter. Then seeing I am come to the last figures, I must write the cipher under them, and the article in a further place after the cipher, thus: Master. So now ye see, that of 848, and ●86 added together, there amounteth 1034. Scholar. Now I think I am perfit in Addition. Mayst. That will I prove by this example. There are two armies of soldiers: in the one are 106800, and in the other 9400: How many are there in both armies say you? Scholar. first I set them one over an other, beginning with the first numbers at the right hand, thus. But the neither number will not match the over number. Master. That forceth not. Scholar. Then do I add 0 to 0, and there amounteth 0, that must I write under the first place, thus. Master. Well said. Scholar. Then likewise in the second place I add 0 to 0, and there ariseth 0, which I writ under the second place, thus. Then I come to the third place saying: 4 and 8 make 12, of which I writ the diget 2, and keep the article 1 in my mind, thus. Then add I 9 to 6, which maketh 15, to that I add the article 1 that was in my mind, and it is 16. I writ 6 under 6 and 9, and keep one in my mind, thus. Master. Why do you not write both figures, seeing you are come to the last couple of numbers? Scholar. Nay reason showeth me, ☜ that I must add that article that is in mind, unto the next figure of the over sum, though there be no more in the neither sum. Ma. That is well considered: then do so. Scholar. Then say I, 0 in the over sum, and in my mind, maketh 1, that I writ under 0: Then followeth there yet one more in the over sum, which hath none to be added to it, for there is none in the neither sum, nor yet in my mind, therefore I think I must write that even as it is. Master. Yea. Scholar. Then doth my whole sum appear, thus. Master. If you mark this, you have learned perfectly the common addition of all sums which are of one denomination: so that ye observe this also, that in Addition you must have two numbers at the least, or else how can you say that you do add? And ever let the greatest number be written highest, for that is the best way, though it be not necessary. And forget not this, that if you have many numbers to add together, you shall have oftentimes an article of a greater value than 10: sometimes 20, sometimes 30, sometimes more, yea, peradventure 100 Therefore, as you did with the article 10, so do with them, reserving them in your mind, and adding to the number next following, so many as their valuer or value certain is: that is to say, 2 for 20, 3 for 30, and so forth of other. But if the article be 100, ☞ then must you not add the article to the next figures following, but to the third figures from them, as I will show you anon by example. And if it chance the number to be such, that it do comprehend two sundry articles, (that is, one of tens, and an other of hundreds) then must you reserve them both in your mind, and add the article of tens, to the figures that follow next, and the article of hundreds, to the figure of the third place from thence. Now take this example for all. I would add these xiii. sums in one, which I set after this manner. Then do I begin and gather the sum of the first figures, which cometh to 107. For first I take 9 there x. times, and that is 90: then 9 and 8 is 17, that is in all 107. of which sum I writ the 7 under the first figures, and then have I an article of an hundred in my mind, which either I must keep in my mind till I come to the third figures, which are in the rooms of hundreds, or else I may for fear of forgetting, writ this one (being of the third place in your of come) under the third row of figures, making two lines, as you see here done. And then must I write the digites under the lowest line: and this is the surest way, when the sum is so great, that the addition of one row passeth 100 When I have so done, I must then come to the second row of figures, and add them together, which doth make 115. of which sum I writ the dygitte 5 under the same second row, and then I have a mixed number remaining of two figures, of which the 1 (that standeth for 10) must be added to the second or next place after them that I did last add. And the other that standeth for 100) must be added to the third place from thence. Scholar. ☞ That is to say, the fourth place from the first line or row of figures. Master. Even so. And then will the sum appear thus. Then add the third row of figures, ☞ with the two unities between the line, and the sum amounteth to 50: of which I writ the cipher under the same third row, and the 5 under the next figures toward the left hand. And with my pen I give a dash to the two unities between the lines, whose value I have already added under the lowest line. Then I add the figures of the fourth row, with the 1 and 5 that are under them between the two lines, & they make 29: then dash I the ●, & the 5, with my pen, as I did before the two unities: & so write under the lowest line the 9 (that is the digit) under the fourth place: & the 2, that is the article, beyond it, toward the left hand. So those sums do make 29057. Scholar. This seemeth somewhat hard, by the reason of so many numbers together. Howbeit I think if I do often prove even with this same example I shall be able to do so shortly, which any other sum. Master. So shall you. For it is often practise that maketh a man quick and ripe in all things: But because of such great sums there may chance to be some error. I will teach you how you shall prove whether you have done well or no. Scholar. That were a great help and ease. Master. Begin first with the highest number, and then to all the other orderly, & add them together, not having regard to their places, but as though they were all unities: and still as your number increaseth above 9, cast away 9 Then go forth, ever casting away 9 as often as it amounteth thereto: and so do till you have gone over all the numbers that you intended first to add, and whatsoever remaineth after such addition and casting away of 9, writ it in some void place by the end of a line for the better remembrance: & then put together the figures that result of the Addition, still casting away 9 also. And then that that remaineth, writ at the other end of that line: and if those two figures be like, then have you well done by likelihood: but if they be unlike, then have you miss. As for example in this present sum: The first figure of the over line is 9, let him go: then 8 and 8 is 16, take away 9, and there remaineth 7, add to it 4 that followeth, and that maketh 11. from which if you take 9, there resteth 2: then come to the next row, whose first and second number are 9, thereore overpass them both, and take the 5 to ●che 2 which did remain in the first row, that maketh 7, put thereto the 4 following, that maketh 11, thence take 9, and there remaineth 2●: next that, go to the third line, whose two first numbers you may let pass, because they are nines: then take the two which with the other two that remained in the second row, make 6: then go to the fourth row, whose two first numbers let go, and take the 6, to the ● that remained, and that maketh ●●, take away 9, and there resteth 3, which with the 3, that is next, maketh 6. And so go through all the other numbers, and you shall find that there remaineth 5, after you have cast away 9 as often as you find it: therefore writ 5 at one end of a line in a void place thus. Then gather all the figures of the total sum which is under the lowest line, and cast away 9 as often as you find it, as thus: seven and 5 make 12, take away 9, and there resteth ●, to that if you add the 2 that is last (for you may let go the 9) then doth it make 5, which you must write at the other end of the line that you made in the void place, and it will be thus. And than you see that those two figures be like, whereby you may know that you have done well, and so may you prove in any other. Scholar. If it please you, I will prove in an other sum. Master. With a good will. Scholar. Then will I take one of your former examples, which was this. First in the highest line, 8 and 6 make 14, than 9 taken away, there remain 5, to which I add the 1 that followeth, and that maketh 6. Then come I to the second line, where I find first 4 which with 6, maketh 10, from that I take 9, and there resteth 1, the next figure is 9, and therefore I let him alone, so find I one remaining, which I set at the end of a line thus. Then I come to the total sum, and there I find that all the figures put together make 10, from which I take 9 and there resseth 1 also, which I put at the other end of the line thus, And because they be like, I know that I have well added. Master. So you know now both how to add two sums or more together: and also how to prove whether you have done well or no: Addition of numbers of diverse denominations. which thing also you may do best by Subtraction. But because you cannot yet skill of it, I will let that pass till anon, and will teach you now how to add sums of divers denominations: which thing can never be but when the one denomination is such that it containeth the other certain times. And yet you shall add them to the other, not after this sort as you did them that were of one denomination, but after such a sort as I will now show you, that is to say. If you have a sum of divers denominations, then look that ye set every denomination by himself, with some note or figure of his denomination, as they be wont to be written. Then writ your other sums so under that first, that every one be set under the other of the same denominations, as for example: if your denominations be pounds, shillings, & pens, writ pounds under pounds, shillings under shillings, and pens under pens, and not shillings under pens, nor pens under pounds. Scholar. Now that you have spoken it, me thinketh it needeth not to warn me of it, for it were against reason so to confounded sums: but yet if you had not spoken of it, peraduenure I should have been deceived in it. Master. If you do say it is so plain, I will speak no more of it, but with an example make the matter to appear evidently. first, one man oweth me 22 lb, 6 s, 8d. An other oweth me 5 lb, 16 s, 6d. And an other oweth me 4 lb, 3 s, I would know what this is altogether. Therefore must I first set down my greatest sum & then the other, every one under his denomination gréeing to the greatest sum, as here you see. Then must I begin at smallest numbers, (which must always be set next the right hand) and add them together, and if the sum of them will make one of the next denomination, then must I keep it in my mind till I come to that place, or else for more easiness writ it under that place between the double line, and under that place must I note the residue, if there remain any of the same denomination, but if there remain none, then need I to write under it nothing. And this is all that you must mark in this Addition: for all other things are like to the other manner of Addition before mentioned. ☞ Therefore the chiefest point of this Addition is, to know the values of common coins and rated sums. As how many shillings be in a pound: how many pence in a shilling, of which and of other like things, I will instruct you hereafter, in teaching of Reduction: But now I may not disturb your wit from the thing that we are about. Therefore let us return to that former example, which I proposed of three debtors, which sums when I had set orderly they stood thus, with a double line under them. Then to add them unto one sum, I must begin at the right hand, where the smallest denomination is, and add them together first, saying: 6 and 8 make 14. Now seeing these 14 are pennies, and that 1● pence make one shilling, which is the next valewer, I take away 12 from 14, and there resteth 2, which I writ under the pennies, and for the other 12, which maketh 1 shilling I writ under the title of shillings, thus: Then do I add all the shillings together, and find them 25, to which I add that 1, between the two lines, that maketh 26, but because that 20 shillings do make 1 pound, I take away 20 from 26, and for that 20 I writ 1 under the pounds between the two lines, and the other 6 that remaineth, I writ under the shillings, as appeareth in the example before. Then come I to the pounds, adding them all together, and find them to be 31: thereto I add the 1 between the two lines, & that maketh 32, which sum I writ down whole, because there resteth no greater denomination, and then my whole sum appeareth thus. So is my total sum, 32 lb. 6 s. 2d. And this may you prove in an other like sum. Sc. Then will I cast the whole charge of one months commons at Oxford with batteling also. Master. Go to, let me see how you can do. Scholerr. One weeks commons was 11d. ob. q and my batling that week was 2d.q. q. The second weeks commons was 12d. and my batling 3d. The third weeks commons 10d. ob. and my batling 2d. q. c. The fourth weeks commons 11 d,q. and my batling 1d. ob. ●. These eight sums would I add into one whole sum, and therefore I will set them one over another, thus. But I had forgotten, I should have set the greatest sum highest. Master. So is it commonly best, howbeit, here it forceth not: and in such sums as this is, that go by order of weeks, days, or years, it is better to keep that order, than to alter them, and to set the greatest number highest, for that serveth for such sums as go not by order. Scholar. Then if I have set them well enough, I will begin to add them thus. first of the smallest valewers' at the right hand, which are called cées, I find ●, and seeing that 2 cées, do make one q, I will write nothing under the cées, but will write 1 q for 2 cées, under the kewes between the lines, as the example showeth. Then come I to the next valewers, where I find 2 q, and to them I add the q that is between the lines, and so are they 3 q: but because 2 q, maketh one q, I writ one q under the farthings between the lines, and the q that remaineth must I write beneath the nethermost line under the kewes, thus. Then come I to the farthings, where I find 3, and the other q that is between the lines, maketh 4 farthings. And because 4 q make just 1 penny, I shall write nothing under the farthings, but must write 1 under the pens, between the lines. Next that must I add the half pence together, of which there are 3. but seeing that 2. ob. make 1 d, I must write 1 under the pens between the lines: but how shall I do it, for there is 1 already? Master. Have you forgotten how I did in addition of the great sum before? ☜ you must set it under the other, so shall they both stand for 2. For if you should set it before or behind the other, they should make 11. Scholar. I remember it now, and I perceive the reason. Then I will write 1 ob, under the halfpences, and for the other two halfepens, which make 1 d, I writ 1 under the pens: Then come I to the pens, & find, that there are of them 52. then put I to them the 2 between the lines, and that maketh 54, which amounteth to 4. s. 6 d: the 6 d I must write under the pens, and the 4 s. I must set (I suppose) farther toward the left hand by themselves. Master. Even so. Scholar. Then appeareth all my addition thus. And the sum is 4 s 6d. ob. q. M. Now have you done this well. But tell me, why did you writ kewe, cée, thus, q, c. & not rather thus q ᶜ, as the fashion is? Scholar Because I thought it was the best way for due gathering of every denomination by himself. Master. So was it in deed. Well now, can you tell how to prove this addition, and such other like of divers denominations, and to try whether you have done well or no? Scholar. I would I could. Master. That shall you do by this means. First as you did begin to add so reckon again every denomination by itself, and when you find so many small that do make any other denomination, let them go, and keep in mind only the residue that will make no greater denomination, and look whether there be any such like value under the neither line, and if there be, you have well done, and so go from one denomination to an other, unto the end. But here must you note, that in gathering of the sums, ye must reckon those figures that are written between the lines, with them that are written above them: as for an example, I will examine the sum that I did last add, which stood thus. first I find 6 and 8, which maketh 14, from which I take 12, because it maketh one of the next denomination, and there remaineth 2, and under that place I see a like figure, therefore I know that well to be done. Then come I to the s, where I find 1, 3, 16, and 6, that maketh 26, I cast away 20, for they make another denomination, that is to say pounds: and the 6 which remaineth, is like to the 6 that is written under them beneath the lowest line, therefore▪ that is well done also. And thence I go to pounds, where I find 1, 4, 5, 22, that is 32, to which sum agreeth another like under it. Therefore I judge all well done. Scholar. I perceive reason in this probation. Now will I attempt the same in the sum that I did add, which when I had ended adding, stood as you may see in the example following. first amongst the cées I find but two, which make one q even, therefore there must nothing be under the line for them: And amongst the kewes I find 3, of which two make 1 q, therefore I let them go, and the one q, that is left, hath an other like under his place, therefore that is well done. Then the farthings are just 4, which make 1d. and therefore I let them go. Amongst the half pence there is one odd (for 2. must I cast away, because they made one penny) and unto it answereth a like sum under it. The pens are 54. from which I take away 48, that makes 4, s, and the 6 remaining agree to a like figure set under them. And last of all remaineth the 4 s, which the abjected pens did make: so I perceive that I have well done. Now this will I not forget. But will this examination serve in all addition? Master. It serveth for all addition of sundry denominations, ☜ if the addition be made with two lines, (as were these) else it will not serve, because that those sums which are here added between the lines, in Addition by one line, are understanded and not written: but I let that way pass, because as it is common, so is it more deceivable than this way, namely if a man's memory be either dull or troubled. Scholar. Yet it were good to know that way also. Master. If you desire to know it, Another form of Addition. this it is in few words. Do every thing as you did in this sort of Addittion, save that where you made here two lines, you shall make there but one: and those sums that you did here write between the lines, you must keep in your memory, and use them (as you did here) each one when you come to his place. Scholar. Then they differ not, but in this, that this addition with two lines leaveth nothing to memory, but writeth down all: and the other way committeth certain numbers to memory, as you taught me in the first examples of addition of small sums of one denomination. But what if a man use it (as you say men do commonly) how shall it be examined? Master. Seeing you are so desirous of it, I will show both an example of the addition, and also the manner to examine it. I propose these three sums to be added, and I gather first the pence, as I did in the other sort, and I find of them 8, 3, 9, that is 20, of which sum I bate away 12, which make 1 s, and keep that 1 in my mind, and the rest, that is 8, I writ under the pence. Then do I add the shillings together, and find of them 6, 7, 8, that is 21, whereof I bate 20, that make 1 lb, which I keep in mind, and to the other 1 that remaineth, I add that one that came of the pens & was in my mind, which make 2, and them I writ under the shillings. Then do I reckon the pounds together, 3, 6, 12, that is 21, and to them I add the 1 in my mind that remaineth of the shillings, which make 22, them do I write under the pounds, and then my sum total appeareth to be 22 lb, 2 s, 8 d. Now to examine this sum and all such like, you shall do thus. first begin at the left hand with the pounds, Another form of proof. and take from them that are above the line, 9, as often as you can: then that that remaineth shall you double, and join it with the shillings, and take away 9 from that as often as you can, and whatsoever remaineth, ye shall take for it three times so much, and put to the pence: then take from all that sum 9, as often as you can, and what so remaineth after you have withdrawn 9 as often as you can, writ that at the end of a line, as I taught you in the other Addition. And then come to the sum under the lyre, beginning with the pounds, and do even as you did with the sums above the line, till you come to your pennies: and if the figure of the sum that remaineth after casting away 9, (as often as you can) do agree with the other that remained before of the other sum, which you did write at the end of the line, then have you done well, else not: and for an example, I will examine that last sum which was thus: First I shall begin at the left hand with the pounds, putting them together, which make 21, in which sum I find 9 twice, (for twice 9 is 18) that I deduct, and there remaineth 3: that 3 must I double (as I said) because it is the remainer of the pounds, and it will be 6. Then gather I the sum of the shillings, which is 21, to the which I add the foresaid 6, and then it is 27, wherein I find 9 three times, and there remaineth nothing. This remayner should I take three times, but three times nothing, is nothing: therefore in this place is there nothing left to be added to the pennies. Wherefore I must take the sum of penies alone, which is 20, from thence if I take 9 twice, there remaineth but 2, which I put unto the end of a line thus. Then I come to the pounds of the under number or total sum, and there I finds 22, from which I take away 9 twice, and there remaineth 4: that 4 I double, and it is 8, then do I add that 8 to the shillings, and it maketh 10, from which I withdraw 9, and there resteth one: then do I take that 1 thráe times, and it maketh 3, which I add to the 8d. and it maketh 11, from which if I bate 9, there resteth 2, which is equal to the number noted at the end of the line: and thereby I perceive that I have done well. Scholar. But I do not see the reason of this. Master. No? The reason of this proof. no more do you of many things else, but hereafter will I show you the reasons of all Arithmetical operations: for this I judge to be the best trade of teaching, first by some brief precepts to instruct a learner somewhat in the use of the Art, The best trade of teaching. before he learn the reasons of the Art, and then may you afterward more sooner make him to perceive the reasons: for hard it is to occupy a young learned wit with both the art and the reasons of it all at once: howbeit he shall never be cunning in deed in an art, that knoweth not the reason of every thing touching it. But for this work, because the reason is easy, I will show it you now. You know that if one pound do remain, it being turned into shillings, would make 20 s, in which number there is 9 contained twice, and 2 s beside. And therefore for one pound you shall take 2 s, and so for every one pound 2 s. Scholar. I see it well, for if there remained 7 lb, after the nines were cast away, I must take 14 s for that 7 lb. And so have I cast away 14 times 9 s, and yet remaineth of every pound 2 s which maketh 14 s. Master. Like ways in shillings, which contain 12 d: for every shilling, if you abate 9 pence there resteth 3 pence. Scholar. It is plain enough. And so if ● shillings do remain, I must take for it 15 d, that is three pence for every shilling, and yet in that so doing, I have cast away five times nine pence. Master. Other works have as good reason, but I will not stand about yielding reasons now. Scholar. Yet one thing more I pray you show me, why did you write your number that remained (after you had withdrawn all the nines) at the end of a line? for I saw no reason why that line did serve. Master. Did you ever mark a Carpenter when he wrought? Scholar. Yea many times. Master. And have you not seen him when he hath taken measure of a board, that he hath pricked it, and hath with a twitch of his hand drawn a line from the prick that he made? Scholar. Yes I have marked that and have seen some mark 3 or 4 lines by the prick, some also have I seen make a cross by it, but that I perceived was for the easy finding of their prick. Master. But there is another sort of proof of Addition; Another kind of proof most usual and aptest of all. to which the cross serveth more meeter: & that is when the addition is of diverse denominations: and I would examine every denomination by itself, which way though it be not much unlike to the first proof that I brought of such diverse sums, yet will I declare it, leasts you should think that I would hide it from you. For the proof of the which, because it containeth three denominations, I must make a cross of three lines, as in the page afore. Then I reckon first at the right hand the pennies: 7, 1, 5, make 13, from which I take 12 for the next denomination, that is to say, a shilling, and there resteth 1, which I must write at one end of the neither thwart line. After that I gather the sum of the shillings, 2, 8, 12, which maketh 22. to them I put one that I took of the pennies, and that maketh 23: from those I take ●0, the quantity of the next greater denomination, that is to say, a pound, and there resteth 3, which I writ at the end of the highest thwart line. thirdly, I add together the pounds, 9, 12, 16, which make 37, to them I add the 1 that came of the shillings, and then there is 38, wherein I find 4 times 9, and 2 over, that 2 I writ on the upright line. That done, I come to the total sum, and examine it, beginning at the pennies, where I find but one, and cannot take 9 from him, therefore I set him at the other end of the neither thwart line: Then I come to the shillings, where I find only 3, which because it is less than 9, I set it at the other end of the line of shillings, that is, the overmost thwart line. Last of all, of the 38 lb, I take four times 9, which is 36, and there remaineth 2, which I writ under the upright line. Then I consider every number, comparing it to the number that is against it, and because I find them to be every one like his match, I know that I have well done. Scholar. This cross I perceive doth serve for those three denominations, pounds, shillings, pennies. But what if I had ob, q, q, and c? Master You think you be at Oxford still, you bring forth so fast your q and c. These lines, as I have said, do serve for three denominations, such as they be: as here they do serve for pounds, shillings, and pennies: but if ye have no pounds in your sum, then may they serve for shillings, pennies, and half pennies: yea for q, q and c, if you have no greater denomination, so that you remember that the upright line serveth for the greatest denomination, and the highest thwart line, for the next, and the lowest for the least, Examples of Addition. The Proofs. An other Example. SUBTRACTION. Scholar. THen have I learned the two first kinds of Arithmetic: now as I remember, doth follow Subtraction, whose name me thinketh doth sound contrary to Addition. Master. So is it in deed: for as Addition increaseth one gross sum by bringing many into one, so contrary ways, Subtraction diminisheth a gross sum by withdrawing of other from it, so that Subtraction or Rebating is nothing else, but an art to withdraw and abate one sum from an other, that the Remainder may appear. Scholar. What do you call the Remainder? Master. That you may perceive by the name. Scholar. So me thinketh: but yet it is good to ask the troth of all such things, least in trusting to mine own conjecture, I be deceived. Master. So is it the surest way. And as I see cause, I will still declare things unto you so plainly, that you shall not need to doubt. Howbeit, if I do overpass it sometimes (as the manner of men is to forget the small knowledge of them to whom they speak) then do you put me in remembrance yourself, and that way is surest. And as for this word that you last asked me, take you this description: Remainder. The Remainder is a sum left after due Subtraction made, which declareth the excess or difference of the two other numbers: as if I would abate or subtract 14 out of 18, there should remain 4, which is called the remayner, and is the difference between those two numbers 14 & 18. Scholar. I perceive then what Subtraction is: Now resteth to know the order to work it, Master. That shall you do by this means. first you must consider, ☜ that if you should go about to rebate, you must have two sundry sums proposed, the first which is your gross sum or sum total: (and it must be set highest) and then the rebatement or sum to be withdrawn, which must be set under the first (whether it be in one parcel or in many) and that in such sort, that the first figures be one just over an other and so the second and third, and all other following, as you did in Addition: then shall you draw under them a line, and so are your sums duly set to begin your working. Then begin you at the right hand (as you did in Addition) and withdraw the neither number out of the higher, and if there remain any thing, writ that right under them beneath the line: and if there remain nothing (by reason that the 2 figures were equal) then write under them a cipher of nought. And so do you with all the other figures, evermore abating the lower out of the higher, and write under them the Remainder still, till you come to the end. And so will there appear under the line what remaineth of your gross sum, after you have deducted the other sum from it, as in this example. I received of your father 48 s, of which I have laid out for you 36 s: now would I know what doth remain? and therefore I set my numbers thus in order: First I writ the greatest sum, and under him the lesser, so that the figures at the right side be even one under another, and so the other, thus. Then do I rebate 6 out of 8, and there resteth two, which I writ under them right beneath the line, thus. Then I go to the second figures, and do rebate 3, out of 4, where there remaineth 1, which I writ under them right, and then the whole sum and operation appeareth thus. Whereby it appeareth, that if I withdraw 36, out of 48, there remaineth 2. Scholar. Now will I prove in a greater sum: And I will Subtract 2367924 out of 3468946. Those sums I set in order thus. Then do I begin at the right side, and deduct 4 out of 6, and there resteth 2, which I writ under them. Then go I to the second figures, and withdraw 2 out of 4, and there remain two, which I set under them also: then I take 9 out of 9, and there resteth 0, which I writ under them: for you say, that if the figures be equal, so that nothing remain, I must write this cipher 0 under them. Master. It was well remembered, now go forth. Scholar. Then I come to the fourth place and draw 7 out of 8, and there remaineth 1, which I writ under them also. Then in the fift place I take 6 from 6, and there resteth nought, for it I writ under them a cipher, 0: Then in the sixth place 3 rebated from 4, there remaineth 1, which I writ under them: and likewise in the seven. & last place, 2 taken from 3, there is left 1, which I write under them: so have I done my whole working, and my sums appear thus. Whereby I see, that if I rebate 2367924, out of 3468946, there remaineth 1101022. Master. This is well done. And that you may be sure to perceive fully the Art of Subtraction. let me see how can you subtract 52984732 out of 8250003456. Scholar. first I set down the greatest sum, and after that I writ under if the lesser number, beginning at the right side: and then my figures will stand thus. Then take I 2 from 6, and the rest is 4 which I writ under them: then do I withdraw 3 from 5, and there remain 2, which I writ under them. Then take I 7 out of 4, but that I cannot, what shall I now do? Master. Mark well what I shall tell you now, Note. how you shall do in this case and in all other like. If any figure of the neither sum be greater than the figure of the sum that is over him, so that it cannot be taken out of the figure over him, then must you put 10 to the over figure, and then consider how much it is, and out of that whole sum withdraw the neither figure, and write the rest under them. Can you remember this? Scholar. Yes, that I trust I shall. Now then in mine example where I should have taken 7 out of 4 and could not, I put 10 to that 4, which maketh 14, from it I take away, 7, and there resteth 7 also, which I writ under them. Master. So have you done well, but now must you mark another thing also: that whensoever you do so put 10 to any figure of the over number, you must add one still to the figure or place that followeth next in the neither line, as in this example there followeth 4, to which you must put 1, and make him 5, & then go on as I have taught you. Scholar Then shall I say: 4 and 1 (which I must put to him for the 10 that I added to 4 before) make ●, which I should take out of 3, but that cannot be, therefore must I put to it also 10, & then it will be 13, from which I take 5, and there resteth 8 to be written under them: and because of that 10 added to the 3, I must add 1 to 8 that followeth in the neither line, & that maketh 9, which I should take out of 0, and cannot, therefore I put thereto 10, and that maketh 10: from 10 I take 9, and there remaineth 1, which I writ under them. Then do I add likewise to the next figure beneath, which is 9, and that maketh 10, that 10 should I take out of the figure above, but I cannot, for it is 0, therefore I put 10 to it, and so take I 10 out of 10, & there resteth 0 to be written under them. Then come I to the next figure which is 2, and to him do I add 1, which maketh 3, that 3 I can not take out of nought, therefore of that nought I make 10, and thence do I take 3, so remaineth there 7 to be written under them. Likewise do I put 1 to 5 that followeth, and then is it 6, that would I take out of 5, and cannot, therefore I add 10 to that 5, and make it 15, from which I rebate 6, and there remaineth 9, which I writ under them. Now have I spent all the neither figures, & what shall I do more? Master. You should have added to the next figure following (if there had been any) because you added 10 to the last figure before of the over line: but seeing there is no figure following, you must add that to the place following, and then deduct that from the number above. Scholar. Then shall I say because I borrowed 10 to the over 5, I must put in the next place beneath, that is under 2: then must I subtract that ● from ●, and there resteth 1, to be written under that, in the ninth place. Now I have no more to subtract, for there is never any figure remaining beneath, neither yet any unite to be added, because I borrowed not 10 to the figure last before, ☞ and yet is there 8 remaining in the over line, which (I think by reason) should be set at the end of the figures in the lowest row which is under the line, for because there was nothing taken from it. Mayst. That is well considered, and reason teacheth so in dáede. Scholar. But sir I beseech you, shall I always when any number so remaineth alone (as this ● did) writ him under the line strait against his own place? M. Yea, what else? Whether they be one or many: and this well remembered, you have sufficiently learned Subtraction. How be it, because of certain things that might deceive you, if you did not take good heed to your working, I will propose to you another example of many numbers to be subtracted, as thus, I received of a friend of mine to keep ●869 Crowns, of which at one time I delivered him again 500, at one time 368, and at another time 440, and an other time 80, and an other time 64: now would I know how many doth rest behind. Therefore first I set down my gross sum, and a line under it: & underneath it I set all the parcels, thus: and under them a double line. Then first I begin at the first place, & gather together the sum of all those lines (save the overmost) in their first figures, and so do I with all the figures of the second place, & so forth as I did in Addition, save that I leave out the highest row of numbers (as the line warneth me) and that sum so gathered between the double line, do I subtract out of the highest row of numbers, and the remainer do I set under the nethermost line: as for example. I set the sums as before: than do I gather the first figures together, where I find but 4 & 8, that make 12, (for three Ciphers increaseth no sum in addition, as you learned before) of the 12 therefore do I write the digite 2, between the double line, and keep the article in my mind, till I come to the second place, where I find 6, 8, 4, 6, that make 24, to them I put the article in my mind, and it is 25, of which I writ 5 under the second place, and keep the digite 2 in my mind for the third place, where I find 4, 3, 5, that make 12, to the which I add the 2 in my mind, and it maketh 14, thereof I writ the 4 under the third place: and because there remaineth no more figures to be added, I writ the digit 1 in the fourth place, as you see in the example. Then come I to subtracting of this sum between the lines, for by Addition it is equal to the five parcels over it. Therefore I proceed to subtract it from the overmost sum, saying: 2 from 9, remain 7, to be written under them beneath the lowest line. Then in the second place I take 5 from 6, and there resteth, to be written under them. Then in the third place, 4 from 8, resteth 4. Last of all in the fourth place, 1 from 2, remaineth 1. And thus I see that after those 5 sums are subtracted from 2869, the Remainder is 1417. Scholar. This I perceive: but is there no shorter way and more spéedier? Mayst. Yes, An abridgement of the former manner of Subtraction. when you are a while exercised in it: for you may as fast as you can gather the numbers together, withdraw them out of the highest sum if so be it, that all the parcels which you do gather, do not exceed nine, but and if they exceed nine, then must you subtract only the digit that is in it, and reserve the article till the next place, where you shall add it with the other figures and so subtract the whole out of the figure above them: but and if in this place the sum of the parcels do exceed 9, than (as I said before) subtract the digit only, and reserve the article to the next place, and so still go forth, till you have ended your working. As for example: In the last sum proposed, I gather first in the first place 4 and 8, that maketh 12, of which I deduct the digitte 2 out of 9, and write under the remayner, which is 7, and the article 1 I keep in my mind. Then in the second place I gather the parcels 6, 8, 4, 6, which amount to 24, to that I add the article 1, which I have in my mind, and then is it 25, then do I take 5 (that is the digitte in this number) from 6, that is in the second place of the highest sum, and there remaineth but 1 to be written under them, and now do I keep the article 2 in my mind still. Then in the third place, 4, 3, 5, maketh 12, and the article 2 in my mind maketh 14: then take I 4 (which is the digitte) from 8, that is over them, and there resteth 4, which I writ under them. Then have I the article 1 yet in my mind, which I should add to the parcels next following, but seeing there is no number following, I take that digit alone, and deduct him out of the next sum above, which is 2, and then is the remainer 1, which I writ in the fourth place under 2. Lo, now have you a shorter way. Scholar. I like both ways well, and I perceive both well, yet as in the one the working seemeth somewhat long, so in the other it leaneth very much (me seemeth) to remembrance, and therefore may cause error quickly, except a man have a quick and an exercised remembrance. Master. What? would you then have such a way that should not be so long as the one, nor so short as the other? Scholar. Yea, if there were any such. Mai. Than do thus: ● still as you gather your parcels, when they exceed a digit, & maketh him 10 or more, take the article, and write him between two lines (as in the first example) under the next place toward the left hand: and then deduct the digitte from the figure that is over him, and write the remayner. And then when you gather the next parcels, you shall add to them the figure that is under them, between the two lines. And if it exceed 9, do as I said before, writ the article under the next place between the lines, and subtract the digitte from the Figure that is over those parcels: and if that all the parcels together and the number between the lines do make but a digit, then deduct it wholly from the figure above: as in this example. I would subtract out of 40308964, these three parcels Therefore I set them first in order due: & then I gather the parcels of the first place, which are 8, 2, 1, that is 11: of which I take away the article, and set him under the second place between the lines: and the digit 1 that remaineth, I deduct out of 4, and there resteth 3 to be written under the first place beneath the lowest line. Then come I to the second place, and gather the parcels of it, 6, 4, 2, and the 1 between the lines, which make 13, of which I take the article, and set him under the third place between the lines, and the digit 3 I take from 6, and there remaineth 3, which I writ under the second place beneath the lowest line. Than in the third place I find 4, 3, 4. which with the 1 between the lines, do make 12, therefore I writ the article again under the fourth place, and the digit 2 I take from 9, and there remaineth 7 which I writ under them beneath the lowest line. And then come I to the fourth place, where I gather 1, 2, 3 & h● 1 between the lines, that maketh 7, which because it is but a digitte I pluck from 8, and the Remayner is 1. and must be written under them in the fourth place. After that come I to the fift place, where are only three cyphers, which make nothing, them should I take that, that is to say nothing, from the figure over them, which is also a cipher, therefore I must say thus: if I take nought from nought, there remaineth nought: so must I write a cipher under them. Then in the sixth place I find but 1, which I take out of 3 over him, and the Remainder is 2, that must be written beneath the lowest line in the sixth place. So I go to the seventh, where I find only cyphers, and in the gross sum over them a cipher also, therefore must I write their remainder (which is nothing) with a cipher also. Then in the eight and last place, I gather 1, 1, 2, that make 4, which if I take out of that 4 that is over them, there will nothing remain. And that must be noted with a cipher beneath the lowest line, as I have often said, and so have I ended my work, and the figures stand thus. Scholar. Sir, I remember you taught me that ciphers should not come in the last place, for because they serve only to increase the value of other Figures which follow them, and serve not for those figures that go before them: and now in your example you have set two ciphers in the two last places. Master. I commend you for your remembrance. And truth it is, I should not have set them here, but only because that I would make you plainly to perceive the art of Subtraction. Therefore seeing that you do now perceive it, whensoever you should write down a cipher, look whether any other figures be yet behind. And if not than let go the cipher also, for it needeth notto write him in any latter places, where no other figure doth follow, except it be (as I did) to teach the use of Subtraction the plainer. Therefore my figures must stand thus when I have ended my work. Scholar. So I would think by that you taught me before. And now I believe I could subtract any sums. Master. So may you if you have marked what I have taught you. But because this thing (as all other) must be learned surely by often practise, I will propound here two examples to you, wherein if you often exercise yourself, you shallbe ripe and perfect to subtract any other sum lightly, for in them is contained all the observances of whole number. And because you shall perceive somewhat both how to do it, and also whether it be well done when you have proved to do it, therefore have I written under them, both the Remayners: And to one of them also adjoined his proof. Scholar. Sir, I thank you. But I think I might the better do it, if you did show me the working of it. Master. Yea, but you must prove yourself to do some things that you were never taught, or else you shall not be able to do any more than you were taught: And that were rather to learn by rote (as they call it) then by reason. And again there is nothing in this example or any other of whole number, but I have taught you the rules of them already. Scholar. Then I trust by practice to attain the use of it. And is this all that I shall learn of Subtraction? Master, Yea, saving that (as you have seen in Addition) there are numbers of divers denominations, in which the working is not much unlike, yet without some instructions be given of it, it might seem to a learner more difficult, then in deed it is. Therefore I will briefly show you the use of it only, by one example or two. A certain man owed to me 14 lb, 12 s, 8 d, of which he paid me at one time 4 lb, 6 s, 8 d: at an other time 3 lb, and at an other 2 lb, 3 s, 4 d, and last of all, 6 s, 8d. Now would I know what remaineth unpaid yet, therefore I set my sums thus. Scholar. Sir, I pray you why do you write 2 lb. for the common speech useth rather to say 40 s. Master. We must here use the denomination that is greatest in any sum, so that we may not write according as we use to speak, saying: 16d. 18 d: or likewise, 7 groats, 8 groats: 24 s. 40 s 48 s. and such other, but we must write every denomination that is in any sum by itself, namely shillings and pounds. So must we write for these sums now named, 1 s, 4 d: 1 s, 6 d: 2 s. 4 d: 2 s, 8 d: 1 lb, 4. s: 2 lb, 8 s: and so forth of other like. Scholar. So that we may not write in Arithmetic pennies, when the sum amounteth to shillings, nor shillings when tho sum maketh pounds. Now if it please you, end your example. May. When my sums are so set as I showed, then must I begin with the smallest denomination, saying: 8, 4, 8, are 20. which sum because it is pence, & 12 pence do make 1 s, I must take from that 20 (which cometh of the parcels) 12. & for them write between the lines under the shillings, than the 8 d, that remaineth I take out of the highest sum, which is 8 also, & then remaineth nought: wherefore under the pence I writ nothing. Then come I to the shillings, and gather the parcels 6, 3, 6. which with the 1 between the lines, make 16, that must I take out of the sum that is over it. But seeing that sum is but 12, I cannot take 16 out of 12, I must borrow one of the 14 lb, and put to the 12, and that maketh 32, for 1 lb is worth 20 s, then take I 16 out of 32 and there resteth 16 to be written under the shillings. Then come I to the pounds, whose parcels are 2, 3, 4. that is in all 9, and one more must I add thereto, because of the 1 that I borrowed before unto the 12 s, and than is there 10, which I must take out of 14, so doth there remain 4 to be written under the pounds: so doth my remainder appear to be 4 lb, 16 s. Scholar. This do I perceive very well, and if there be none other thing to be learned in Subtraction, then may I come to Multiplication, for that you reckoned to be in order next. Master. We have done in deed with the art of Subtraction, as touching the working. But yet before we go to multiplication, A proof of Subtraction in numbers of one denomination. I will instruct you how to examine your work whether it be well done or no, and that is by casting away 9 as often as you can find it, as you did in Addition, saving that you must here examine the highest number alone, and note the residue of it at a lines end, as you did in Addition. And when you have done with the highest number, then examine all the other together, casting thence 9 as often as you can: and if the last remayner be like the other, then have you done well. But if you have diverse denominations in your sum, A proof in Subtraction among diverse denominations. yet for them all shall you make but one several line, as you did in Addition, remembering to begin the examination at the greatest denomination, and to double the remayner of pounds, and triple the remainder of shillings, as you did also in Addition. As for a proof, I will examine this work where in the highest line I find of pounds 14. from thence I bate 9, & there resteth 5, which I do double, because they are pounds, and then are they 10 thereto I add the 12 & it maketh 22 from which I take 9 twice, & there resteth 4, which because they are shillings, I triple, and then are they 12, thereto I add the 8, & then are they 20 thence take I twice 9, and yet resteth 2, which I writ at the one end of a line thus, 2— Then I examine all the other parcels & the remainer together, every denomination by itself. And first of pounds I find 4, 3, 2, 4 that is 13. from which I take 9, & there resteth 4. that do I double, and it maketh 8, to it do I put the shillings, 6, 3, 6.16. that is 31 (for the one between the lines must not be reckoned nor none in that space) and that maketh in all 39 Where hence I take 9, four times, and there remaineth 3. that do I take three times, and it is 9, wherefore I cast it away: then do I take the pennies 8, 4, 8. that maketh 20, from which I take 9 twice, and there resteth 2. which I writ at the other end of the proof line. And because I see that those two numbers are equal, I say that I have well wrought. And if you will you may make for every denomination a line, as you learned in Addition: but then must you begin your examination at the smallest denomination, as you did in Addition, for their proof is altogether like, saving that in Addition you examine the nethermost sum alone, and all the other together: and in Subtraction ye must examine the highest number alone, & all the other together. And if you mark it well, it is even all one, for that sum that in Addition is lowest, in Subtraction is highest: Gross or total sum. and that sum is called the Gross or total sum. Therefore if you mark what I said in Addition, you may easily perceive what is to be done for the proof of Subtraction. And to the intent that you may perceive it the better, I will show you an other proof of Subtraction, and that shall be by Addition, thus. Draw under the lowest number (which is your remayner) a line: then add that number, another proof of Subtraction. and all the other that you did subtract before, together, and write that that amounteth, under the lowest line: and if the sum that cometh thereof, be equal to the highest of the subtraction, than was the subtraction well wrought, or else not. As for example: in the last sums, which stood thus. First I add 8, 4, 8, that maketh 20. whereof I take 12 away, because they make one shilling, and write for them under the shillings: and the 8 that is left, I writ beneath the lowest line, then add I the shillings 6.3.6.1.16. that make 32: from which I take 20, and for it I writ 1 under the pounds, and the 12 that remaineth, I writ under the shillings. Then come I to the pounds, adding them together, which are 4.3 2.1.4. that maketh 14: then do I writ 14 under the lb, and so have I ended the Addition. And I see that the lowest line of number and the highest be like, wherefore I know that I have well done. For my figures appear as you may plainly see in the page following. And thus now have I taught you the art of Subtraction, and the means to prove whether it be well wrought or not. And this last proof of Subtraction is most aptest and best allowed of any other proof: whether it be of lb sd. or any other gross some whatsoever. Now and you remember, I omitted in teaching the proof of Addition one way why: i I said was by subtraction. Scholar Truth it is, and then was it deferred, because that I had not then learned the feat of Subtraction, whereby I should have proved it, but now I thank you, I have well learned the art of Subtraction, & the proves of it, both by 9 and by addition. And now I would be glad to know, how I may prove Addition by Subtraction. Master. Then mark you this. The proof of Addition by Subtraction. When you have ended your addition: take the numbers all that you did add, to the highest sum, and deduct or subtract them from the gross sum that doth result, and if the remainer be like to the highest number, then have you done well, else not. As for example. I take one of the sums that I did add before, which was this that followeth here. Then do I come to the middle number (because here in this example are only three numbers) and subtract that from the neither number, beginning at the right hand, and first I say, 0 out of 0, there remaineth 0: that writ I under an other line. Then again, 0 in the second place from 0, remaineth 0 under it I writ 0 also. Next that in the third place, 4 out of 2 will not be, therefore I add to that 2, 10, and make it 12 from that I take 4 and there resteth 8. Then say I farther: 9 in the fourth place, and 1 (which I must add for the 10, borrowed before) make 10, that must I take from 6: and because I can not, I add to the 6.10, and then is it 16: from than I take 10, and there resteth 6. to be written under them. Again in the fift place where I find nothing written, I must set 1 for the 10 last borrowed, and that 1 do I take from the 1 under him, and so remaineth nought, wherefore I writ down a cipher 0. Now have I done with the subtraction: and yet in the gross sum remaineth 1, which I must set right in the same place, in the remayner, and so the remayner appeareth to be like unto the highest sum of the Addition, as here appeareth. Wherefore I say that the Addition was well wrought. And note, that if you had subtracted the uppermost from the product or total sum, than the residue thereof would be equal to that middlemost number. But if the parcels which you added, be more than two: (as three four, five, six, or more) than from your gross or total sum subtract first one of the parcels: and note that new residue. Out of that new residue, subtract an other of your parcels, (which you will) and Note that second new residue. And if you have no more parcels added, but three, than is that second new residue equal & alike to the third parcel, which you have not (as yet) subtracted, if you have wrought well: both in your first Addition, and now in your subtracting. And so in this wise, (if you have four, five, or more parcels) may you proceed to make yourself sure of your total sum, first, by Addition of the said parcels, produced and gathered. And thus may you do in any other sum of one denomination or many: Saving that I will tell you by the way, the last manner of proof that was showed you in Addition, is the best, and the aptest proof for the Rule of Addition. Scholar. Sir, I thank you most hearty, Master. Therefore now will I make an end of Subtraction, and will instruct you in Multiplication. ☞ MULTIPLICATION. MVltiplication is such an operation, that by two sums produceth the third: which third sum so many times shall contain the first, as there are unities in the second. And it serveth in the stead of many Additions. As for example. When I would know how many are 30 times 48: if I should add 48, thirty times, it would be a long work. Therefore was this work of Multiplication devised, which shall do that at once, that Addition should do at many times. Scholar. I perceive the commodity of it partly, but I shall not see the full profit of it, till I know the whole use of it. Therefore sir I beseech you, teach me the working of it. Master. So I judge it best, but because that great sums can not be multiplied, but by the multiplication of digits, therefore I think it best to show you first the way of multiplying them: As when I say, 8 times 8, or 8 times 9 etc. And as for the small digits under 5, it were but folly to teach any rule, seeing they are so easy, that every child can do it. But for the multiplication of the greater digits, thus shall you do. first set your digittes one over the other right, then from the uppermost downward, and from the nethermost upward, draw straight lines, so that they make a cross, commonly called Saint Andrew's cross, as you see here. Then look how many each of them lacketh of 10, and write that against each of them, The difference. at the end of the lines, and that is called the Differences, as if I would know how many are 7 times 8, I must write those digittes thus. Then do I look how much 8 doth differ from 10, and I find it to be 2, that 2 do I write at the right hand of 8, at the end of the line, thus. After that, I take the difference of 7 likewise from 10, that is 3, and I write that at the right side of 7, as you see in this example. Then do I draw a line under them, as in Addition, thus. last of all I multiply the two differences, saying: 2 times ● make 6, that must I ever set under the differences, beneath the line: then must I take the one of the differences (which I will, for all is like) from the other digit (not from his own) as the lines of the cross warn me, and that that is left, must I write under the digits. As in this example. If I take 2 from 7, or 3 from 8, there remaineth 5: that 5 must I write under the digits: and then there appeareth the multiplication of 7 times 8, to be 56. And so likewise of any other digits, if they be above 5, for if they be under 5, then will their differences be greater than themself, so that they cannot be taken out of them. And again, such little sums every child can multiply, as to say: 2 times 3, or 4 times 5, and such like. Scholar. Truth it is. And seeing me seemeth that I understand the multiplying of the greater digits, I will prove by an example how I can do it. I would know how many are 9 times 6. Master. It is all one in value to say 9 times 6, or 6 times 9: but yet the order is best to put the less sum first, saying: 6 times 9, and so of all other sums. Sch. Then would I know, how many are 6 times 9: therefore I set the digits thus, & make the cross thus. Then do I set their differences at the right side: the difference of 9 which is against it, and the difference of 6, which is 4 against it also, as in this example. And under them I draw a line. Then do I multiply the digites together, saying: one time 4 maketh 4, that 4 do I write under the differences thus. Then take I one of the differences from the other digite, as one from 6, or else 4 from 9, and each ways there resteth 5, which I do write under the digits thus. And so appeareth the multiplication of 6 times 9, to be 54. Thus I see the feat of this manner of multiplication of digittes. Master. Now might you go strait to the multiplication of greater numbers, save that both for your ease and surety in working, I will draw you here a table, whereby shall appear the multiplication of all digits, and this is it that followeth in the next page. MULTIPLICATION. 1 1 2 3 4 5 6 7 8 9 2 4 6 8 10 12 14 16 18 3 9 12 13 18 21 24 27 4 16 20 24 28 32 36 5 25 30 35 40 45 6 36 42 48 54 7 49 56 63 8 64 72 9 81 In which figure, when you would know the producte in any multiplication of digits, seek your first or last digit in the greater figures, and from it go right forth toward the right hand, till you come under the number of your second digite, which is in the highest row: and then the number that is in the meeting of the rows of little squares (which some directly from both your propounded digits) is the multiplication that amounteth of them. As if I would know by this table the multiplication of 7 times 9, seek first 7 in the greater figures, and then go right forth toward the right hand, till you come under 9 of the highest row, in which place where y●● so come under the other digit (as here for example you come under 9) is always contained the of come or product, which you seek and that place we term to be in the common angle, in respect of the two numbers 〈◊〉 taken on the outsides, as here in that common angle, where the rues of little squares (directly proceeding from 7 and 9) do meet▪ you have 63, which 63 is the sum of the multiplication of 9 by 7. Scholar. This is very good and ready. And so may I find the multiplication of any digits. But now how shall I do in greater sums? Master. When you would multiply any sum by an other, you shall mark that it is the meetest order to set the greatest number highest, which is the place of the number that must be multiplied: and likewise the lesser number under it, for that is the place of the Multiplier or multiplicator, that is to say, the number by which multiplication is made: and is in English always put before this word, times: in such speaking when I say, 20 times 70. And the number that followeth this word, Times, is that which must be multiplied. Therefore when I would multiply one number by another, I must write the greatest highest, and the lesser under it, as in Addition. And under them must I draw a line. As for example: If I would multiply 264 by 29, I must set them thus. Then must I multiply every figure of the higher row by every figure of the neither row: and that that amounteth, I must set under the line, as thus. First I do multiply 4 by 9, saying: 9 times 4 (or 4 times 9, which is all one) and that maketh 36, as the table before of digittes doth declare, of that 36 I must write the 6 that is the digitte, under the 9, and the 3 in the next place toward the left hand. Then come I to the second figure of the higher row, and say: 9 times 6 make 54, of which I writ the 4 under the 3, and the 5 under the next place (as the reason willeth me) thus. After that come I to the next figure, which is 2, and do multiply it by 9, & that maketh 18: whereof I writ 8 under the third place, and the article 1 in the fourth place, thus. And so have I ended the first figure of the multiplier. Wherefore I give it now a fine dash with my pen. Then begin I with the next figure, and multiply it into all the higher figures, as thus, ☞ First, 2 times 4 make 8, that do I write under the second place: for evermore the digite or first figure of multiplication that amounteth of the first figure of the higher number, must be set under the multiplier of it, and the other in their order, toward the left hand. Scholar. I understand you thus: that the digit of the sum amounting of the multiplication of the first figure of the higher row, by the first figure of the lower row or multiplyer, must be set under the first place: & that that amounteth of the same first figure by the second multiplier, must be set under the second place, & so of the other, if there be more multiplyers. Master. So mean I indeed: and if there amount but a digit, then must it be set under the multiplyer. And now to go forth: I multiply by the same 2, the second figure of the higher row, which is 6, saying: 2 times 6, make 12: whereof I writ the digit 2 under the third place, & the artic 1, I writ under the fourth place. Then do I multiply the last figure of the higher sum, by that same 2, saying: 2 times 2 is 4: which I writ under the fourth place. And so have I ended the whole multiplication: wherefore I also give the 2 a dash with my pen, thus: and so I do ever assoon as I have dispatched any digit by which I multiply. And the sums stand thus. Than must I draw a line under all those sums that amount of the multiplicatication, and must add all them into one sum, as in this example you may see. Where in the first place I find but 6, and therefore writ I it under the line. Then in the second place 843, make 15, whereof I writ 5, and keep one in my mind, and so forth, as you learned in Addition. And so appeareth the whole sum to be 7656, which amounteth of the multiplication of 264 by 29. Scholar. If there be no more to be observed in it, then can I do it, I suppose, as by this example I shall prove. I would multiply 1365, by 236, wherefore I set them thus. Then do I multiply 5 by 6, saying: 6 times 5 make 30: of which I writ the ciphre in the first place, and the article 3 in the second place. Then do I by the same 6. multiply the second figure of the higher sum, which is 6, saying: 6 times 6 make 36: of which I writ the 6 under the second place, and the 3 under the third place. Then do I multiply the third figure which is 3, by the same 6, and that maketh 18: of that I set the 8 under the third place, and the 1 in the fourth place. Then come I to the last Figure of the higher sum, and multiply it by 6, saying: 6 times 1 make 6: that do I write under the fourth place. And so have I ended the first multiplier, and dash him slightly with my pen. Then begin I with the second multiplier, and say first 3 times 5, that maketh 15, of which I set the 5 under the second place, because that the multiplier is there, and the 1 I set under the third place. Then come I to the second figure, that is 6, and multiply it by 3, which maketh 18, of which I set the 8 under the third place, and the article 1 in the fourth place. Than come I to the third figure, which is 3, and multiply it by 3, saying: 3 times 3, make 9, which because it is but one digit, I set under the fourth place. And then coming to the last figure 1, I multiply it by 3, and it maketh 3. which I set in the fift place, and then have I ended two of the multiplyers, and the sums stand thus. And then I give 3 his dash. Then come I to the third multiplier, and multiply it into every figure of the higher sum, and first I say: 2 times 5 makes 10, of which I set the cipher under the multiplyer in the third place, and the article 1 in the fourth place. And so multiplying the second figure 6, by that same 2, there amounteth 12: whereof I writ the digitte 2, under the fourth place, and the article 1, under in the fift place. Now do I multiply by the third figure of the higher sum, which is 3, and that maketh 6: which I set under the fift place, as appeareth in the example following. Than come I to the last place, and multiply that 1 by 2, and there amounteth 2, which I set in the sixth place, and then doth the sum stand thus. And so have I ended the whole multiplication. But now (as you taught me) to know what this whole sum is, I must add all those parcels together, and then under the line will appear as you may see the gross or total sum, that is, 322140. Mai. That is well done. Sch. Then me thinketh I would call it well done, when I knew whether I had well done or no. Master It may be tried by 9, as addition was, but the surest proof is by Division, and therefore I will reserve that till you have learned the art of Division. And before we pass from Multiplication I will yet show you another way of multiplication, which is counted of some men, and is in deed, both more readier, and more certain, which differeth nothing from this that I have taught you, save that it doth understand always the articles, and join them to the next sum, and therefore I will declare it only by an example. If I would multiply 1542, by 365, I must set them as I said before, and then do I multiply 2 by 5, and it maketh 10, of which I writ the article under the first place, and keep the digit 1 in my mind. Then say I forth: 5 times 4 do make 20, and the 1 in my mind, are 21, thereof I writ the 1 under the second place, and keep the 2 in my mind. Then come I to the third figure 5 saying: 5 times 5 make 25, & the 2 in my mind make 27, whereof I writ the 7 under the third place, and keep the article 2 in my mind. Then coming to the last figure, I say: 5 times 1 make 5, and 2 in my mind make 7: that do I write under the fourth place. And then have I ended my first multiplier, and therefore I dash it. Then do I likewise with the second multiplier, saying: 6 times 2 make 12, thereof I writ the digit 2 under the second place, and keep the article ● in my mind. Then say I forth: 6 times 4 maketh 24 and 1 in my mind make 25. so I set that 5 under the third place, and keep the 2 in my mind. Then multiply I forth, saying: 6 times 5, maketh 30, and 2 in mind make 32, whereof I writ the 2 under the fourth place and, keep the 5 in my mind. Then do I multiply the last figure 1 by 6, and it maketh 6, to the I add the 3 in my mind, and it maketh 9, which I writ in the fift place. And so have I ended ij. figures of the multiplier. Than with the third & last multiplier, do I likewise, and say first: 3 times 2 make 6: which I writ in the third place under the multiplyer. Than by that 3 do I multiply likewaies the second figure 4, and it maketh 12, whereof I writ the digitte 2 under the fourth place, and the article 1 I keep in mind. Than come I to the third figure 5 saying: 3 times 5 maketh 15, and the 1 in my mind make 16, thereof I writ the 6 under the fift place, and keep the article 1 in my mind. Then come I to the last figure, which is 1 and multiply it by 3, & it maketh 3, thereto I add the 1 in my mind, and it maketh 4, which I writ in the 6 place. And then have I ended the multiplication, and the figures stand in order thus. Which parcels if I add into one sum, it will be 562830, which is the gross or total sum of all that multiplication Scholar. Well, this manner of multiplication I perceive: but what other sorts have you? Mai. There is one way that is wrought by a chequer Table, made thus. Look how many places your sum hath that you would multiply so many squares must you make in your table, from the right side to the left: Another way of multiplication. and so many from the higher part to the lower, as there be places in your multiplier. Then set down your greatest sum first on the top of the table, every figure in due order, in a square alone: I mean in those squares that be open and uncrossed. And likewise in those like squares at the right hand, set down your multiplicator or multiplier, the last figure in the highest place, and so downward, that the first figure may be in the lowest place. Scholar. Sir if it please you, me thinketh than I understand you best, when you do not stand long in telling the rule before examples: But propose, some example, and then in declaring it, bring in the rules withal. Mai. In deed, that way is easiest for a young learner, therefore will I even so do. Take this example: now I would multiply 2●36 by 2, Then set down your first or greatest sum on the top, & your multiplyer on the right side in the open squares thus. Then begin to multiply the first figure of the higher sum, by the highest of the multiplier, saying: 2 times 6 make 12, that 12 must you write in the square that is against the 2 and the 6, but in such manner that the digit be set in the neither corner of the square, & the article in the higher corner: as you may see in this example. And so of every other multiplication, what ever amounteth you must write in the common square, which is against both those figures, by which you do multiply. And if that sum do make but one digitte, than must it be set in that lower corner of the square, but if it make an article, than writ the article in the higher corner, and let the cipher go (if you will) evermore, for here it serveth for nothing, seeing the lines do distinct the places: but if the sum amounting of such multiplication do make a mixed number, then writ the article in the higher corner, and the digitte in the lower corner, as I did by that 2. Then when you have multiplied and ended the first figure, come to the next, & multiply it in like manner, as in saying: 2 times 3, is 6: that 6, because it is but a digitt, you shall set in the neither corner of the square, next under 3, thus. Then go forth, saying: 2 times 0 is 0: set that under the bar (if you list) in the third square. Then forth and say: ● times 2 make 4, that set in the last square under the bar, so have you ended the first multiplyer: Dash him. Come now to the second multiplier, and say: 3 times 6, make 18, of which sum, the article 1 must be set above the bar, in the square that is next to the 3 (as you see here) and the 8 under the bar. Then say 3 times three make 9, set it in the next suqare beneath the bar. Then 3 times 0 is 0. writ it in the next square, or let it go, for all is one. Scholar. I perceive it well ● for here the lines distinct the places, wherefore ciphers do only serve, and therefore here they need not to be. Master. Then say farther: 3 times 2, make 6: writ that in the last square, then will the whole figure stand thus. Sch. Now could I (me seemeth) do like again. But how shall I do now to gather the sum? Mai. Mark first the order of the places in this figure, and so shall you perceive the reason of gathering them into a sum. The slope bars do part the places; so that the first place is the lowest corner in all such figures (of the nethermost square next the right hand: & all the half squares between that bar and the next, standeth for the second place, and so the room between that & the next bar, is the third place: & so forth. Now if you perceive this, then must you add all the figures of one place together, as if you had an Addition of divers sums. Scholar. If I understand you right, then must I take here in this example 8 to be in the first place: 9, 1 and 2, in the second: 0, 6, 1, in the third: 6, 0, in the fourth: 4 in the fift: and the sixth place hath no figure. Mai. You say well, and the reason is because the multiplication serving to the square, made but a digit. Scholar Then it is all one, as if they stood thus. Master. Even so it is: and now add this sum, and there will appear the total of the multiplication to be 46828. A proof without squares. And if you will see the agreement of this manner of multiplication, and the other that you learned before, then multiply those two sums (that is 2036, by 23) after the first manner without squares. Scholar. You mean to set them thus in order. And then multiply 3 into 6 make 18: 3 times 3 make 9, 3 times 0 is 0 then 3 times 2 make 6: which must be set thus. Then do I likewise with the second multiplier, saying: 2 times 6 make 12, 2 times 3 make 6 2 times 0 is 0, & 2 times 2 make 4, which when I add to the other than will the whole multiplication stand thus. Mai. So that you may see in every place the same figures, as they were in the multiplication by squares, though they differ in height and lowness of places, but being added together, they make one sum. And thus now ye have learned three sorts of multiplication, which liketh you best, that may you use. Yet are there other forms, but sith they nothing differ from these three in effect, but only in setting of the numbers, I will overpass them till a more meeter place and time. And now will I instruct you in Division, so that you think yourself sufficiently to perceive what I have taught you. Sch Yes sir I thank you, but I do not perceive how to examine my work, to try whether I have well done or no. Mai. That is commonly used by the proof of 9, as you learned before in Addition and Subtraction, save that it hath this ways divers from them. Then must you examine your sum that should be multiplied, and look what remaineth after casting away of 9, that set you at the one side of the cross: then examine the multiplier, and whatsoever remaineth in it, after casting away 9 as often as you can, writ that at the other side of the cross: then must you multiply those two numbers together, and look what amounteth thereof, if it be under 9, writ it at the higher part of the cross: but if it be above 9, then take thence 9 as often as you can, and write the rest at the head of the cross. As in the last example of multiplication, the number to be multiplied is 2036, wherein is once 9, and 2 remaineth, which I write at one side of a cross thus. Then do I examine the multiplier, which is 23, wherein there is no 9 but 5 in all, that 5 therefore I set at the other side of the cross, thus. Then do I multiply by 2 and it maketh 10, from which I withdraw 9 and there resteth 1 that 1 do I set at the head of the cross, then do I examine the gross sum, amounting of the multiplication, which is 46828. wherein I find 9 three times and 1 remaining, that .1. I set at the foot of the cross and then I see it to agree with the other 1 at the top of the cross, and so know I that I have done well: for if they two did differ, than were my work vain, and the multiplication false. This is the common proof, but the most certain proof is by Division, of which I will anon instruct you. Scholar Sir, what is the chief use of Multiplication? Master. The use of it is greater than you can yet understand: howbeit, these plain commodities it hath, that if you would resolve any great and whole valour into many small and less portions: as if you would change pounds into shillings, pence or any other greater or smaller parcels, by multiplication, ye shall do it speedily and easily. Also if you should need to add one sum to itself, or to any other oftentimes, you shall do it by Multiplication much more speedily, readily, easily and surely, then by often and sundry Additions. Take you these commodities grossly showed for an answer at this time, and hereafter I will more abundantly make you to perceive the use of it. DIVISION. Scholar. WEll sir, then in Division I pray you to instruct me. But me thinketh by the name of it, that it should be all one with Multiplication: for I call that Division, when any thing is parted into diverse & many parts. Master. You take it as it is taken commonly, howbeit, if you mark well, you shall perceive that it is quite contrary to Multiplication, and doth not part one thing or few things into many, but contrary ways it bringeth many parcels in to few, but yet so, that these few taken together, are equal in valour to the other many: for by Division pence are turned into shillings, and shillings into pounds: as for example of 1●0 shillings, it maketh 6 pounds, so are 120 turned into 6, which is a smaller number: but then if you consider the denominatours, you shall see that they are such, that one of the latter is equal to 20 of the first, and so in value the sums are one, though in number they do far differ, and the latter sum is the lesser, and so is it always in division how be it, yet in the working, the sum is parted by an other, & thereof doth it take the name. Scholar I think I shall better understand the reason of the name, when I know the use of the work, therefore now would I gladly learn that. Master. Division what it is. Division is a distributing of a greater sum by the unities of a lesser Or Division is an Arithmetical producing of a third number, in respect of two propounded numbers: which third number shall so often contain an unit, as the greater of the two propounded numbers doth contain the lesser. So that, even as Multiplication did seem to serve in steed of many Additions, so Division may seem to be in place of many Subtractions: Because that third number briefly expresseth, how many times the lesser of your two propounded numbers may be Subtracted, from the greater: As in practice will more plainly appear. Therefore (as you may perceive) unto Division are required three numbers: the first, which should be divided, and that must (generally) be the greater: and the second, by which the other must be divided, and that is (generally) the lesser, & is called the Divisor. And the third which answereth to the question, How many times: and therefore is called the Quotient. The first must be first written and the second so set under it that the last figure of the lower number be right under the last of the higher, A general rule for placing the figures. contrary ways to the work of the other kinds of Arithmetic: for in them the two first figures were set ever meet one under the other, but in Division the last figures must be set meet, except it chance so, An exception. that the last figure of the Divisor be greater than the last of the higher number, for than you shall set the last of the Divisor, under the last (save one) of the higher number, as for example. If you should divide 365 (whithe are the sum of the days of a year) by 2● which are the days of a common month, then should you set them thus. But if you would divide those 365 days, by 52. which is the number of weeks in one year, then should you set them thus. likeways if I would divide the same 365 by 4, which is the sum of the quarters of a year, then must I set them thus. Scholar. Sir, this do I understand, but how now should I do to divide the one by the other? Master. You must begin with the last figure next the left hand, ☞ and see how many times the last figure of the divisor may be taken out of the last figure of the over number, and that shall you note within a crooked line toward your right hand. As for example. I would divide 365 by 28, then set I those two sums thus. And I look how many times I may find 2 (which is the last figure of the divisor) in 3, (which is the last of the number to be divided) and considering that I can take 2 out of 3 but once, I make a crooked line at the right hand of the numbers, & with in it I set 1, and that is called the Quotiente number, as I told you. Quotient number. Then because that when 2 is taken out of 3, there remaineth 1, I must write that 1 over 3, and deface or cancel the 3 and the 2, then will the figures stand thus. Then must I go to the next figure of the divisor, and take it likewise so many times out of the figures that be over it, and look what doth remain, that I must write over them, and cancel them, as in this example. Therefore now I do take once 8 out of 16, and there remaineth 8, which I must set over the 6, and cancel or cross out the 15. and the 8 of the divisor: And then will the figures stand thus. And so have I once wrought. Scholar. So I perceive that you take the neither figure not only out of the other that is right over him, but out of that with the other also that remaineth before, and are written toward the left hand. Master. So must you do: for you must so take the divisor out of the over number, that there remain not over it so great a sum as itself is, for than were your work in vain. But yet again here must you mark, that when you seek how many times the last figure of the divisor may be found in the number over him, Note. that you look also whether you may as often find all the figures following in those that are above them, (considering all the remainders if there be any) if not, take your Quotient less by one and then prove again, and so still, till you find a moete Quotient: And by that moete quotient must you always multiply your divisor, and the product set under your divisor, so that his first figure stand under the first figure of your divisor, and the second under the second, and so forth: and then subtract that product from the number to be divided, that standeth directly over it, as you have seen me do. ☞ When you have thus wrought once, than must you begin again and write your divisor a new, nearer toward the right hand by one place, as in this example, you shall set 2 under 8, and 8 under 5, thus Then (as before) seek how many times you may take your divisor out of the number over him now. Scholar. That may I do here 4 times. Master. Truth it is that you may find 2, four times in 8: but then mark whether you can find the figure following so many times in the other that is over him: Can you find 8 four times in 5? Scholar. No, neither yet once. Master. Therefore take ● out of ●, once less. Scho. That is 3 times. Ma. Well, then 3 times 2 make 6: Mark how to consider this kind of remainder. if I take 6 out of 8, there remaineth 2: which 2 with the 5 following, make 25, in which sum I find 8 in times also, and therefore I take 3 as a true quotient, and write it within the crooked line of the quotient, before the one thus. Then say I: 3 times 2 make 6, then 6 out of 8 resteth 2, therefore I cancel the 2 and the 8 and writ over it the 2 that doth remain, thus. Then do I take 8 as many times out of 2●, saying: 3 times 8 make 24, and if I take 24 out of 25, there remaineth 1: so than I cancel 25 and 8, and over the 5 I set 1, thus. Or you mought (after you found 3 to be a fit quotient) strait way have multiplied the whole divisor 28, by that 3, at once: which giveth 84, which being set under 28, and duly subtracted from 85, of the number divided, giveth 1, the remainder of the whole division: as before you had. Work which way you list: here you see also the form. And now have I done with dividing, for I can find my divisor 28 no more in the over sum. Scho. No, except you would part the 1 that remaineth, into 28 parts. Mai That is well said, and so must we do in such cases, when there remaineth any thing: but I will let that pass now, and will make you perfect in division of whole numbers, and will hereafter teach you particularly of broken numbers, called Fractions. Now if you do perceive the order of division, then do you divide this sum, 136280 by 452. Scho. first I set down the number that should be divided, then do I set the divisor under it, so that the last figure of it be right under the last figure of the over number. Then will it be thus. Master. Can you take the last of your divisor (which is 4) out of 1, which is the last of the over number? Sch. I had forgotten, because the last of the divisor cannot be taken out of the last of the over number, in so much as it is the greater, therefore must I set the divisor one place more forward, toward the right hand thus. And then must I look how often I may find the last figure of the divisor (that is 4) in 13, which thing I may do 3 times, therefore do I say: 3 times 4 is 12, which I take out of 13, and there remaineth 1. Then do I make at the right hand of my sums a crooked line, and write before it my quotient 3: and I cancel 13 and 4, and over the 3 I set the 1 that remaineth, and then the figures stand thus. Then do I multiply the same quotient into every figure of the divisor, and withdraw the sum that amounteth out of the numbers over them, as first I say: ● times ● make 15, which I take from 46, and there resteth 1, I cancel therefore 16 and 5, and write over the 6 that 1 that remaineth, thus. Then do I say lykewaies, 3 times 2 makes 6, which I take out of 12 and there resteth 6, therefore I cancel the 12 and the 2. and over the 2 I writ 6 that remaineth, thus. Then should I set forward the divisor, into the next place toward the right hand, thus. Master. But you may see, that over the 4 is no figure, therefore must I set the divisor yet forewarder by an other place. And mark, when soever it chanceth so, that you should set forward the divisor, and that it can not stand there, because there is no number over the last place, or if there be any, it is lesser than the last figure of the divisor, ☜ then must you remove the divisor yet once again: and because that his first place of removing served not to subtract him so much as once, therefore shall you write in the quotient a Cipher 0. And if you should by chance need to do so oftentimes, for every time write a cipher in the quoitente. The reason of this, will I show you hereafter. Scholar. Then must I set my sums thus. And because I removed the divisor, so that I overskipped one place, A must write a cipher in the quotient: & then must I seek a new quotient, as in this example I must say, How many times 4 is there in 6? and sith it can be but once, therefore do I writ 1 in the quotient, and then say I: 1 time 4; taken out of 6, remaineth 2, I cancel the 6 and the 4, & write 2 over them thus. Then say I again, once 5 out of 28, remaineth 23, I let the 2 stand as it did, and over the 8 I set 3, canceling the 8 and the 5 under it, thus. Maist. You might as well have said, once 5 out of 8. & so remaineth 3 but now go forth. Scho. Then once 2 out of 0, can not be, what shall I now do? Ma. Borrow of the next number that is behind (for there is 230) and do as you learned in Subtraction in a like case. Scholar. Then must I borrow 1 of the 3 coming behind next, and make that 0 to be 10: and then take I 2 out of 10, and there resteth 8. And because I borrowed one of the 3. I must cancel the 3, and write 2 over it: then doth the figure stand thus. Master. Now have you done, and yet remaineth 228, and your quotient show you, that if you divide 136280 by 452 you shall find your divisor in your greater number 301. that is CCC times & once and 228 remaining. And in the other example, where I divided 365 by ●●, the quotient was 13, & 1 remained: whereby I know that in a year (which containeth 365 days) there are 13 months reckoning 28 days (or 4 weeks) just to a month, and 1 day more. Sch. Why then do we call a year but 12 months. Ma. Of that at a more convenient time will I fully instruct you: but now it is not convenient to entangle your mind with other things, than do directly pertain to your matter. Therefore if you remember what you have heard, you have learned a short mone● of division, which I would have you often to practise, so that you may be perfect in it, and hereafter I will show you certain other proper points touching it, Scholar. Then I pray you, yet tell me, how I shall examine and try my work, whether I have done well or no, that though no man be by me to tell me, yet I may perceive it myself. Master. Some men (yea and commonly) do try it by the rule of 9, as in all the other kinds, save that their order is this. First they cast away 9, as often as they can, out of the divisor, and that remaineth, they set at one side of a cross: As in our first example, the divisor was 28, from which you may take 9 three times and 1 remaineth which they set by a cross, thus. Then do they likewise examine the quotient (which in our example is 13) and from thence they cast away 9 as often as they can, and the remainder they set at the other side of the cross, and then multiply they together those ij. remainers: and to it that amounteth they add the remainder of the division; if there were any, from that whole sum they withdraw 9 as often as they ●an, and the rest they set at the head of the cross: as in our example the quotient is 13, from which take 9, and there remaineth only 4 and therefore must you set 4 at the other side of the cross, thus. Then multiply 4 by 1, and it yieldeth but 4 thereto add the remainder of the division (which was 1) and it will be 5, which sum doth not amount to 9, and therefore must be set wholly at the head of the cross, as you see here. And this number on the head of the cross, is the first proof, to which if you find an other like in the number that was divided, then have you done well. Therefore now shall you likewise examine the whole sum that was divided, and take away 9 as often as you can, and that that remaineth, set at the foot of the cross: and if it be equal to that in the head of the cross, then have you well done, else not. As in our example the whole sum was 365, which maketh 1●, from that take 9, & there resteth 5, which set at the foot of the cross, thus. And you shall see that they agree: therefore have you well done. Scholar. Now will I likewise examine our second example, where the divisor was 452 which maketh 11: from thence I take 9, and the 2 that remaineth I set at the right side of a cross, thus. Then examine I the quotitient, which was 301, where I find but only 4. that do I set at the other side of the cross thus. Then do I multiply 4 by 2, and it maketh 8, to that do I add the remayner of the division (which was 228, and maketh 12) and they two make 20, wherein I find twice 9, and 2 remaining, that 2 must I set at the head of the cross, thus. Then do I examine the whole number to be divided, which was 136280, where I find twice 9, and 2 remaining, which I set at the foot of the cross, thus. And because that it doth agree with the figure at the head of the cross, I know that the division was well wrought. Mai. This is the common proof, howbeit, the more certain working is by the contrary kind, as to prove Division by multiplication, thus. Multiply the quotient by the divisor, and if the sum that amounteth be equal to the sum that should be divided, then have you well divided, else not. Howbeit, this must you mark, that if there remained any thing after the division, that must you add to the sum that amounteth of the multiplication: as in our first example the quotient was 13, and the divisor was 28: Now multiply the one by the other, & the sum will be 364: to that if you add the one that remained after the division, than will it be 363, which was the sum that should be divided, & therefore I know that I have well done. Scholar. Now will I prove the same in the second example, whose divisor was 452, and the quotient 301: these do I multiply together, and there amounteth 136052, to which if I add the 228 that remained, then will it be 136280, which was the whole sum to be divided, and therefore I perceive that I have well done. Mai. This is the surest way to examine Division by Multiplication: and contrary ways the surest proof of Multiplication, is Division. And therefore now will I show how you may prove Multiplication by Division. When you have ended multiplication, A proof of Multiplication by Division. and would know whether you have well done or not, set the gross sum that amounteth of the Multiplication overmost, and divide it by the multiplier: and if the quotient be the same number that should be multiplied, then have you well wrought, else not: as in that example where we multiplied 264, by 29, the gross sum was 7656. Now if you will know whether that multiplication be true, you shall divide that 7656 by the multiplier, 29: and you shall perceive that the quotient will be 264, and that is a token that you have well wrought. Scholar. By your patience I will prove that: and first I set down the gross sum & the multiplier, not after the rule of multiplication, but after the rule of division, for now that number is become the divisor, that was before the multiplier, I shall set them therefore thus. Then shall I seek how many times 2 in 7, that may be 3 times, and 1 remaineth: but then may not 9 be found so often in 16, therefore must I take a lesser quotient, that is to say, 2● then say I twice 2 maketh 4, which I take out of 7, & there remaineth 3, then do I cancel 7 and 2, and over 7 I writ 3 and in the quotient I set 2, so the figures stand thus. Then say I forth, 2 times 9 make 18 which I bate out of 36, and there resteth 18 then cancel I 3 and over him set 1, and likewise I cancel 6 and 9, and over them I set 8, so that thus stand the figures. Then do I set forward the divisor by one place, and seek a new quotient, that is to say, how many times 2 are in 18, which I find to be 9 times, but then can I not find 9 so many times in 5, therefore I take a lesser quotient, as to say, 8: but yet is that to great, for if I take 8 times 2 out of 18, there remaineth but 2 and I can not find 8 times 9 in 25, therefore yet I take a lesser quotient, that is 7, which is also to great, for if I take 7 times 2 out of 18, there resteth 4, but now I cannot take▪ 7 times 9 out of 45, therefore yet I seek a lesser quotient, as to say, 6: then say I▪ 6 times 2, make 2, that I take out of 18, and there remaineth 6, so I cancel the 18 & the 2 and write 6 over 8, thus Then say I forth: 6 times 9 maketh 54, that take I out of 65 and there remaineth 11 and the figures stand thus. Then must I set forth the divisor again, and seek a new quotient, which will be 4: for though I may find 2 in 11 five times, and 1 remain, yet I cannot find 9 so often in 16, therefore I set the figures thus. And the 4 in the quotient I multiply into the figures of the divisor, saying: 4 times 2 maketh 8, which I take out of 11, & there resteth 3, therefore I cancel the 11 & the 2 and set 3 over the first place of 11, thus. And then do I say forth, 4 times 9 maketh 36, which I take from 36, and there remaineth nothing, so that the quotient of this division, where 7656 is divided by 29, is 264, which doth declare that if 264 be multiplied by 29, the sum will be 7656. And thus I perceive now how both Multiplication is proved by Division, & division also by Multiplication. Master. Now have I ended the five common kinds of Arithmetic: for as toucing Mediation, Duplation, triplation, and such other, they are no several kinds of Arithmetic, but are contained under the other: for Mediation is contained under Division, & is nothing else but dividing by 2: and so are duplation and Triplation contained under Multiplication: for Duplation is nothing else but multiplying by 2, and triplation is multiplying by 3, of which I will only propose an example, An example of Mediation. for the rules you have heard already. If you would mediate or divide into ●, this sum, 4531010, you shall set 2 for the divisor, & work as you learned before, as thus. Then I find 2 in 4 two times, therefore my quotient must be 2: so I cancel 4 & 2, and remove the divisor forward, thus. Then again I find 2 in 5 twice, and 1 remaining, so I writ 2 again for my second number of the quotient, and cancel 5 and 2, and over 5 I set 1, thus. Then remove I the divisor forward and seek a new quotient, which is 6: then say I, 6 times 2 make 12, take that out of 13, and there resteth 1, so I cancel 2 and 13, and over 3 I set 1 thus. Then remove I the divisor forward; and seek a new quotient which is 5. then take I twice ● out of it; and there resteth 1, so I cancel the 2 and the last figure of 1, and let the first stand thus. Then remove I the divisor forward, and seek a new quotient, which is 5: then take I 2 five times out of 10, and there resteth nothing. Then remove I again the divisor forward, thus. But because I can not find the divisor in the number over it, I must set a cipher in the quotient, and remove the divisor to the next place, as appeareth in the figure before. Then seek I a new quotient which I find to be 5, for so many times may I have 2 in 10. Then have I fully ended this Mediation or Division by 2, and the quotient is this, 2265505, which is the half of 4531010, as you may try by Duplation: for double that quotient, Duplation. or multiply it by 2, and the same number will amount. I will no longer tarry about these, seeing they are but members of the other kinds. Easy forms of multiplication. But here now will I teach you certain easy forms both of Multiplication & of division, and first of multiplication. If you would therefore multiply any sum by 10, you shall need to do no more but ad a cipher before his first place, as for example: 36 multiplied by 10, make 360. Likewise if you will multiply any sum by 100 put two ciphers at his beginning. So if you would multiply any sum by a thousand, ad three ciphers to the beginning of it. Scholar. This do I well perceive, and also the reason of it. Mai. I will omit all reasons till our next meeting, when I shall tell you the reason of all other parts of Arithmetic also: and as to our matter now, look (as I have told you) that you both remember it, and also often practise it. But if you would multiply any number by 5, mark first whether the number be even or odd: and if it be even take the half of it, and write a cipher at the beginning of it, as for example: I would multiply 2564 by 5, I take the half of it, which is 1282 (as you may know by Mediation) and before it I set a cipher, thus, 12820, and this is five times 2564. And thus may you do with any other even sum, that you would multiply by 5. But if the sum be odd, as for example 2563, then must you take the lesser half of it, or (if you will) take away 1 from the first figure (as here take 1 from 3) & then take the half of the rest, and set before it 5: as of 2563, the lesser half is 1281, for here I take but 1 for the half of 3: and if I put 5 before that lesser half, then have I multiplied it 5 times, as thus, 12815. Scholar. What mean you by the lesser half. Mai. There is no just half of any odd number, therefore if we divide an odd number into 2 parts, as nigh equal as can be, yet will the one half exceed the other half by one, as for example. The two nearest halfs of 9, are 5 and 4: and likewise of 15, be 7, & 8, where you see that the one part still is greater than the other by 1. Now it is easy to know which is the greater half, and which is the lesser half. Scho. Then I perceive you, and can do likewise (I doubt not) with any sum. For if it be not very easy to part into halfs, then will I do it by Mediation easily enough. Mai. That is a sure way. And now have you learned how to multiply easily by 5, 10, 100, 1000: and of like manner may you do with any other of that sort. But now if you will multiply by 20, 30, 40, and so forth: or by 200, 300, and such like, where there is one cipher in the first place, or many orderly in the first places, you shall take away those ciphers, and multiply the sum only by the other figure or figures, (if they be many) & then at the beginning of the sum that amounteth, shall you set so many ciphers as you took away. Example of 2873, which I would multiply by 300. First I cast away the 2 ciphers from the multiplier, & I multiply the sum by only 3 that is left, and it amounteth to 8619: before which I put the two ciphers that I took away before, and then is it 861900. And that is the sum that amounteth, when 2873 is multiplied by 300. Scholar. And if there were two or more figures beside the ciphers, I must only take away the ciphers, and multiply by the other figures, as I learned before: as if I would multiply 93648 by 25000, I should take away the three ciphers, and multiply the same by 25, and then at the beginning of that total sum, should I add the three ciphers again. Mai. Even so: but and if it chance the number that should be multiplied, or both the sums as well the number that should be multiplied as the multiplier, to have ciphers in their first places, evermore cast away the Ciphers, and work by the rest. But remember to restore as many ciphers to the amounting sum, as you bated before, as in this example: 30200 shall be multiplied by 206: I shall only take away the two ciphers from the greater number, and then multiply 302 by 206, and afterward add the two ciphers again. But if I would multiply the same 30200 by 2060, I shall not only take away the two ciphers from the number that should be multiplied, but also I may take away the one cipher from the multiplier, and then must I add 3 Ciphers to the sum that amounteth: but take heed that you take away no cipher that cometh after any signifying figure, as in this last example, you may not take away that in the fourth place of the higher number, neither that in the third place of the multiplier: how be it, yet this you may do: If one cipher or more come in the midst of your sums, you may multiply by the other figures, and overskip them, but so, that you give every figure his due place, as thus: I will multiply 3026 by 2004, therefore I set them thus. And thus do I multiply then: first 4 times 6 make 24: I set the 4 under the first place, and keep the 2 still in my mind: Then say I again: 4 times 2 maketh 8, and the 2 that is in my mind, maketh 10, I set down the cipher 0, and keep the article one in my mind. Then 3 times 4, is 12, and the 1 in my mind maketh 13, I set down the Figure 3, and keeping the 1 still in my mind having no more places of the upper number to multiply it withal: I put it down next 2 in the fift place. But now when I come to the next place being a Cipher 0: I let it go, because it multiplieth nothing: And likewise the second cipher. But then when I do come to the 2, and multiply it into the 6 of the over number, you must take heed (according as I taught you in multiplication) that the first number amounting of the multiplication, be set right under the multiplier, and the other orderly toward the left hand, according as you may see in this example. which being finished with the Addition thereof gathered together, will stand as this example showeth. Which is indeed wrought so much sooner and shorter by overskipping of the two ciphers which else would have had two workings more as by the same example here also set down doth appear. Scholar. Sir, I thank you: for I see great ease in this way of Multiplication, and if you can show me such like in Division, you shall greatly further me. Master. Yes, I will teach you some easy ways in division also, and first this: If you would divide any sum by 10, you shall only with your pen make a square line, Easy forme● of Division. between the first figure of your sum and the second, and than have you done: for the whole number that followeth the line, standeth for the quotient and the figure that is before the line, is the remainder: as for example, 3648 divided by 10, will stand thus. Where 364 is the quotient, and betokeneth that so many times are 10 in 3648: and the 8 after the line, is the remainder, which cannot be divided into 10, but by breaking it into fractions, wherewith I will not meddle yet. And so likewise if you would divide any sum by 100, with your pen, you shall cut away the two first figures, & if ye would divide by 1000, you must cut away the 3 first figures, & so of any other divisor, whose last figure is 1, & the other be ciphers, look how many ciphers the divisor hath, and so many figures at the beginning shall you cut away with the squire line, and they stand always for the remainder because they are less than the divisor, and cannot be divided by it, and the other figures that be behind the line, stand for the Quotient. But now if your divisor have any other figure in his last place than 1, and in all his other places have ciphers, look how many ciphers there be, cut away so many of the first figures of the number that should be divided, and divide the rest that followeth the line, by that figure that is in the last place, as if it were the whole divisor. Example of 64284, which I would divide by 300, here must I cut away the two first figures, (for so many ciphers my divisor hath) and must divide the rest by 3 which is the figure in the last place of the divisor. First therefore I part away the two first figures, & the sum standeth thus. Then do I divide 642, by 3, and the quotient is 214, for in 6 I find twice 3, & in 4 once, and 1 remaining, which 1 with the 2 next before, doth make 1●, wherein I find 3 four times: and this is a ready way to turn shillings into pounds: for sith one pound doth contain 20 shillings, I must divide the whole number of shillings by 20 therefore easily to do it, I see that my divisor hath one ciphre, and therefore I cut away one figure from the beginning of the whole sum of shillings, & then I do mediate or divide by 2 the other figures or sum that followeth. Scho. I will put an example. If you would divide 64287 shillings by 20, that is to say: If I would turn so many shillings into pounds, I must cast away the first figure, that is 7. & divide the rest, that is 6428 by 2 so shall the Quotient be 3214, whereby I know that 64287 shillings, make 3●14 pounds, and 7 shillings remaining. Mai. Now prove by Multiplication whether you have well done or no. Scho. The quotient is 3214. which I do multiply by the divisor 2, and it doth amount to 6428. Maist. Hereby may you perceive not only that you have well done, but also how by division you may turn shillings easily into pounds: And contrary ways, by Multiplication you may turn pounds into shillings. But here shall you see amongst divers men, diverse forms of such division, but if you mark what I have told you, you shall perceive easily all their ways: for some men do not cut away so many of the first figures of the sum that they would divide, another manner of the abridgement. as there are ciphers in the first places of their divisor, but they set all their ciphers orderly under the first places of the number that they would divide, and then with the other figure (or figures if they be many) they divide the rest of their sum. Example. If they would divide 725931, by 3400, they set their sums thus. And then do they divide orderly till they come to the ciphers: for there they stay & end their work, as in this example: They seek how often 3 may be found in 7, which is 2 times, and one remaining, therefore they set 2 in the quotient, and cancel 3 and 7, & over 7 they set the 1 that remained, thus. Then do they go forth saying: two times 4 maketh 8, which they take out of 12, and there remaineth 4, thus. Then renew they the divisor forward, and seek how often 3 may be found in 4, which is but once, and 1 remaineth, than set they 1 in the quotient, and cancel 3 and 4, & over them they set that 1, thus. Then take they once 4 out of 15, & there resteth 11. Or else more easily: Take once 4 out of 5, & there resteth 1, so they cancel the 4 & 5, and set 1 over them, thus. Then set they forth the divisor again, & seek how many times 3 are in 11, which they find 3 times, and 2 remaining: so they set 3 in the quotient, and cancel 11 and 3, and over them setteth 2, thus. Then do they multiply 4 by 3, which maketh 12, that with draw they out of 29, and there resteth 17. of which the 7 must be set over the 9, and the 1 over the 2, thus. And now are the two ciphers next ensuing, so that the divisor can no more be set forward, and therefore is the division ended, and the remainder is 1731. Now the quotient, which is 213, doth declare, that if you divide 725931, by 3400▪ you shall find it therein 213 times, and there remaineth 1731, so shall you find it, if you work as I taught you, by cutting away the two first figures, ☞ because of the two ciphers. But this must you mark (as you may perceive by this last example) that if there be left any other remainder in the sum that was behind the squire line, that the remainer must be set to the latter end of the first remayner, which was cut away with the squire line: as if you would divide 725931 by 3400, after the form that I taught you, than would your sums appear thus. So that 17, which remaineth after the line, must be set to the 31 (that was cut away with the line) in higher places, as you see here: where that 17 with the 31, do make 1731. And here would I make an end of Division, saving that there cometh to my mind one late invention of easy Division which I will briefly set forth to you, another invention of easy Division. so that if you find ease in it, you may use it. Because that the hardest point in Division, is the ready and easy finding of the quotient number: and again, if that be truly known, all the rest is but light to be done: therefore this ways shall you quickly and truly find the quotient. Firsts writ the nine figures of number: I mean 123456789, not a long as I have set them now, but up and down as in this form. And at the left side of them draw a long line, as you see here: Then consider the divisor, by which you intend to work, and set it on the left side of the long line, right against 1, and for a distinction draw a line under it: then multiply your divisor orderly by each of those figures, beginning with 2, and so go downward till you have ended all. And look what doth amount of the multiplication of each figure into the divisor, then writ it against the figure whereby you did multiply. Sc. By example I may perceive it better. Master. Take this example 263845 to be divided by 64 then must I set the 9 figures as I said before, and the divisor must I set against the 1, thus. Then must I multiply the divisor byech figure orderly: first by 2 and it maketh 128, which I must set against 2 at the left hand. Then multiply 64 by 3, and it maketh 192, which is set against 3. Then 4 times 64, make 256, the set I by 4 Then say I, 5 times 64 make 320, that set I against 5. Then 6 times 64, make 384, that set I against 6. Then 7 times 64 make 448, which I set against 7. Further I say: 8 times 64, make 512, which I set by 8. And last of all I say: 9 times 64, make 576, which I set against 9 And then the will stand thus. And so is the table ended, by which you may easily find the Quotient, as you shall see by example now. Do you set down the numbers (as you learned before) according to the order of division. Sch That is thus. Mai Now look what number standeth over the divisor, reckoning thereto all them that be behind it toward the left hand. Scholar Then are there over the divisor, 263. Master. That is just: now seek in the table on the left side, whether you can find 263. Sch. It is not there. Master. Then take the number that is next to it, beneath it: I mean a lesser number than 263, but of all the lesser numbers that the table hath, take you that that goeth nighest to 263. Sch. That is 256. Master. So is it: and mark this evermore, when you can not find justly in the table that sum that is over your divisor, than note that that is next beneath it of any sum that is in the table, ☞ and look at the right hand of the line what figure or digit that is against that sum, and take that digit for your quotient, and then work on, as you learned before: for now have I told you the whole use of this table. Howbeit, yet that you may be sure to understand it, I will see you end this example of Division by it. Now therefore begin again. Scholar. First I set down the sums after the common manner, thus. Then do I look over the divisor, and ●nd there 263▪ Now to know how many times 64 may be taken out of 263; I resort to the table aforesaid; and seek for the number 263; but it is not there, therefore as you ●ad me, I take a lesser number, the next to it that I can find in the table, and that is 256, which number hath against it on the right hand this digite 4 which I must take for the first figure of my quotient. Then do I (as I learned before) multiply that quotient into eneme figure of the divisor orderly, withdrawing the sum thereof, amounting out of the over sum: as here I say first: 4 times 6 make 24, so I take that out of 26, saying: 4 out of 6 remaineth 2, which I writ over the 6: then 2 out of 2 remaineth nothing, then cancel I 2 and 6, and also 6 in the divisor, and the sum standeth thus. Then do I likewise say forth: 4 times 4 make 16. which I take out of 23, & there resteth 7 to be set over 3, & that 3 with the 2 behind it and the 4 under it, must be canceled, as you see here. Then have I done with the figure of the quotient. Master. Now set forward your divisor, and seek a new quotient, as you sought this. Scho●●●▪ Then thus standeth the figures so that over the divisor I see 78, which I seek in the table, and cannot find it: therefore I take the next lesser, and that is 64 the divisor itself. Mai. So must you do when there is none other. Sch. Then against it I find this digit 1, which I must set in the quotient before 4, thus. Then multiply I 6 by 1, and it is but 6 still. Master. Note. You need not go about to multiply when the Quotient is 1, for 1 doth neither multiply nor divide, but in such case only subtract the divisor out of the number that is over it, Scholar. Then I take 4 out of 8, and there resteth 4, & 6 out of 7 there remaineth 1, so I cancel those numbers, & write the remainers over their places, thus. Then set I forward the divisor again, thus. Where I see over the divisor 144, which I seek in the table, and find it not: therefore I take the number in the table that is next thereto, beneath it, which I find to be 128, against which in the right side I find 2, which I take for my quotient, and that do I multiply first into 6, and thereof cometh 12, which I take out of 14, and then remaineth 2, that 2 I set over 4, and cancel the other figures, 1, 4 and 6, thus. Then say I for the: 2 times 4 are 8, which I take out of 24, and there remaineth 16, of which I writ the 6 over 4, & the 1 over 2 and cancel 2, 4 and 4, thus. Now again I set forward the divisor thus. And seeing over it 165 I seek that in the table, but find it not, therefore I take the next lesser, which is 128, against which I find ●: that do I set into the quotient, and by it I multiply first 6, & thereof cometh, 12, which I take out of 16 and there resteth 4 them cancel I 1, 6, and 6, & over 6 I set 4 thus. Then do I multiply 4 by 2, and it maketh 8, which I take out of 45, & there remaineth 37, as in this example. And now have I done. Mai. Well, now I see that you can work by this kind of division, as far forth as I taught you. Scho Yea sir, I thank you, and I find in it much ease and certainenes. M Yet one thing I doubt whether you perceive: what if you did find in the table the number that standeth over the divisor, what would you next do? Scho. I think I should take the digitte against it on the left hand for the quotient. Ma. So is it: and as often as you seek in the table and find your number just, Mark the diversity between a true quotient and a just quotient. the digit against it is your true and just quotient. I call that a true quotient also, if it be the right quotient that you should take●● though your divisor multiplied by the same, do not clearly subtract the number over it, but there doth somewhat remain, as it chanced in all the examples that you did work by. But if it should chance (as it doth often) that your divisor multiplied by your quotient, do subtract clean the number over it, them call I that quotient not only a true quotient, but also a just quotient, because it doth justly consume the number over the divisor: and that chanceth evermore when the number over the divisor is justly found in the table. Scho. This I shall remember. Mr. But yet one easy point more I will tell you in this sort of division, therefore mark it well. When you have found in the table, other the same sum that is over the divisor, other the next beneath, (for lack of the other) them look what digit standeth against it, take that for your quotient. And because it is some pain to multiply the divisor by the quotient, you shall not need to do it, but only take the number that you found in the table, and subtract that from the over number: for if you do multiply the divisor by the quotient, that will be the number that shall amount, therefore is this way more easier. S. So is it, and also more cetainer for such as I am, that might quickly err in multiplying, especially being smallly practised therein. M. Then prove in some brief example whether you can do it, and so will we make an end. Scholar. I would divide 38468 by 24, therefore first I set the table as here followeth: Then set I the two sums of Division thus. And over the divisor I find 38, which I seek in the table & find it not, therefore I take the next beneath it which the table hath, & that is 24, the divisor itself: against which is set 1, which I take for the quotient, which I set in his place. And now I need not to multiply the divisor by it, but only to withdraw the divisor out of the 38, that is over it, & so remaineth 14, as thus. Then set I forward the divisor, and find over it 144, as appeareth ●: then seek I the number in the table and find it, and against it is 6, therefore I set 6 before 1 for my quotient, and I take that 144 for the just multiplication of the divisor by that quotient, and therefore without any new multiplication I do subtract the 144, from the other 144, and there resteth nothing, as you may see. Therefore I set forward the divisor: but seeing it will not be in the next place, (for then over 2 would be nothing) I set it forward twice, as you see here. And for because that I could not set it in the next place following, therefore I set a cipher in the quotient, as you see. Then look I over the divisor, and find 68, which I can not find in the table, therefore take I the next beneath it; which I find in the table, and that is 48, and against it standeth 2, which I take for the quotient▪ And then without any multiplying of the quotient into the divisor, I do subtract that 48 from 68, & there resteth 20, as here appeareth. And so have I ended the whole division. Master. In very great sums to be divided by great divisors; I think there is no better way than this for any man to use, though he be never so expert. And that especially, if one great divisor be often to be occupied about dividing many and divers great sums. As commonly happeneth in Astronomical workings, and Geometrical, about the signs, both strait and reversed: as if it be your fortune and desire to wade to the profoundness of Geometrical and Astronomical calculations demonstrative, you will soon confess. Whereof, an other time shall better serve to speak. Now can you sufficiently skill in these kinds of Arithmetic. And now for the farther use of these two last, that is multiplication and division. I will briefly show you the feat of Reduction, by the way. REDUCTION. REduction is, by which all sums of gross denomination may be turned into sums of more subtle denomination. And contrary ways, all sums of subtle denomination, may be bronught to sums of grosser denomination. Scho What call you gross denomination, and subtle denomination? Mai● That I call a gross denomination, Gross denomination. which doth contain under it many other subtler or smaller: As a pound in respect to shillings, is a gross denomination: for it is greater than shillings, and containeth many of them. And shillings in comparison to pounds, Subtle denomination. or a subtle denomination, for because they are lesser than pounds, and many of them are contained in one of the other: as so, likewise of other things, whatsoever thing is compared to other, if it be greater and containeth many of them, it is a grosser denomination: but if it be lesser, so that many of them are in the other, then are they called subtle denominations: whereby you may perceive, that one denomination may be called a gross denomination, and also a subtle (that is to say, a great and small) in divers comparisons. For shillings compared to pounds are a Subtle or small denomination: but compared to pennies, they are a gross or great denomination. Scho. Now I understand the name, I pray you teach me the use. Mai. The use is easily learned, if you remember what you have learned before. For if you will reduce any sum of a gross denomination, To reduce gross denominations to subtle. into a sum of a smaller or subtler denomination, you must consider how many of that subtler denomination do make one of the grosser denomination, and by that number or numerator do you multiply the other sum: as if you would reduce 20 pounds into shillings, you must consider that in a pound are included 20 shillings, therefore multiply the one 20 by the other 20, and there will amount 400, whereby you may know, that in 20 pound are contained 400 shillings. Likewise if you would reduce 30 shillings into pennies, considering that in 1. shilling, are 12 pennies, you must multiply 30 by 12, and it will be 360: whereby you find, that in 30 shillings, are contained 360 pennies. And thus may you reduce any gross denomination into a more subtler, by multiplication, if you know how many of the lesser do make the greater: of which thing I will anon give you a brief table for the most accustomed kinds of money, weights, measures, and time, and such like, whereby you may know how often each subtle denomination is contained in the Grosser, when you shall need it for the foresaid kind of Reduction. And also the same shall serve you, if you would reduce any sum of a subtler denomination, To reduce subtle denomination to gross. into a sum of a grosser denomination: For in such Reduction you must consider (as in the other form) how many of the smaller do make the greater, and by that number must you divide the other sum, and the quotient will declare how many of the greater denomination, are comprehended in that sum, as for example: If you would know how many shillings are contained in 3240 pence, consider that 12 pennies do make 1 shilling: you must divide that 3240 by 12, and your quotient will be 270, whereby you know that so many shillings are in 3240 pennies. But and you would know farther, how many pounds are in those 270 shillings, séing that every pound containeth 20 shillings, divide that 270 by 30, and it will be 13, and 10 remaining, whereby you may, know that in 3240 pennies, or 270 shillings, are 13 pounds and 10 shillings. For evermore the remainder must be named by the name or denomination of the sum that was divided, which in this place were shillings. ☞ And thus may you do with any other kinds of denominations. Wherefore to the intent that you may have a light knowledge in the common Coins, weights, measures, and such other, I have prepared here a brief table, which shall suffice to you at this time, till hereafter at more convenient opportunity I may instruct you more exactly in the same. Note (gentle Reader) these values of English comes, as they were when this Author first published his Book. But in our time (namely An. 1582)▪ they are much diverse. Therefore something to pleasure thee in this purpose. I have for thy benefit at the end of Reduction, set down and annexed a Table, not only of our coins, but also of the most part of Christendom, with their just weight and values, currant in this Realm of England, as by the same shall plainly appear. A Table for English coins. An, 1540 A Sovereign. Half a Sovereign. A Royal. Half a Royal. A quarter Royal. An old Noble. Half an old Noble. An Angel. Half an Angel. A George Noble. Half a George Noble. A quarter Noble. English Coins. A Crown. Half a Crown. A Crown. A Grote. A harp Grote. A penny of 2 pence. A dandie pratte. A penny. An half penny. A Farthing. The value of English Coins. The value of English Coins. A Sovereign is the greatest english coin, and containeth 2 Royals, or 3 Angels, either 9 half Crowns, or 4 Crowns and an half, that is to say, 22 s 6 d. Half a Sovereign is equal with a Royal. A Royal containeth an Angel and a half, that is to say: 11 s, ● d. Half a Royal containeth 5 s, 7 d, ob. A quarter of a Royal containeth 2 s, 9 d, ob. q. An old Noble, called an Henry, is worth 2 Crowns, or a Noble and half, that is 10 s. Half an old Noble is worth 5 s. An Angel containeth a Crown and half, or 3 half Crowns, that is 7 s 6 d. Half an Angel is worth 3 s, 9 d. A Noble called a George, is worth 6 s 8 d. Half a Noble is worth 3 s, 4 d. A quarter of a Noble (which in the old Statute is called a Farthing) containeth 20 d. A Crown containeth 5 s: and the half crown 2 s 6d. Howbeit there is an other crown of 4 s. 6 d which is known by the rose side, for the Rose hath no Crown over it, as in the other Crown, but it is environed on the 4 quarters with 4 flower deluces, whereby you may best know it. But I will return to speak of the value of the coins, for I intent not now to describe the forms of them. Now of gold are there no more common coins. In silver the greatest is a Groat, which containeth 4 pennies. Then is there another Groat called an Harp, which goeth for 3d. Then next is a penny of 2d. And then a Dandiprat, worth 3 half pence. next it a penny, then half a penny, and last and least of all a Farthing, whose coin is on the one side a cross, & on the other side a purculles. This I tell you, because I see many that cannot know a farthing from a small half penny. Now have I told you all the English coins both of gold and silver, but yet of the three most common valewers of money spoke I nothing, that is to say, of pounds, Marks, and shillings, which though they have no Coins, yet is there no name more in use than they: of which the shilling containeth 12 pennies or 3 groats: and the pound 2 old Nobles, 3 George Nobles, or 4 Crowns, that is to say, 20 s. A Mark, two George Nobles, that is 13 s. 4 d. Here would I now express the values of sundry other coins of divers countries, but for three causes I now refrau●e. The first and chiefest is, because they are not corrant by the Statutes of this Realm. another cause is, by reason they are so uncertain, that they be never long at one rate. And again they are so different in so many places, that it were matter enough for a great book, to speak sufficiently of them all. Howbeit, yet because you shall not be altogether ignorant of them, I will show you the values of some that are most in use, and first of France. French coins. The most common money are Deniers, Soulx and Franks, 12 Deniers make 1 s. 20 soulx make 1 Frank, so that as you see, these three kinds are like in the rate: to penies, shillings, and pounds with us, but that this is the difference, that their Denier is but the 9 part of our penny, and so their soulx (commonly called sowses) go 9 to our shilling, and 9 of their franks to an English pound of money: So that 3 of their Franks make a noble. And by those 3 may you practise how to reduce French money into English money. And as for the rest of their coins I will omit till an other time, when I intent to show you the rate of sundry other kinds of money. But now as for the coins of Flaunders they be so changeable, Flaunde is Coins. that you must know them from time to time, else you cannot reduce them into our money certainly. But yet that you may have an example of their money to exercise you withal, you shall take those that be most common as Stivers both single and double, Groats Flemish, Carolus, and Gyldens: A Flemish Grose is a little above 3 farthings English. A single stiver is 1 d, ob, q. The double Stiver is ● dq. The silver Carolus single, 2 dq. q. c. The double stiver Carolus is 4 d ob, q, q. Then is there also the Carolus Gylden which is worth 20 Stivers And the Flemish noble is worth 3 Carolus Gildens, and 12 Styvers. Touching Dansk money, they have their soulx whereof 20 is a Liver, which is 2 s sterling. They have also their Grashe whereof 30 makes a Gylderne, which is four s. sterling: They have also Dollars, and their common or old dolor is 35 Grashe: new Dollars they have, which be divers, some valued at 24 Grashe, some at 26, and some at 30: and thus much I thought good to add to the Author touching Dansk money. But I will let them pass now, exhorting you to practise to reduce those kinds into English money, according as I have set forth here following: 2100 deniers, make 240 d: or 20 s: 3240 deniers, make 360 d, or 30 s 8352 deniers, make 928 d, or 2 lb 17 s, 4 d: 2160 soulx, make 240 shillings: and so of other the● in ●ilie rate. But if you will reduce Flemish money justly, you must reduce it first into the smallest part of English money that is in that come, as for example. If I would reduce 368 double stivers into English money, considering that a double Stiver containeth 131 d q, you shall first look how many q be in the double Stiver, and you shall find them 12, therefore multiply the sum of the stivers by 13, and then have you their value in farthings, which is 4784. Now if you divide that by 4, then will there appear the number of pence: but better it were to divide it by 48 (for so many farthings are in 1 shilling) and then will the quotient declare the sum of the shillings. Likewise, if you would reduce any sum of single styvers into English money, you must multiply the sum first by 13, & then have you a certain sum, which sum if you divide by 8, then will amount the sum of pence: or if you divide it by 96, the sum of shillings will appear. But this mark in all division, when ye do reduce to bring one denomination into an other, ☜ if there be any remainer after the division, that must be named by the denomination of the gross sum that was divided: as for example: I would bring 254 q into pence, therefore I do divide that 254 by 4, for so many farthings make 1 penny, and the quotient is 63, which is the sum of the pence, and then remaineth yet 2, which are farthings still, as one may prove by dividing. And this must be marked in all Division, namely when it is done for Reduction. Yet two words more added to the Author Concerning Spanish money, whereof the most common money are Cornadoes, Marueides rials and Ducats: 6 Cornadoes make a Marueide, 34 Marueids maketh one Ryall, and 11 Rials maketh one Duckate, so the Ducat containeth 374 Marueids, which to reduce into sterling money English, 34 Cornadoes maketh our penny: or 5 Marueides & 4 Cornadoes etc. Weights. Thus much have I said of Money, now will I show you in like sort the distinction of weyghts, after the statutes of England, where the least portion of weight is commonly a grain, meaning a grain of Corn or wheat, A Grain. dry and gathered out of the middle of the ear. A Penny of Troy. Of these grains in time passed ●2 weighed just 1 penny of Troy, and then was but 10 pennies in an Ounce. An Ounce. But now are there 46 pennies in an Ounce, so that there are not fully 14 grains in one penny. But now of Ounces after Troy rate (which is the standard of England) 12 do make 1 pound. Haberdepoise weights. But commonly there is used another weight called Haberdepoise, in which 16 ounces make a pound. Therefore when you would reduce ounces into pounds, you must consider whether your weyghts be troy weight or Haberdepoise: and if it be Troy weight you must divide your ounces by 12, to bring them to pounds: but if it be Haberdepoise, you must divide them by 16. Now again, there be greater weights which are called an hundred, half a hundred, A hundred wa●ght. & a quartern, and also a half quartern etc. Scholar. Why? so there may be reckoned 20 pound, 40 pound, 200 pound and such innumerable. Master. All these are numbers of weight, but they have not common weights made to their rate, as the other have. And again, these that I did name are not just in number as they seem by their name, for an hundred is not just 100, but is 112 pound. And so the half hundred is 56: the quarter 28, and the half quarter 14. And these be the common weights used in most things that are sold by weight. Howbeit there are in some things other nams', as in wool, Wool● weights. Todde Stone· 28 pound is not called a quartern but a Todde: and 14 pound is not named half a quartern but a Stone, and the 7 pound half a Stone. Other names because they differ in many places, and agree in few, I let them pass. Sack. But a Sack of Wool by the Statutes, is limited to be 26 Stone. Cheese weights. Now in cheese, though it be sold by the hundredth, and by the Stone in some places, yet the very weights of it are Cloves and ways. So that a Clove containeth 8 pound: and a Weigh 32 Cloves, which is 256 pound, that is 12 score and 16 pound: And so much weigheth the Wey of Suffolk cheese: And the like weight is or should be the Barrel of Suffolk Butter. The Wey of Essex Cheese containeth 16 score, and 16 pound: And so much is also the barrel of Essex Butter. Measures for liquor. Now of weights are made other measures, both for grain and liquor. For a pound in weight maketh a Pint in measure, A Pint. Gallon. pottle. Quart. so that 8 pound (or 8 Pints) do make a Gallon: half a Gallon is named a Pottle: and half a pottle is called a Quart, which containeth two pints. Fyrkin. Tertian. Kilderkin. barrel. Now above a Gallon the next measure is a Firken: then a Tertian, a Kilderkin or half Barrel and a Barrel. And by those measures are sold commonly, Ale, Beer, Wine, and Oil, Butter, and Soy● Salmon, Herrings, and Eels. But as these be unlike things, Ale measures so the measure of their vessels do differ: for the measures of Ale are as followeth: Of Ale. the Fyrken the Kilderken the Barrel containeth 8 16 32 gallons Of Béer the Fyrken the Kilderken the Barrel containeth 9 18 36 gallons Soap measures, both Firken, Kilderken, Soap measures. and Barrel, should be equal to all Ale measures. Moreover the Statutes do limit the weight of every of those three vessels being empty. A Barrel Half Barrel A Firken to weigh empty 26 13 6 ½ pounds. Herrings also be sold by the same measures that Ale and Soap be sold by. Herring. Herrings also are sold by the tale, 120, to the hundred, ten thousand to the last. Salmon & eels have a greater measure. Salmon and eels. Salmon & eels the butt the barrel half bar. the firken holdeth 84 42 21 10 ½ Gallons. Howbeit, some Statutes did limit Eel vessels equal with Herring vessels. Wine measures. Now as for wine vessels seldom are smaller than Hogs heads, which are of 63 gallons: every Hogs head is two Barels: yet there are many other wine vessels, but of them all, see this table, and mark the measures one to an other. Of wine and oil. the Rondlet the Barrel the hogs head the Tertian the Pipe the Ton holdeth ●8 ½ A 1 ½ 63 84 126 252 Gallons. Tertians. But you shall mark, that there be other kinds of Tertians: for there be Tertians (that is to say) Thirdles of Pipes, of hogs heads, and of Barrels, as well of other things as of wine. B●tte. Also of Malueseys', and Seek, etc. the half Ton is not called a Pipe, but rather a Butt. And thus much have I thought meet to tell you at this time. Scholar. And is that always true? Master. I have told you how it should be, but how it is I may not say: how they do differ daily from their just measure, that Gagiers can tell you better than I. But I will let this pass now, and speak briefly of the other measures. And as of weights there did spring the liquid measures, (whereof I spoke last) so of the same springeth dry measures: as Pecks, Dry measures. Bushels, Quarters, and such like, whereby are measured corn and like grains: also salt, lime, coals and other like. And this is the order and quantity of them. A Peck is the measure of two Gallons. A Peck. A Bushel containeth four Pecks. A Bushel. A Quarter holdeth eight Bushels. Quarter. A Wey containeth six quarters. These are the common names & measures, Wey. but in divers places there be divers sorts. The bushel in many places is 2 bushels: but then is the bushel there called a Strike. Strike. And in some places half a quarter is called a a Cornoke. But these diversities are to many to tell you briefly them al. And again, sith they are against the law and Statutes, I count them unmeet to be used. But now remaineth yet an other kind of Measures, Measure to meat length, breaden, & thickness. whereby men met length & breadth, and thickness, and those are an Inch, a foot, and such other: whose names and quantities this table showeth. An inch. 3 Grains of barley in length, make an inch. 12 inches make a foot. Foot. Yard. Elle. Perch. 3 Foot make a yard. 3 Foot and 9 inches make an Elle. 5 yards and a half, make a Perch. 1 Perch in breadth and 40 in length, do make a Rod of land, which some call a rood, some a yard land and some a Farthendele. Acre. 2 Farthendels, make half an acre of ground. 4 Farthendels make an Acre. More 40 Rods in length do make a furlong: 8, furlongs make a mile which containeth 320 Perches. So that an English mile grounded upon the Statute is in length 1760 yards, 5280 foot, and 63360 inches. Somewhat greater than the Italian mile 1000 paces, and 5 foot to a pace. Here might I tell you many things else touching measure, and also how to reduce strange measures to our measures, but because it can not well be done without the knowledge of Fractions, which as yet you have not learned, I will let them pass till an other time, when I shall instruct you in Geometry, wherein I should be enforced else to repeat the same often again. Scholar. But yet sir of the parts of time I pray you tell me somewhat. The parts of time. A Day. An hour. week. month. Year. Master. You know that a natural day hath 24 hours, and every hour hath 60 minutes. It needeth not to tell you, that 7 days make a week, and 4 weeks make a common month, and 13 months make a year, lacking 1 day, & certain hours, and minutes: But of that I shall instruct you hereafter. Here will I make an end of Reduction for this time, which though it be counted no kind several of Arithmetic, yet you see it is no less needful to be known, or easier to be done, than of any of the other. Scholar. Marry sir, it seemeth unto me much harder than any other sort, for it requireth the knowledge of so many things: but now sir when you see time, I am ready to learn forth: for as much of Reduction as you have taught me, I remember, but and if I do at any time forget, I shall have recourse to the tables which you have set forth for me. Master. So do you, for it will not be remembered without exercise. And now according to promise here followeth the Table which I have added to this Author for thy utility, intending at the latter ends of my Addition to this Book, to write of the ordinary money used in most parts of Christendom: and their common values currant for traffic in those places, with the manner of their exchanges, as also their usual weights and measures, which I hope will be as gratefully taken of Gentlemen, merchants, and other my Countrymen in general as I of good will set forth the same. A Table of the names and valuation of the most usual Gold coins throughout Christendom, with their several weight of Pence and Grains: and what they are worth of currant money English. The names and titles of the gold. The weight. Penny. Grains. The value. Shil. Pence. Royal 4 23 15 0 Half Royal 2 11 8 6 Old Noble 4 9 13 4 Half old Noble 2 4 6 8 Angel 3 7 10 0 Half Angel 1 15 5 0 Salute 2 5 6 4 2. parts of Salute 1 11 4 2 George Noble 3 0 9 0 half George noble 1 12 4 6 First crown K. H. 2 9 6 4 Base crown K H. 2 0 5 0 Great Sovereign 10 0 30 0 Sovera. K. H best 0 0 10 8 Edward Sovera 3 14 10 0 Sovera. K.E. 3 14 10 0 Vnichorne of Scot 2 10 6 6 Elizab. Sovereign 3 14 10 0 Elizab. crown 1 19 5 0 Scottish crown 2 5 6 0 French Nobulle 4 16 13 4 All the sorts of French crowns 2 5 6 0 Old French cro. 2 5 6 0 Flaunders Rider 2 6 6 6 Gelder's Rider 2 2 3 6 philip's Royal 3 10 10 0 philip's crown 2 5 5 0 colyn Gilden 2 2 4 8 New Andr Gyld: 2 3 5 0 Flanders noble 4 10 12 Flem. Angel best 3 6 9 Fland. real, or Key 3 10 10 Carolles, Gylden 1 21 3 6 Flanders Royal 2 6 5 Saxon, Gylden 2 2 4 8 Flanders crown 2 5 6 Phillips Gild. 2 3 4 2 Golden Lion 2 16 7 8 3. parts of gol. Ly. 21 2 5 2.3 parts gol. Ly. 1 19 4 11 David's Gylden 2 2 ● Horn Gylden 1 12 4 11 Old Andre, Gyld. 2 4 4 10 Crusado long cros. 2 6 6 Crus. short cross 2 6 6 2 Mill rays 4 20 13 4 Half Mill rays 2 10 6 8 Portigu. 1. ounce 2 16 3 lb. 8 s. Portigu. 1. ounce 2 18 Golden Castilio 2 23 8 10 Ducat of Castille Ducat of Arags 2 6 6 6 Hungary Ducket 2 7 6 4 Double Pistolate 4 8 11 8 Single Pistolate 2 4 5 10 Ducat of Valens 2 6 Ducat of Floren. 2 5 6 4 Double Ducat 4 11 13 Single Ducat 2 6 6 6 Dou. duke. of Rome 4 13 12 8 Of silver coins currant in this Realm The Edward Crown of 5 s The Edward half crown of 2 s 6 d. The Edward shilling, half shilling, and the 3 d. Philip and Mary's shilling and half shilling. The Marry groat, and Marie 2 d. Queen Elizabeth's shillings 6d. 4d. 3d. 2d. 1d. 3. ob. and 3 q. It is to be understood (gentle Reader) that whereas the weight is called by the name of a penny, Note. it is not meant a penny of silver money, but a penny of goldsmiths weight, which is 24 Barley corns dry. And xx. of these pence make an ounce: and twelve of these ounces make a pound Troy: So that if a man have not the weight where with to weigh any piece that may come to his hand, he may do it with the Barley grains or corns being dry and taken out of the middle of the ear. Now to Progression. PROGRESSION. Although until this day the most part of writers have defined Progression as a compendious kind of Addition, yet truly it is not so: for progression (as the very nature of the word doth inform any man) is a going forward and proceeding in numbers, and that regularlie and orderly, whose place is aptly chosen to be very ●eare, or rather next after the exposition of the four principal parts of Arithmetic, for in it after a most easy manner, are all the four former parts exercised and practised: and not only Addition, as customably is done. Which custom hath been the cause, why it hath so specially been named a kind of Addition▪ and defined to be a quick and brief Addition of diverse sums, proceeding by some certain and reasonable order. You shall also understand, that there are infinite kinds of progressions, but for you (as yet) two are sufficient to be exercised in: of which the one I call Arithmetical, and the other Geometrical. Arithmetical progression. Arithmetical progression is a rehearsing or placing down of many numbers, number after number, in such sort, that between every two next numbers rehearsed or placed down, the difference, diversity, or excess, be equal and alike. Scho. Sir, I thank you for that you have both opened unto me what Progression is, truly, and also why it is here placed. But I pray you with an example make plain your definition. Ma. Examples can not want, seeing all reasonable creatures naturally use the order of one kind of Arithmetical progression (which therefore is also named Natural) when so ever they distinctly do count or number any multitude one by one, saying: 1.2.3.4.5.6. whereby the proceeding from number to number, and every one surmounting and exceeding his fellow next before by a like quantity (which here is 1) declareth the same to be Arithmetical progression. And for the more plainness, I set it down in this manner. Sc. This is most evident. And I think that I am able to tell you now of any progression Arithmetical propounded, what is that common excess or difference whereby it proceedeth if this order be kept in it. M. What say you of 3.6.9.12.15? Sc. They exceed each other by 3. And that may I set down in such evident order, as you did your example of Natural progression, in this wise. Master. And do you not also now perceive, that the whole table of Multiplication may be made by the order of progression Arithmetical? either if you will begin at the first number of any of them on the left hand, and so proceed right overthwart: or at any of the first numbers of the upper row, and go directly downward? Sch. I pray you let me consider the thing a little, and I will answer you. 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 6 9 12 15 18 21 24 27 30 4 8 12 16 20 24 28 32 36 40 5 10 15 20 25 30 35 40 45 50 6 12 18 24 30 36 42 48 54 60 7 14 21 28 35 42 49 56 63 70 8 16 24 32 40 48 56 64 72 80 9 18 27 36 45 54 63 72 81 90 10 20 30 40 50 60 70 80 90 100 By this trial I perceive it now very well: for the common excess or difference between any two next, is continually as much as the first number of every row, either from the left hand overthwart taken, or from any of the uppermost overthwart rows downward. Ma. Now than if of any such progression you would speedily know the total sum, much quicklier than by common additions rules: To know the total sum of an Arithmetical progression. first tell how many numbers there are (which numbers here we call places or parcels) and if they be odd, writ their sum down by itself, as in this example, 2, 4, 6, 8, 10, 12, 14, where the numbers are 7, as you may see, therefore set done 7 in a place alone: then add together the first number and the last, as in this example: add 2 to 14, and that maketh 16, take half of it, and multiply by the 7, which you noted for the number of the places, and the sum that amounteth, is the sum of all those figures added together, as in this example: 8 multiplied by 7 make 56: and that is the sum of all the figures. Scholar. That will I work by an other example. I would know how much this sum is, 5, 8, 11, 14, 17, 20, 23, 26, 29. I tell the places and they are 9, that I note. Then I put the first number 5 and the last 29, together, and they make 34. I take the half of it, that is, 17, and multiply by 9, and it maketh 153. That you say is the sum of all the numbers. Ma. So shall you find it if you try it. Sc. How shall I try it? Master. By your common addition: for if you add all the parcels together, you shall see the same sum amount, if you did work well. And that manner of Addition trieth all kinds of summing any Progression. Sc. Then can I sum a progression, if the numbers of the parcels be odd. But what if they be even? as in this example, 1, 2, 3, 4, 5, 6, 7, 8? Ma. When the number of the parcels is even, then note that also as you did before, and likewise add the first sum to the last, and by the half of the number of the places do you multiply it: as in your example, the parcels are 8, that note I? then adding the first sum to the last, there amounteth 9, that do I multiply by the half of parcels, that is by 4, and it maketh 36, which is the sum of the 8 parcels. ☞ But if you will take one rule for these both, do thus. Multiply the half of the one by the other whole, and the sum will amount all one. For sometime it chanceth that the number of the parcels be odd, so that their half can not be taken: and sometime it chanceth the Addition of the first number and the last, to bring forth an odd number, so that the half of it can not be taken: but they will never be both odd. Sc. Then I perceive this, if there be no more longing to it. Master. As accustomably it hath been taught, this hath been the chief and only exercise in Progression used. But that you may perceive how diverse ways and to how great profit so simple a thing (as this Arithmetical Progression is) may be considered and used, I will here propound you six propositions, of which four of them were invented by a friend of mine, and never before this published: and the first two, were never to my knowledge written of, but by three men. Sch. This doth greatly encourage me to be attentive unto your words, seeing I shall not only be instructed at your hands in the common known rules of this excellent art, but besides that, so abundantly in other new rules informed, as my very entrance shall seem to pass a great many men's farther study, and longer continuance. Therefore sir, I beseech you, let me know your six propositions. Ma. These they are. 1 To know the last number without proceeding by continual addition, till you come unto it, ●o that the common excess, the first number and the number of the places be known. 2 The first number of the Progression and the last being known, with the common excess, to find the number of the places. 3 The excess being given, and the first or last, to know the quantity of any middle number, whose place is given from the first or last. 4 The total sum being given, and the first and last, to find out the number of the places. 5 The total sum of any Arithmetical progression being given, and the first and last, to find out the common excess. 6 The total sum being given, and the mutual excess, with the number of the places, to give the first or last number of the same progression. Many more considerations could I propound you in these Arithmetical progressions, but these are sufficient to give you occasion to think, that rules of knowledge and arts are infinitely capable of enlargement. Sen. Happy were I, if I did but well understand that which is already invented and written. And yet in my simple fantasy. these things offer themselves (in manner) to be studied for about Progression, therefore I pray you to proceed to the rules answering to these propositions. Master. I will orderly for every of these six propositions give you rules, and with every one an example, unless the plainness and easiness need no further exemplefying. For the Solution of the first. Multiply the excess by a number less by 1 than the number of the places, and the offcome add to the first number, so shall you have the last number, which is sought for. As (for example) if there were seven places in a progression Arithmetical, whose continual increase, or mutual excess were 5 and the first number were 5, and I would know what the last & seventh number is, I multiply 6, which is less than 7, (the number of the places by 4, thereof cometh 24, which I add to 5, that maketh 29: and that is the last number, which I desired to know. And this you may straight way prove, by continual proceeding from 5 till the seventh place, increasing every one by 4, as thus. 5 9 13 17 21 25 29. Lo here, the last, being also the seventh, is 29. Sc. I perceive already one good property in this rule, which in all works is to be desired: that is, it will ease one from great labour, if a progression were propounded of 100 or 200 places, or more, And also it is very easy to work, and most necessary for the total sum finding, in a very long progression. Mai. The second rule is this. From the last subtract the first, the remainder divide by the common excess, to the Quotient ad 1, and you have the number of the places, which you would know: As in this progression. 6 11 16 21 26 31. If I know only 6 and 31, and that they increase by 5, than according to the rule, from 31 I subtract 6, there remaineth 25, which 25 I divide by 5 (the common excess) the quotient cometh forth 5, to which I add 1, that maketh 6: and so many are the places, as you see. Scholar This rule is so easy, that I were much to blame, if I could not remember it. Ma. The third proposition may always thus be soluted: Multiply the excess by a number less by 1. than the distance of the place is from the first or the last number given: the of come ad to the first, if the distance be reckoned from the first, and the first also known or subtract from the last: if the distance be from the last counted, and the last given also, and that which cometh forth, either in that addition to the first, or subtraction from the last, is the number sought As for example I propound you this progression. 8 15 22 29 36 43 50 57 And for the apt considering the manner of this question, I will note over every place his distance from the first and under every place his distance inclusively from the last, thus. 1 2 3 4 5 6 7 8 8 15 22 29 36 43 50 57 8 7 6 5 4 3 2 1 Now, if that excess whereby this Progression standeth, be known to be 7, and the first number given, being 8, I would know what number standeth under 4, that is to say in the fourth place. I multiply 7 by 3 (which is less by 1, than the number of the place propounded) that yieldeth 21, to which I add 8 (the first number) so cometh 29: which I say to belong to the fourth place, as ye see in the example it also doth: or if in the third place from the last, you would know what number in this example should stand, the last number being known to be 57, and the common excess 7, than by 2 (which is less by 1 than the place propounded) I multiply 7, that giveth 14: which I subtract from 57, so remaineth 43: which appertaineth to the third place inclusively reckoned from the last, & so my example giveth you. Scholar. I perceive right good use of this rule: for if I had forgotten what the first number were, and remember still but the last, the common excess, and the number of the places, than might I come by the knowledge of my first number again. And me thinketh, that it differeth not much from the first proposition saving that which you make here a middle number, there was made the last: and also in this point it differeth, that in it the last was only sought, and no consideration had in numbering the places from the last, as here I mark in your numbers noted under your progression. Master. And think you not the middle numbers of a Progression standing of a hundred or three hundred places or more, may as much cumber a man to come to the knowledge of them by continual increasing from the first (by the common excess) or abating from the last continually (the common excess) as the very final numbers in a shorter Progression would do? Scholar. Yes sir, that I think right well, and therefore I am glad of this new framed proposition, and the manner of the working of it. Master. The rule of the fourth is this. Add the first and the last together, and by the ofcome divide the total sum. Double the quotient, and that will be the number of the places. Scho. Then if in a Progression, whose sum were 207, and the first number 12, and the last 57, if I add 57 and 12 together, that maketh 69: and by it I divide 207, the Quotient will be 3, which I double, and so I have 6, and so many must be the number of the places, that this progression standeth on. Mai. Whether it be so or no, how will you try? Scho. Half 6, which is 3, being multiplied by 69, must make 207, the total sum, if 6 be the number of the places. For so the whole work of your rule in summing any Arithmetical progression did inform me. I will than multiply by 3, thus. It cometh forth justly. Maist. I must much herein commend your promptness, both in memory and in well applying your rule: although in manifest words if did contain no such matter. Scho. Sir, I pray you hear me frame one example more. Maist. I am well pleased, so that ye be short, for you make me more longer here, then willingly I would have been: but I can not perceive how I could have omitted any thing as yet, without your great lack thereof. Sch. If I had received 85 pounds of certain men but of how many▪ I have forgotten, yet I remember that the first gave me 7 lb, and the last 27 lb, and every payment after other did rise by a like sum. And the man for whom I received this money, conditioned with me, that of every payment I should have 12 pence for my labour: now unless I can by art find the truth of this case, I am like to lose the most part of my reward. Master. I perceive you can handsomely frame an example, which should concern your own gain: I pray you let me see how you would do justice in this point. Scholar. I add the first and the last together that maketh 34: by which I divide 85, thus. Why how now? Sir, here is a remnant of 17, in which 34 cannot be had, so that now I am in the briers for doubling of my quotient, and farewell then both my justice, and a good lump of my gains. Ma. Ye are never the farther from the matter, though it fall into a fraction. For you shall understand that the fraction which of any such work proceedeth, is ever half of one such, ☞ as the vnits of the quotient before are. And that you may try, if you double that which so remaineth, for than it will be equal to your divisor, as if ye double 17 (the remnant) it maketh 34, and your divisor also was 34, this noteth the remainder to be half of one. Sch. Now I am glad of this hard example. For with it I have a general rule for the Fraction that may hap in this work. So that the quotient being two & a half, I double that, & it maketh 5, therefore should my gain be 5 shillings. And to be sure (by your leave) I will try it for I will multiply half of 34, (which is the first and last number joined together) by 5 thus. It is most true (I see) that I should lose nothing by the former working. Mai. The fift proposition hath this rule appertaining unto it: By the fourth rule find the number of the places, that being done, from the last subtract the first, and the residue divide by a number less by 1, than the number of the places, and the quotient will show the excess which is sought for. An example hereof shall be this: If ye had disbursed 685 lb. to a certain number of men, you neither can tell how many they were or how much the ones money exceeded his next before, but you are sure that the excess was equal between every two next: & also you remember that the first had 19, and the last 118 pounds, how would you find both the number of the men & the excess, continually observed in the succession of their reiments. Scholar. Your rule doth plainly bid, first to find the number of the places, which I will do according to she fourth rule. I add 19 and 118 together, thus. By this 137, I divide 685, thus. Seeing there is no fraction, but a whole number, being 5. I double that, and than must the number of the places be 10 Now from the last I subtract the first, as 19 from 118, thus: And so remaineth 99 This 99 I divide by a number less by 1 than the number of the places, and seeing the places were 10, I divide 99 by 9, thus. The quotient is 11, and so was the excess, if I have followed your rule right. May. You have wrought every part of this question both well in order and truly in the practice of your rules. Sch. I will than set it down also formablie, so that the number of the places, the excess and the total sum may straight appear, as your first example stood. The common excess. The Progression. 11 11 11 11 11 11 11 11 11 11 19 30 41 52 63 74 85 96 107 118 That the places be 10, and that from the first to the last the common excess is 11, I perceive most evidently, but whether the total sum be 685, I have not yet proved, which I will now do. I adds 19 and 118 together that maketh 137: I multiply that by half the number of the places thus. All things agree most exactly, so that I am perfect enough in these rules, if I forget them not again. Mai. Use maketh all things perfect. Your sixth rule is this. By the number of the places divide the total sum, double the quotient, and that will be the first and the last joined in one sum. Than by a number less by 1, than the number of the places multiply the excess, that ofcome subtract from the first doubled quotient, and the half of the residue is the first number. The last number you may diversly find out, as by the first of our six rules, or by subtracting this first number from the sum which here contained both the first and the last jointly (or thirdly) by continual adding the excess. Scho. I pray you make this somewhat more plain with an example. Mai. If every month in the year (counting them now as thirteen) you gained clearly 40 shillings more than you did the month next going before, and at the years end you find the whole gain 5720 shillings, but ye remember not how much either the gain of the first month, or the last was, by this rule it may be tried out. Sc. So that here ye seem to apply the 13 months to 13 places the 40 shillings every one more than the other next before it, to be the common excess, and 5700 s. to be the total sum. Mai. It is true: by 13, then I divide 5710, in this manner. I double this quotient, so have I 880 for the first, and the last sin joined together, by 12, which is less by one than the number of the places, I multiply 40, (the common excess) so cometh 480. This 480 I subtract from 880, so remaineth 400: half whereof is the first number which we desired to know, that is 200. And as for the last number, I can give you it 3 ways: As by the first of my sire rules, I multiply the excess by a number less by 1 than the number of the places: as 40 by 12, that giveth 480, which I add to the first being 200, so shall the last be 680. The same sum cometh forth, if ye subtract 200 from 880. And thirdly, if I begin 200, and so proceed, increasing by 40, I shall at the thirteenth place have 680, as thus. 200. 240. 280. 320. 360. 400. 440. 480. 520. 560. 600. 640. 680. Sc I thank you most heartily for these 6 rules. Now if it be your pleasure, I would hear and learn somewhat of Progression Geometrical. Mai. There are yet very many rules and propositions, which fall into this Arithmetical progression: but these shall suffice for this time. And in Geometrical progression I will be more brief, both because I have been so long in this part of Arithmetical progression, and also for that it would require the knowledge of Roots, and numbers surde, (whereof ye have yet learned nothing) if I should frame the like propositions in them as I have done in these. Therefore I will only teach you two practices, about it, and so end the considerations & works of these progressions. Progression Geometrical. Progression Geometrical is when the numbers increase by a like proportion, that is, if the second number contain the first, 2, 3, or 4 times, and so forth: then the third containeth the second so many times also: and so the fourth the third, and the fifth the fourth: wherefore I set these three examples. Here in the first example you see, that every number containeth the other (that goeth next before him) 2 times: and in the second example 3 times in the third example 5 times. Now if you will know how to find easily the sum of any such numbers, do thus. Consider by what number thoy be multiplied, whether by 2, 3, 4, 5, or any other, and by the same number do you multiply the last sum in the Progression. Sc. I pray you work it by this example, 2, 8, 32, 128, 512, 2048, which I have framed by proceeding from 2, and continually multiplying by 4. Mai. Then must I multiply the last sum (which is 2048) by 4 also, and it will be 8192. Now must I bate from this sum the first number of the progression, which here is 2, then resteth 8190, which sum I must divide by 1 less than was the number that I multiplied by. Seeing then I multiplied by 4, I must divide by 3, so dividing 8190 by 3, the quotient will be 2730, which is the sum of all the Progression. And now to prove whether you can do the same, I give you these numbers to add by this rule, 3, 15, 75, 375, 1875, 9375, 46875. Sch. I cannot well tell by what number this Progression doth increase. Master. In any such doubt, do thus: Divide the second number by the first, ☜ and the quotient will show you the number that engendereth the Progression. Sc. Then is that number in this example 5, for so many times is 3 in 15. Master. So is it. Now work as I taught you. Scholar. The last number is 46875, which I multiply be 5, and it yieldeth 234375, from which I abate the first number of the Progression, that is 3, and there resteth 234372, which I divide by 4, for that is one less than 5, & the quotient is 58593, which is the whole sum of the progression. Mai. Now that you know the summing of Geometrical Progression, I will show you a compendious manner either to proceed by, or to find out the quantity of a number whose distance from the first may be very great, An abridgement in progression. which to do by continual multiplication, would be very tedious, if the numbers be great and the places many. Sc. Nothing can pleasure me more than brevity, if it be plain. Master. I think I am not yet in any point so dark or hard, that you need to fear any obscurity now. The manner is this: set down of your progression four or five of the first places, and under the first put a cipher, under the second 1, under the third 2, etc. as if ye had a progression increasing by a five fold quantity: as here, 2, 10, 50, 250, 1250: then under 2 I put a cipher, and under 10 the figure of 4: and so forth if ye will: but to a wise and wary worker, a few places were sufficient to proceed by to any number of places in this sort, if any two of your numbers progressional be multiplied the one by the other, and the ofcome divided by the first of your progression, the quotient is one of your numbers progressionall, and belonging to the place of your under numbers, that is equal to that sum, that is made of Addition together of your two numbers which stood under these two of your Progressionall numbers; that were multiplied the one by the other, as in this example. If I multiply 10 by 50. thereof cometh 300, which I divide by 2, (the first number of the Progression,) and the quotient is 250: which 250, must stand in the third place, because the number which standeth under 10 is 1, and that under 50. is 2: and 2 and 1 maketh 3. Therefore I say, that 250 belongeth to the third place of this progression, as ye see also here it doth. Moreover If I multiply 50, into itself, thereof cometh 2500: that 2500 I divide by 2, the Quotiente is 1500, which must be set in the fourth place: because 2 added to himself again, maketh 4, and in our example 1250 occupieth the fourth place. Sc. Then for that fifth place I multiply the Progressionall numbers over 2 and 3 one by the other: and for the sixth, I multiply that over 3 in itself. etc. M. ye must well remember that these places that we now speak of, belong to the under numbers, for the true places of the upper numbers is ever one place more. Scholar. That I see the reason of, because the under numbers begin one after, and against the first place of my progression standeth a cipher, so that the 250 which you said before did belong to the third place, I see belongeth to the number of 3 among your under numbers, but from the true progressions beginning, it is the fourth. Master You understand me as I mean. Therefore for your exercise of both the rules here given for Geometrical progression, I will ask you a question, much used among the common people, (as they have a great many the like) If I would sell you a Horse, having 4 shoes, and in every shoe six nails, with this condition, that you shall pay for the first nail 1 ob, for the second 2 ob, for the third 4, and for the fourth 8, and so forth doubling until the last nail. Now I demand of you how much the price of the Horse would amount unto? Scholar. Seeing the Horse hath 4 shoes, and in every shoe 6 nails, I perceive here will be 24 places. If I could now have the last number, I would quickly dispatch this question. I will therefore with as few multiplications as I can devise, to come to the knowledge of the last number of this progression. In double I set forth than a few of my progression, thus. If I now multiply the numbers over 5 and 6, the one by the other, I shall have the number of the eleventh place for the under numbers, but of the twelfth for the upper numbers in which my progression standeth, and then that of the eleventh place under, if I multiply in itself, I shall have for the 22 place under, but for the 23 of that above, which I multiply by that over 1 of my neither places, and I shall have the 23 of my neither places, and the 24 of the upper, which is the number I seek for. Master. Me thinketh you have forgotten your rule for abridging your multiplications: for in it, the ofcome over of any multiplication, is to be divided by the first of the progression. And you now speak of no division. Scholar. Sir I need not, as my progression beginneth now: for if I should divide by 1, it maketh no other quotient, than the number is, it doth divide. Master. It is very well remembered and noted of you, to your work then according to your prescribed manner, which I like well. S. I multiply 64 by 32 as here. And it maketh 2048. which is the eleventh place under, but the twelfth above, and this, I multiply in itself, in this manner. And this is the 22 place under but the 23 above. I multiply this then by 2, as here. And this of come 8388608, is my four and twentieth place, which I have found now by 3 multiplications. Then do I resort to the rule of summing this Progression, where I consider that the increase of this sum proceedeth by multiplication of 2, and therefore I do multiply the last sum by 2 also, and it yieldeth 16777216. from which 1 I abate the first number which is 1, and then resteth 1677715, which I should divide by 1 less than I did multiply: but seeing that it is 1, I need not to divide it, for 1 (as I have before said) doth neither multiply nor divide, therefore I take that sum 16777215 for the whole sum of the pence, which by Reduction I find to be 699050 s, and 7 d, ob: that is 34952 lb, 10 s, 7 d, ob. Master. That is well done, but I think you will buy no horse of the price. Scholar. No sir if I be wise. Yet for my assurance will I take so much pain, as to come to this last 8388608 by continual multiplication by 2, as in this example you may behold my work till I have done. Master. Well, are you not almost weary? Scholar. Well far my short rule, for in troth it hath more cunning and more ease. Master. Well, then answer me to this question. A Lord delivered to a Bricklayer a certain number of loads of brick, whereof he willed him to make 12. walls, of such sort, that the first wall should receive 2 thirdeles of the whole number: and the second 2 thirdles of that that was left. And so every other 2 thirdles of that that remained: and so did the bricklayer: And when the 12 walls, were made, there remaineth one load of Brick. Now I ask you, how many load went to every wall, and how many load was in the whole? Scholar. Why sir, it is impossible for me to tell. Master. Nay, it is very easy, if you mark it well. Mark well that I said, that every wall should receive ● thirdeles of the sum that was left. Now take away 2 thirdels from any sum and you must needs grant that that which remaineth is 1 thirdle of the sum last before: example of ●▪ from which if you take 2 thirdels, there will remain 3, which is one thirdel of 9 Likewaies' from 3 bate 2 thirdels: and there will remain 1. Scholar. This is true, and now I perceive, that the least wall had but two load of brick. Master. And by the same reason may you know how many load every wall had, according as this figure following doth show, and likewise what the whole sum of briekes was: for if you make 1● sums, multiplying by 2, still from the last remayner, as you may see here on the left side of the table, there will appear all the remayners of every wall: and if you multiply the last of those 12 sums by ●, also, then will that be the sum of the loads which were delivered to the bricklayer. Again, if you do double every remainder, as you may see at the right side of this table, those numbers will show the sum of loads that went to each wall: whereby you may perceive, that each wall was 3 times so great as the next lesser. Sc. Lo, now it appeareth easy enough. Now surely I see that Arithmetic is a right excellent art. Ma. You will say so when you know more of the use of it: For this is nothing in comparison to other points that may be wrought by it. Scholar. Then I beseech you sir, cease not to instruct me further in this wonderful cunning. The remainder after every wall. 1 12 2 Loads due to each wall. 3 11 6 9 10 18 27 9 54 81 8 162 243 7 486 729 6 1458 2187 5 4374 6561 4 13122 19683 3 39366 59049 2 118098 177147 1 354294 Some of the 531441 Loads delivered. THE GOLDEN Rule. Master. BY order of the science (as men have taught it) there should fellow next the extraction of Roots of number, which because it is somewhat hard for you, yet I will let it pass for a while, and will teach you the feat of the rule of Proportions, which for his excellency is called the Golden rule. Whose use is, by three numbers known, to find out an other unknown, which you desire to know: as thus. If you pay for your board for three months 16 shillings, how much shall you pay for 8 months. To know this and all such like questions, you shall consider which two of your 3 numbers be of one denomination, and set those two the one over the other, so that the undermost be it that the question is asked of: as in my question 3 and 8 be both of one denomination, for they both be months, and because 8 is the number that the question is asked of, I set them one over the other, and 8 undermost, thus, with such a crooked draft of lines. Then do I set the other number which is 16, against 3, at the right side of the line, thus. And now to know my question, this must I do: I must multiply the lowermost on the left side, by that on the right side, and the sum that amounteth I must divide by the highest, on the left side. Or in plainer words thus: I shall multiply the number of which the question is asked (which is called the third number) by the number of an other denomination, The third number. The second number. The first number. (which is called the second) and that sum that amounteth must I divide by the sum of like denomination, which is called the first. Then for the knowledge of this question, I multiply 8 into 16, and there amounteth 128, which I divide by 3, and it yieldeth 42 shillings, and 2 s remaineth, which I turn into pennies, and they be 24 d, of which the third part is 8 d, so the third part of 128 s is 42 s, 8 d: which sum I writ at the right hand of the figure against 8, thus. Hereby I know, that if thee months boarding do come to 16 s, that 8 months boarding will come to 42 s, 8 d: and likewise of any other like question. But here must you mark, that the first number and the third be of one denomination, and also the second and the fourth, the which you seek: or else be of such denominations, that you in working may bring them into one. As if a man should ask me this question. * Twelve weeks journeying cost me fourteen French Crowns at 6 s. the piece, how many pounds is that in one year. Here you see no two numbers of one denomination, But yet in working, you may turn them into like denominations, as thus: turn the one year into 52 weeks, and the fourth sum will be French Crowns, by the order of the working: Then to know this question, multiply the third sum 52 by the second 14: and the sum will be 728: that divide by your first number 12: and the quotient will be 60. Crowns: And 8 Crowns remaining: which if you turn into shillings they will be 48 shillings which if you do divide by your first number 12 the quotient will be 4: which signifieth 4 s. put these 60. French Crowns (which make 18 pounds) with the 4 shillings: for the sum that answereth to the question: And it is the just expenses of a year: And the sum will be thus. And take this evermore for a general Rule touching this whole Art, That the doubtful or unknown number, that you would be resolved of, shall always be set in the third place, note also the first number and the third, must ever be of one nature and denomination, or else must in working be brought to like denomination and then of necessity must the other number be in the second place. Remember also, that the place of the first number is the highest on the left side: and the place of the second right against it on the right side: the place of the third number is under the first, as by those examples you have seen. Sc. This I trust I can do. Master. But and if the question be asked thus: In 8 weeks I spend 40. s how long will 105 shillings serve me? Here you see that 8 weeks answers himself, and saith 40 shillings. But how long time 105 shillings will serve, you know not. Therefore you shall set 105 in the third place, according as I told you even now. And the first place must always be of the same nature or Denomination, that the third is of, which here is 40. Then must 8 needs be that other. Now multiply 105 by 8 and it will be 840 which if you divide by 40, it will yield 21, which is the Fourth number, and showeth how many weeks 105 s. will serve, if you spend 40 s. in eight weeks. The figure of this question is this: as if you should say: If 40 s. serve for 8 weeks, 105 serve for 21 weeks. Other diversities there be of working by this rule, but I had rather that you would learn this one well, than at the beginning to trouble your mind with many forms of working, sith this way can do as much as all the other, and hereafter you shall learn the other more conveniently. And for your further aid and instruction to make you better acquainted with this Goldon Rule, I have here proponed 6 questions, and their answers, which I think most convenient and meet to prefer the desirous to perfect understanding. The first four are all branches of one Question sprung out of the best tree, (for a young learner to taste of) that groweth in this Ground of Arts, for that no manner of Question in the Rule of 3 what so ever it be, can be proponed, but it must be comprehended, under the reason or style of one of these four. The Questions be these. If 15 else of Cloth cost 7 lb. 10 s: what comes, 27 else too at that price: Answer. 13 lb. 10 s. If 27 else cost 13 lb. 10 s: what are 15 else worth. Answer. 7 lb. 10 s. If 27 else cost 13 lb. 10 s: how many else shall I have for 7 lb. 10 s. Answer 15 else. If I sell 15 else for 7 lb. 10 s how many else are to be delivered for 13 lb. 10 s. Answer. 27 else, If 8 pound of any thing cost 16 s. 6d. what money is to be received for 49 pound: Answer. 2 lb 4 s. 11 d. If 4 lb. of any thing cost 17 d: what money will 8765 pound of that commodity cost. Answer. 155 lb. 4 s. 3d. farthing. Of all which questions I omit the work of purpose, you should whet your wit there by at convenient leisure, to climb each branch and gather the fruit of them: And do mind now, before we make ad end of this Rule, to give you some Instructions of the Backer Rule of 3. whose order is quite contrary to this that you have learned. For in this Rule hitherto evermore look how much the third number is greater than the first, so much the fourth number is greater than the second. And contrary ways: look how much the first sum is greater than the third, (if it do chance so) so much is the second sum greater than the fourth. But in this Rule, there is a contrary order, as this: That the greater the third sum is above the first, the lesser the fourth sum is beneath the second: and this rule therefore you may call the Backer rule, The backer rule. as in example. If I have bought 20 yards of cloth, Question of buying cloth. of 2 yard's breadth, and would buy canvas of 3 yards broad to line it withal, how many yards should I need? Sc. Why, there is none so broad. Mai. I do not care for that, I do put this example only for your easy understanding: For if I should put the example in other measures, it would be harder to understand. But now to the matter: If you would know this question, set your numbers as you did before: but you shall multiply now the first number by the second, and that ariseth thereof, you shall divide by the third: which thing if you do here, I mean if you multiply 30 by 2, it will be 60: which sum if you divide by 3, there will appear 20: whereby I know, that if 30 yards of cloth of two yards broad, should be lined with canvas of three yards broad, 20 yards of canvas would suffice, as this figure showeth. And now because ye found fault at my example, how say you, perceive you this? Sc. Yes sir. I suppose. Mai. Then answer me to this question: how many else of canvas of elle breadth, will serve to line 20 yards of Say, of three quarters broad. Sc. In good faith sir, I cannot tell, for I know not how to bring the sums to like denominations. Master. Then will I tell you: sith there is mention here of quarters, and again every one of the measures both else and yards may be parted into quarters, do you part them so both in the breadth and length, and then put forth the question by quarters. Scholar. Then I shall say thus. How many quarters of canvas of five quarters broad, will line 80 quarters of 3 quarters broad. Master. Now answer to the question. Sc. First I will set them down in their form thus, for 5 is joined with the question, and is therefore the third number: then is 3 the number of the same denomination, I mean because they be both referred to breadth. Now I multiply 80 by 3, and it is 240, which I divide by 5, and it yieldeth 48. Then say I, that 48 quarters of 5 quarters broad, will suffice to line 80 quarters of 3 quarters broad. Ma. Turn the quarters again into else and yards. Sch. Then I say, that 9 else and three quarters of a yard of elle broad will serve to line 20 yards of three quarters broad, as this figure showeth. * M. Now what say you to this question: I lent my friend 400 lb for 7 months, now how much money ought he to lend me again for 12 months, to recompense my courtesy showed him. Can you answer to this. Sc. Yes sir I suppose, for I will set down my numbers thus where I multiply 7 into 400, and it maketh 2800, which I divide by 12, and it yieldeth 233 lb and there is 4 lb. remaining of my Division, what shall I do therewith. Mai. Turn that same 4 lb into s and then divide it by 12 as you did before. Sc. Well sir it shall be done, so have I 6 s. for my quotient, and yet remaineth 8 s. upon my division. Ma. You must also reduce 8 s. into pence, which maketh 96, and divide that also by your first divisor. Sc. So have I done, and I find 8 pence for my quotient and nothing is left. Ma. This must you always do, when any thing remaineth upon your division: whether it be money, weight, measure, or any kind of thing whatsoever. This rule is so profitable for all estates of men, that for this rule only (if there were no more but it) all men were bound highly to esteem Arithmetic. By this Rule may a Captain in war work many things, as Master Digges in his Stratiaticos doth notably declare: Only now in this my simple addition, for a taste and encouragement I will enlarge the Author with a question or ij. more, wishing you, & every my countrymen or Gentlemen, whatsoever, that by nature be any thing given to Military affairs, to be familiar and well acquainted with this Exceliente Art, the which he shall find not only at the Sea, but also in the Camp and Field services, abundantly to aid him, either in fortification, or in paying of Soldiers wages, how different soever their pay be: Charges of Ordinance powder, shot, Munition and instruments, whatsoever, but now to the question. If it should chance a Captain which hath 40000 soldiers, Question of provision touching a●a●●ate. to be so enclosed with his enemy that he could have no fresh purveyance of victuals, and that the victuals which he hath, would serve that army but only 3 months, how many men should he dimisse, to make the victual to suffice the residue, 8 months? Sc. As you taught me, I set the numbers thus, saying: If 3 months suffice 40000, to how many will 8 months suffice? To know this I multiply the first number 3 into second 40000, & it yieldeth 120000, which sum I divide by 8, & there will be in the quotient 15000, which if I do subtract from 40000, the remainder will declare that he must dismiss 15000 as this figure showeth * Maist. Now answer me to this question: If 136 Masons in a month be able to build a Fort, to preserve the Soldiers from the enemy: And such expedition requireth that I would have the same finished in eight days, how many workmen say you is there to be appointed. Sch. As you taught me I set the numbers thus, saying: If 28 days require 136 Masons, what number of men by proportion will 8 days bring forth. To know this I multiply the first number 28 into 236: And it yieldeth me 2808: which I divide by 8. And my quotient is 476, which is the just number of Masons that shall supply this work. And now me think these questions are very easy. Ma. Truly if you take delectation herein you shall find this Art not only easy, but wonderful pleasant & profitable: Now answer me this question, & so will I make an end of this rule, in whole numbers hasting the sooner to broken numbers. For had you that understanding of them perfectly, not only in this Rule, but in all other: the question in sight might have been 10 times more harder to absolve, & yet as easily & as soon wrought as this. Sc. Your words doth greatly encourage me to be studious to attain whole numbers, which me think are wonderful. But might I once attain to be a practitioner in broken, I should think myself a happy lad. M. Now what say you to this, If ●● carpenters in 2 days can make ●00 staves: esteeming they work but 12 hours a day: And such need requireth that 384 carpenters are set to the finishing of these 200 staves, in what time say you will they make them up Scholar. I see here that I must turn my ● days into hours, And so doing I set my numbers thus Saying, if 48 men are 24 hours 384 men will make an end quickly. For it is grounded upon an old Proverb, many hands make quick speed. I multiply 48 into 24, and it amounteth to 1152, which I divide by 384: and my quotient is 3 hours which is my desire. Note. I take this for a note worthy the marking either in the Rule of three, forward, or backward, when the two numbers art multiplied together, the Producte is of the same nature, and denomination that the second number is of. Master. Well, sith you perceive now the use of this Rule, The double Rule. I will show other which ensue of the same, & first the double Rule, which is so called, because there is in it double working, by which thing only it differeth from this. Sch. Then by an example I shall understand it well enough. Ma. Of carriage. So shall you, and let this be the example: If the carriage of 100 pound weight 30 miles, do cost 12d. how much will the carriage of 500 weight cost, being carried 100 miles? Scholar. I pray you show me the working of it. M. You must make 2 workings of it: the first thus. If 100 pound weight cost 12d. how much will 500 lb. cost? Set your figure thus. And multiply 500 by 12 and thereof amounteth 6000, which if you divide by 100, the quotient will be 60, that is the price of 500 for 30 miles. Then begin the second work, saying: if 30 mile's cost 60 d, how much will 100 mile's cost? Set your figure thus. Than multiply a 100 by 60, whereof amounteth 6000 which being divided by 30, will yield 200d. Than you may say, that so many penies shall cost the carriage of 500 pound weight 100 miles, after the rate of 12 pence for the 100, carried 30 miles. Scholar. Now I perceive it also. Mai. These and such other like questions, are to be answered much quicker, at one working by the Rule of 3 composed of five numbers, which here I will not trouble you withal. But at the end of this Rule will show you the work thereof: not only of this and the next question, but also I will there deliver three or 4 other examples, wishing you then to make a comparison the one with the other: And so to use which way you think good. Sc. Sir I thank you much for your courtesy, and I long now till this rule be ended, that I shall see how I may behave myself with that new Rule of 5 numbers, for that I have ever since you taught me hitherto, in the Golden Rule both forward and backward wrought but with 3 numbers only. Ma. Question of sowing. Till we have done with this, let us go on forward: and answer me to this question: 30 bushels of wheat sowed, yielded in one year 360: how many will 80 bushels yield in 7 year. Sc. First I say, that if 30 bushels will yield 360 in 1 year, than 80 bushels will yield 960 in 1 year. Then for the second work I say: If one year yield 960, than 7 year will yield 6720: as these two figures do show. Question of Corn. But now sir, if I set forth 30 bushels of corn to another man for 7 year, agreeing so that he shall sow every year the whole increase of the corn, and I at the end of those seven years to have the half of the whole increase: I would know how many bushels will there amount to my part supposing the increase to be after the rate of the last question, for 30 bushels in one year, 360. Mai. In such a question you must have so many several workings, as there be years, as for example: In the first year ●0 bushels yield 360: then to know the yielding of the second year, I must say: If 30 yield 360, how many yieldeth 360? Work by your rule, and you shall find 4320. Then say for the third year: if 30 yield 360, how many will 4320 yield? you shall have 51840, and so every year multiplying the whole increase by 360, and dividing it by 30, the increase of the next year will amount, as these 7 figures (in the next page) do orderly declare: where I have set 7 letters for the 7 years, of which the first is set without art because that is the increase which you do presuppose: & the last number of each other doth show the increase of the year that it standeth for; which the letters do declare, so that the increase of the seven year, is 1074954240 bushels: how many quarters that is, and also how many ways, you may by Reduction soon find. Now with one question more I will prove you. If 6 Mowers do mow 45 acres in 5 days, how many mowers will mow 300 acres in 6 days? Sch. If 45 acres do require 6 mowers, than 300 acres requireth 40. Now again: if 5 days require 40 mowers, than 6 days needeth but 33 mowers. Mai. Why do you not make mention of the 2 that remaineth in the last division? for the last part of the question is wrought by the Backer rule, where the first number 5, is multiplied into the second that is 40, whereof amounteth 200, which if you divide by the third number 6, the quotient will be 33, as you said, but then will there remain 2, which cannot well be divided into 6 parts: how be it, you may understand by the sixth part of 2, the third part of one man's work, which you must put to the 33, or else you may say, that 33 workmen will end all the 300 acres in 6 days, save two men's work for one day, or 2 days work for one man. But such broken numbers called Fractions, you shall hereafter more better perceive, when I shall wholly instruct you of them. Mai. Yet one question more of field matters I will propone, and so I will make an end of this double Rule of 3. Sc. With all my heart sir I thank you, and I will dispatch it as soon as I can, because would feign see the order of the next Rule of 5 numbers. Maist. Then this is my question, If 300 pioneers in 8 hours, will cast a trench of 200 Rods: I demand how many Labourers will be able with a like trench in three hours to entrench a Camp of 3500 Rods. Sch. I think I am now in the Backe-house diche, for I know not well which way to go about it: And besides that truly I think I shall never come to prefermente that way my growth is so small. Ma. You know not how God may raise you hereafter by service, into the favour of your Prince, for the avail of your Country. Example, Sir Francis Drake, as worthy a man as ever England bred, is not the tallest man, and yet hath made the greatest adventure for the honour of his Prince & Country, that ever English man did. Sc. Sir, I thank you for your good encouragement, my mind, though I be little, is as desirous of knowledge, as any other: I have pondered now a little of it, & thus I set forth the work. Saying if 200 Rod require 3400 300 men what shall 3400 Rod require: I multiply 3400 by 300: and it yieldeth 1020000: which I divide by 200 and my quotient is 5100 men. Then must I say for my second work, if in 8 hours 5100 men be able to discharge it, how many shall perform the same in 3 hours? now if I should work by the Golden Rule of proportion forward, I should find a less number of men, because 3 hours is less than 8 hours: but because reason teacheth me that the lesser the time is, wherein the french must be made, the more labourers I ought to have, whereupon I use now the backer Rule as in example. And I have in my quotient 5000. So many pioneers must I have, to entrench the camp in 3 hours. Ma. You have answered the question very artificially: And truly I commend you for your diligence and apt understanding: and now according to my promise, I will (in whole numbers) give you a little taste of the Rule of 3 compounded of 5 numbers. The Rule of 3 compound of 5 numbers. THis Rule of 3 composed is distinct for most needful questions into two several parts or workings: And there belongeth unto it always 5 numbers, whereof in this rule being the first part: the second number and the fift are always of one nature and like denomination, which rule is to be wrought thus: you must multiply the first number by the second: And that shallbe your divisor: Then again, multiply the other three numbers, the one by the other and their product shall be your dividend. And now according to my promise, we will first work the question of weight and carriage which I delivered you in the double rule of 3: to be absolved by this Rule, which was this. If the carriage of 100 lb. weight 30 mile's cost 12 d, what will the carriage of 500 lb. weight stand me in being carried 100 miles. Then mark well how these numbers stand: multiply 100 by 30 as this figure showeth: And that number keep for your divisor: Then multiply the other 3. numbers the one by the other, and they amount as you see to 600000: which you shall divide by 3000: your quotient is 200 d: Now you see it agreeth with the conclusion of the double Rule. S. Sir I thank you most heartily: it is even so. Ma Yet note this for a generality in this Rule, Note. look what nature or denomination your middle number is: and of the like denomination or nature is always your quotient. Scholar. Well now and it please you by your patience, I will see how I can end, the question than next following of 30 Bushels of Wheat sowed, yielded in one year 360, how many then will 80 bushels yield in seven year: and according to your reasons, I set my numbers thus: which 201600 I divide by 30: and my quotient is 6720: bushels my desire. M Yet one question more I will propound unto you and so leave this rule, till it please God hereafter, that I may make you work it in broken numbers. What comes the interest of 258 lb for five months after the rate: of 8 pound taken in the 100 lb: for 12 months: Sch Sir as this is a question of gains So will I warely work this question in hope one day to reap something for my pains: and thus I propone it. But I beseech you if it be not well set down to show me mine error. Ma. Proceed you have done very well Sch. Then I doubt not by the grace of God but to end it: I multiply 100 by 12 it yieldeth 1200: and the 3 other numbers multiplied together produceth 10 20: which I divide by 1200: and my quotient is 8 pounds. Then according as you have taught me heretofore, I turn the 720 lb. that is left: into shillings: and dividing it by my first number my quotient is 12 s. So I answer that the lone of 258 lb for 5 months, after the rate of 8 lb. in the 100 lb for a year, comes to 8 lb 12 s. Mai. You say true, I commend your diligence, now behold the manner of the second part of this rule. M. In the second part of this rule of 3 composed: the third number is like unto the first. And the rule is to be wrought as thus: you shall now contrary to the last rule multiply the third number and the fourth together: and that product shall be your devisor: Then multiply the fift by the second and the producte thereof by the first: and that is the number that shall be divided. For example I propond this question: for a proof of my last question of interest. A Merchant hath received 8 lb 12 s. for interest for 5 months' term, which he received after the rate of 8 lb. in the 100 lb. for a year. The question is now how much money was delivered to raise this interest: Behold therefore the manner how the question is set forth. Scholar. Sir I perceive it very well: and according to the doctrine which you prescribed for the working thereof: if it please you now it is set down I think I can follow the work. M. Nay stay a while, and afore you work mark well how I deliver a reason, for the perfect understanding of this rule which is thus: Note. if 8 lb. in 12 months do yield me 100 lb To take 8 lb— 12 s. for 5 months, must needs yield a great deal more. So upon the knowledge that I have in this Art, The first part of this rule is answerable to the rule of 3 forward: And this latter part accordeth to the rule 3 backward. S. Sir I yield you most hearty thanks for these your last instructions, they have given me great light into these two Rules, whereby I may the better by deliberation conceive how to use them hereafter, when occasion shall require. M. You say well, go too now if you will, and try your cunning in the question: Note. But this note take with you by the way, in as much as here is mention made of shillings: turn all your money as you work, for your more ease in work. S. If it please you to behold me a little, I will quickly end it: for I have but my first: my second: and my last number to be multiplied together for my dividend: And my third into my fourth for my divisor: Which I divide by 800. and my quotient is 5160 shillings, which in pounds yieldeth 258 my desire. M. I will here for this time in whole numbers end this rule, and will instruct you in the rules of Fellowship. You may at your convenient leisure, for your exercise work the same, by the rule of 3. at twice: And for your aid and encouragement therein, I set down here a proffer how to apply it. The Rule of Fellowship. But now will I show you of the rule of fellowship or Company, which hath sundry operations, according to the divers number of the company. This rule is sometime without difference of time, and sometimes there is in it difference of time. first I will speak of that without difference of time, of which let this be an example. Four merchants of one company made a bank of money diversly, for the first laid in 50 lb, the second 50 lb, the third 60 lb, and the fourth 100 lb, which stock they occupied so long, till it was increased to 30●0 lb. Now I demand of you, what should each man receive at the parting of this money. Sch. I perceive that this rule is like the other, but yet there is a difference, which I perceive not Ma Then will I show it to you. first by Addition you shall bring all the particular sums of the merchants into one sum, which shall be the first sum in your working by the Golden rule, and the whole sum of the gains by that stock shall be the second sum. Now for the third sum, you shall set the portion of each man one after an other, and then work by the Golden rule, and the fourth sum will show you each man's gains: as in example. The parcels of those four merchants make in one sum 240 lb: set that in the first plare, the gains in the second, and the first man's portion of stock in the place thus. Now multiply the second by the third, and it will be 90000, which you shall divide by 240, and there will appear 375 lb. thus. And that the gains for the first man. Now for the second man, set the 50 lb. that he brought, in the third place, and work as before: and his part will be 625 lb. as this figure showeth. Likewaies' for the third man set his money which was 60 lb, and his part of gains will be 150 lb, as here appeareth. And so for the fourth man, if you set his sum which is 1000 lb, his gains will be 1250 lb, as the proof will declare. Scholar. This I perceive: but is there any way to examine whether I have well done oer no? Master. That must you do by one common proof which serveth to the Golden rule and all other ensuing of the same: and that is this: Change the standings of the numbers, and set the third in the first place, the 4 in the second place, and the first in the third place, and they work by the Golden rule, and if you have done well, the fourth number now will be the same that was the second before. As for example, I will take the last work which was this. Which to examine I alter as I said, thus: Now if I multiply the second number by the third, and divide that that amounteth by the first, then will the fourth number be 3000, which was the second before, as you see here, which is a token, that I have well done. But as in a single rule one proof thus is sufficient, so in a rule where many operations be, you must turn every of them as I have done with this one. Scholar. Then for the proof of the first work of this rule, I should turn the numbers thus. And the second thus. And for the third thus. And in each of them if the working were true, the fourth number will be still 3000. Ma. Well, now an other example will I put to you, not of gains, but of loss: for one reason serveth for both. If three merchants in one ship and of one fellowship, had bought merchandise, so that the first had laid out 200 lb, the second ●●00 lb, and the third 500 lb, and it chanced by tempest that they did cast over board into the sea merchandise of the value of 100 pound, how much should each man boar in this loss? Scholar. If I shall do in this as you did in the other question, then must I join their three portions together, 200, 300, and 500, which maketh ●000. Then say I, if 1000 lief 100, then shall 200 lose 20, and 100LS shall lose 30, & 500, shall lose 50, as by these three figures it doth appear plain. Ma. Well sith now you have done these I will propound a question of more importance, which shall make you not only the abler to understand this Rule, but also it will greatly aid you in the next rule of fellowship with time, if such need be that your money be of diverse denominations. For this may not be forgotten in all such questions, if the number be of diverse kinds: you must by Reduction bring it into one kind, that is to say to the least value that is named in the question. And likewaies shall you do, if the time be of diverse kinds, as some years, some months, weeks and days, you shall make all months, weeks or days according as the lest name of time in the question is: As for example. First in diversity of money. Three companions bought 2000 sheep, and paid for them 241 lb. 13 s, 4.d. of which sum one paid 101 lb, Question of Sheep. 10 s. The second paid 82 lb. 17 s, 10d. And the third paid 57 lb, 5 s, 6 d: How many sheep must each of them have? Answer: The first shall have 840. The second 686. And the third 474. And that must you work thus. Solution. first considering that your money is of diverse denominations, you shall (by Reduction) bring it all into the smallest denomination which is in it, that is to say, pence, and so will the total sum be 58000. pence. Now, if you turn each man's money into pennies also, the first man's sum will be 24360 pence: The second man's sum 19894. pence. And the third man's money will be 13746. pence. Now to know how many sheep every man shall have, let the whole sum of money that is 58000. pence, in the first place & in the second place set the number of sheep, and then orderly in the third place set each man's money, and then multiplying the third and the second sums together, and dividing that that amounteth by the first, there will appear the number of sheep that each man ought to have: as these three figures do show, Scholar. Why do you set the money in the first place, seeing in the question you say, 2000 sheep cost 58000 d? & not thus, 58000 d cost 2000 sheep. Master. You remember, I taught you at the beginning of this Golden rule, that the first and third numbers must be of one name, and of like things: and evermore the number that the question is asked of, must be set in the third place. Now is the question plainly this: If four men bought 2000 sheep for 58000 pence, how many sheep shall each man have? But seeing in this question there ought more respect to be had to the sum of money, than to the sum of the persons, (for in the sums of money is there proportion toward the sheep, and not in the number of persons) therefore must we turn the question thus. If 18000 pence bought 2000 sheep, how many did 24360 s buy? Again, how many did 19894 d buy? & how many bought 13746 pens. Scholar. I percive it reasonable, and so shall I do in all like questions. Master. Even so, But for easiness of the work mark this: Note. When soever the first and second numbers have ciphers in the first places, you may both in the multiplication and in the division leave out those ciphers, so that you leave out like many out of both sums, as in this question the first number 58000 hath three ciphers, and so hath the second that is 2000: therefore cast away their ciphers, and so will the first number be 58, and the second 2: set them in their places, and work according to the rule, and you shall perceive that it will be all one, saving that this is the shorter and easier way, as these three figures do show. And this you see is both easier, and also the more certain way to know the answer to this question. S. Truth it is as you say: but sir, me seemeth I might ask a further question here, not only how many sheep each man should have, but also what every sheep cost. Master. That question doth not only belong to this rule, but may also be discussed by Division, especially if the questions number be one only: as thus. Divide the total sum 58000 pence, by 2000 (or 58 by 2, omitting the ciphers) and the quotient will be 29 pence, that is 2● s, 5 d, howbeit, by this rule you may do it, and best when the number of the question doth exceed 1: as if I should ask this question, 2000 sheep cost 58000 d, ☜ how much did 20 cost? Then shall I set my figure thus. And doing after the rule, there will amount 580 pence, that is 2 lb 8 s 4 d: the price of one score: But if you will use that easy way that I did teach you, you may change the first and second number thus. Thus do you perceive the use of the rule without time. The rule of fellowship with time. And that you may as wel● perceive the same with diversity of time, I propose this example. Four Merchants made a common stock, Question of a bank. which at the years end was increased to 35145 lb. Now to know what shall be each man's portion of gains, you must know each man's stock and time of continuance. The first man of these four laid in 669. lb which he did take from the stock again, at the end of 10. months. The second man laid in 810. lb. for 8. months. The third laid in 900. lb. for 7. months. And the fourth laid in 1040. lb. for 12. months. This question shall you examine as you did the other before, saving that where as in the third place of the figure you did set each man's sum alone, ☞ here you shall set the same being multiplied by the number of their time & likewise in the first place of the figure, you shall set that number which amounteth of their whole sums so multiplied by their time, & added into one whole sum as thus. The first man's sum is 669 lb. which I multiply by 10. (that was the number of his time) and it maketh 6690. The second man's sum 810. lb. multiplied by 8, (which was his time) make 6480. The third man's sum 900. lb. multiplied by 7 (for that was his time yieldeth 6300. The fourth man's sum was 1040 lb and his time 12. multiply the one by the other, and and it will be 12480. These four sums thus multiplied by their time, must be set orderly in the third place of the figure: and in the first place must be set the whole sum of all four, which is 31950, and the gain must be in the second place, which is ●5145. Now to end the question, I say first: If 31950 did get 35145, what did 6690 get? Answer, 7359 lb, as by this figure appeareth. Likewise the second man had to his part 7128 lb, the third must have 6930 lb And the fourth man shall have for his part 13728 lb as these figures do partly declare. Scholar. This I like very well: but what proof is there of this work? Mai. The same that I taught you for the other. Howbeit, Another proof. there is used both for this work and the other also this manner of proof, to add all the portions together, and if they agree to the whole sum, then seemeth it well done: but this is no sure rule. Sch. Yet will I prove it in this example. The four parcels are these, which if I add together, there will amount 35145, and that was the whole sum: so is this rule true here. Master. And so will it be still when the work is truly done. Note the imperfection of this kind of proof. But if you lift to see it proved false, take 10000 lb from the fourth man, and put it to any of the other 3, and then be ye sure that you have not done well, and yet will the proof allow it, for the Addition will still be all one. Sc. It must needs be so: but what have I now to learn? Maist. There are many other excellent parts behind, of which I will not, as now, make mention because that without the knowledge of Fractions, they cannot be duly taught, and much less understanded. Therefore will I propose to you two or three questions more, whereby you may practise the better the feat of the rule of fellowship, (that thereby you may better perceive the use of all other) & so make an end for this time. There is in a Cathedral Church ●0 Cannons, and 30 Vicars, Question of Canons. those may spend by year 2600 lb, but every Camnnon must have to his part 5 times so much as every Vicar hath: how much is every man's portion say you? Sc. I pray you make the answer yourself, so shall I perceive best the means to answer to such other like. Mai. In this question you must do as in those that have diversity of time, for here is diversity of portions: Therefore shall you multiply the number of the persons by their difference of portion: (as you did in the other by time) Then must you multiply the ●0. (which is the number of Cannons) by 5, (for that is the number of their portion) so will it be 102: Then ●0, (that is the number of Vicars) by 1, (that is the number of their portion) and it will be 30: put those two sums together, and they make 100L: then say thus: If 130 spend 2600 lb, what may 100 spend? The rule showeth 2000 lb. Again for Vicars: If 130 spend 2600 pound, what may 30 spend? Answer 600 lb as these figures show, But if every Camnon should have so often times 4 lb as the Vicar should have 3 lb then should I multiply 20 by 4, (that were 80) and 30 by 3 (that were 90) and then both were 170. Then should the figures be set thus. But this sort is to hard for you, by reason of the Fractions, therefore I will let it rest to that place. And by this rule you see what the 20 Cannons may spend, which sum if you divide by 20, you shall see each Cannon's portion: and so of the Vicars, if you divide their sum by 30, the quotient will declare every Vicar's portion. The second Dialogue. The accounting by Counters. Master. NOw that you have learned the common kinds of Arithmetic with the pen, you shall see the same art in Counters: which feat doth not only serve for them that cannot write and read, but also for them that can do both, but have not at some times their pen or tables ready with them. This sort is in two forms commonly: The one by lines, and the other without lines. In that that hath lines, the lines do stand for the order of places: and in that that hath no lines, there must be set in their stead so many counters as shall need, for ●che line one, and they shall supply the stead of the lines. Sc. By examples I should better perceive your meaning. Mai. For example of the lines, lo here you see six lines, which stand for six places, so that the nethermost standeth for the first place, and the next above it for the second, and so upward, till you come to the highest, which is the sixth line, and standeth for the sixth place. Now what is the value of every place or line you may perceive by the figures which I have set on them, which is according as you learned before in Numeration of Fi-figures by the pen: for, the first place is the place of units or ones, and every Counter set in that line betokeneth but one: and the second line is the place of 10, for every counter there standeth for 10. The third line the place of hundreds, the fourth of thousands, and so forth. Sc. Sir, I do perceive that the same order is here of lines, as was in the other figures by places, so that you shall not need longer to stand about Numeration, except there be any other difference. M. If you do understand it, then how will you set 1543? Sc. Thus as I suppose. Ma. You have set the places truly but your figures be not meet for this use: for the meetest figure in this behalf, is the figure of a Counter round, as you see here, where I have expressed that same sum. Scholar. So that you have not one figure for 2 nor 3, nor 4 and so forth, but as many digits as you have, so many Counters you set in the lowest line: and for every 101 you set one in the second line: and so of other. But I know not by what reason you set that one counter for 500 between two lines. Maist. You shall remember this, that whensoever you need to set down 5, 50, or 500 or 5000, or so forth any number whose Numerator is 5, you shall set one counter for it in the next space above the line that it hath his denomination of: as in this example of that 500, because the numerator is 5, it must be set in a void space: and because the denominator is hundred, I know that his place is the void space next above hundreds, that is to say, above the third line. But this shall you mark, that as you did in the other kinds of Arithmetic, set a prick in the places of thousands, in this work you shall set a Star, as you see before. Scholar. Then I perceive Numeration: but I pray you how shall I do in this art to add two sums or more together. ADDITION. Master. And then if you list, you may add the one to the other in the same place: or else you may add them both together in a new place: which way, because it is most plainest, I will show you first. Therefore will I begin at the units, which in the first sum is but 2, and in the second sum 9, that maketh 11. Those do I take up, and for them I set 11 in the new room, thus. Then do I take up all the Articles under a hundred, which in the first sum are 40, and in the second sum 50, that maketh 90: or you may say better, that in the first sum there are 4 articles of 10, and in the second sum 5, which maketh 9, but then take heed that you set them in their right lines as you see here. For it is all in one sum as you may see, but it is best never to set five counters in any line, for that may be done with one Counter in a higher place. Scholar. I judge that good reason, for many are unnéedefull where one will serve. Master. Well, then will I add forth of hundreds: I find 3 in the first sum, and 6 in the second which maketh 900, them do I take up, and set in the third room, where is one hundred already, to which I put 900 and it will be 1000, therefore I set one counter in the fourth line for them all, as you see here. Then add I the thousands together, which in the first sum are 8000, and in the second 2000, that maketh 10000: them do I take up from those two places, and for them I set one counter in the fift line, and then appeareth as you see to be 11001, for so many doth amount of the Addition of 8342 to 2659. Scholar. Sir, To add 8 sums together. this I do perceive: but how shall I set one sum to an other, not changing them to a third place? Mai. Mark well how I do it: I will add together 65436 and 3245, which first I set down thus. Then do I begin with the smallest, which in the first sum is 5, that do I take up, and would put to the other 5 in the second sum, saving that two Counters cannot be set in a void place of 5 but for them both I must set 1 in the second line, which is the place of 10: therefore I take up the five of the first sum, and the 5 of the second, and for them I set 1 in the second line, as you see here. Now do I take the 200 in the first sum, and add them to the 40 in the second sum, and it maketh 600, therefore I take up the 2 counters in the first sum, and 3 of them in the second sum, and for them 5, I set 1 in the space above, thus. Then I take the 3000 in the first sum, unto which there are none in the second sum agreeing, therefore I do only remove those three Counters from the first sum into the second, as here doth appear. And if you have marked these two examples well, you need no further instruction in Addition of 2 only sums: but if you have more than two sums to add, you may add them thus. first add two of them, and then add the third and the fourth, or more if there be so many: as if I would add 2679 with 4286 and 1391. First I add the two first sums thus. And then I add the third thereto thus. And so of more, if you have them. Scho. Now I think best that you pass forth to Subtraction, except there be any ways to examine this manner of Addition, than I think that were good to be known next. Master. A prooef. There is the same proof here that is in the other Addition by the pen, I mean Subtraction, for that only is a sure way: but considering that Subtraction must be first known, I will first teach you the art of Subtraction, and that by this example. SUBTRACTION. I Would subtract 2892 out of 8746. These sums must I set down as I did in Addition: but here it is best to set the lesser number first thus. Then shall I begin to subtract the greatest numbers first (contrary to the use of the pen) that is the thousands in this example: therefore I find amongst the thousands 2, for which I withdraw so many from the second sum (where are 8) and so remaineth there 6, as this example showeth. Then do I likewise with the hundreds, of which in the first sum I find 8, and in the second sum but 7, out of which I can not take 8, therefore this must I do: I must look how much my sum differeth from 10, which I find here to be 2, then must I bate for my sum of 800, one thousand, and set down the excess of hundreds, that is to say, 2, for so much 1000 is more than I should take up. Therefore from the first sum I take that 800, and from the second sum (where are 6000) I take up one thousand, and leave 5000, but then set I down the 200, unto the 700 that are there already, and make them 900, thus. Saving that here I have set one counter in the space, in stead of 5, in the next line. So that if I subtract 2892 from 8746, the remainder will be 5854 And that this is truly wrought you may prove by addition: for if you add to this remainder the same sum that you did subtract, then will the former sum 8746, amount again. Scho. That will I prove: and first I set the sum that was Subtracted, which was 2892, and then the remainder 5854, thus. Then do I add the first 2 to 4, which maketh 6: so take I up 5 of those counters, and in their stead I set 1 in the space, and 1 in the lowest line, as here appeareth. Then do I add the 90 next above to the 50, and it maketh 140, therefore I take up those 6 counters, and for them I set 1, to the hundreds in the third line, and four in the second line thus. Then do I come to the hundreds, of which I find 8 in the first sum, & 9 in the second, that maketh 1700: therefore I take up those 9 counters, & in their stead, I set 1 in the fourth line, and 1 in the space next beneath, and 2 in the third line as you see here. And thus you may see, how Subtraction may be tried by Addition. Scho. I perceive the same order here with Counters, that I learned before in figures. Ma. Then let me see how you can try Addition by Subtraction. Scholar. first I will set forth this example of Addition, where I have added 2189, to 4988. And the whole sum appeareth to be 7177. Now to try whether that sum be well added or no, I will subtract one of the first two sums from the third, and if I have well done, the remainder will be like that other sum, as for example. I will subtract the first sum from the third, which I set thus in their order. Then do I subtract 2000 of the first sum from the second sum, and then remaineth there 5000, thus. And so have I ended this work, and the sum appeareth to be the same which was the second sum of mine Addition, and therefore I perceive I have well done. May If I should abate 1846 from 2378, they set the sums thus. First they take 6, which is the lower line, and his space, from 8 in the same rooms in the second sum, and yet there remaineth 2 counters in the lowest line. Then in the second line must 4 be subtracted from 7, and so remaineth there 3. Then 800 in the third line, and his space, from 300 of the second sum can not be, therefore do they bate it from a higher room, that is from 1000: and because that 1000 is too much by 100L, therefore must I set down 200 in the third line, after I have taken up 1000 from the fourth line. Then is there yet 1000 in the fourth line of the first sum, which if I withdraw from the second sum, then doth all the figures stand in order, thus. How be it, as some men like the one way best, so some like the other: therefore you now knowing both, may use which you list. MULTIPLICATION. But now touching Multiplication: you shall set your numbers in two rooms (as you did in those other kinds) but 2 so that the multiplier be set in the first room, then shall you begin with the highest numbers of the second room, and multiply them first, after this sort. Take the overmost line in your first working, ☞ as if it were the lowest line, setting on it some movable mark (as you list) and look how many counters be in him, take them up, and for them set down the whole multiplier so many times as you took up counters: reckoning (I say) that line for the Unites. And when you have done with the highest number, then come to the next line beneath, and do even so with it, and so with the next, till you have done all. And if there be any number in a space, then for it shall you take the multiplier 5 times: and then must you reckon that line for the Unites, which is next beneath that space. Or else after a shorter way, you shall take only half the multiplier, but then shall you take the line next above that space for the line of Unites. But in such working, if by chance your multirlyer be an odd number, so that you can not take the half of it justly, then must you take the greater half, and set down that, as if that it were the just half: and further you shall set one Counter in the space beneath that line, which you reckon for the line of Vnits, or else only remove forward the same that is to be multiplied. S. If you set forth an example hereto, I think I shall perceive you. Maaster. Take this example: I would multiply 1 5 4 2 by 2 6 5, therefore I set the numbers thus. Then first I begin at the 1000 in the highest room, as if it were the first place, and I take it up, setting down for it so often (that is once) the multiplyer, which is 365, thus as you see here: where, for the one counter taken up from the fourth line, I have set down other 6, which make the sum of the multiplier, reckoning that fourth line as if it were the first, which thing I have marked by the hand set at the beginning of the same. Scholar. I perceive this well, for in deed this sum that you have set down is 265000: for so much doth amount of 1000, multiplied by 365. Master. Which sums if you do add together into one sum, you shall perceive that it will be the same that appeareth of the other working before, so that both sorts are to one intent: but as the other is shorter, so this is plainer to reason for such as have had small exercise in this art. Notwithstanding you may add them in your mind before you set them down: as in this example you might have said, 5 times 300 is 1500, and 5 times 60 is 100LS also 5 times 5 is 25, which all put together, do make 1825, which you may at one time set down if you list. But now to go forth, I must remove the hand to the next counters which are in the second line, and there must I take up those 4 counters setting down for them my multiplier 4 times severally, or else I may gather that whole sum in my mind first, and then set it down: as to say, 4 times 300 is 1200: 4 times 60 are 240: and 4 times 5 make 20, that is in all 1460, that shall I set down also, as here you see. Then to end this Multiplication, I remove the finger to the lowest line, where are only 2, them do I take up, and in their steed do I set down twice 365, that is 730, for which I set one in the space above the third line for 500, and 2 more in the third line with that one that is there already, and the rest in their order, and so have I ended the whole sum, thus. Whereby you see, that 1542 (which is the number of years sith Christ his incarnation) being multiplied by 365 (which is the number of days in one year) doth amount unto 562830, The sum of the days sith Christ's incarnation which declareth the number of days sith Christ's incarnation unto the end of 1542 years, (beside 385 days and 12 hours for leap years.) Sc. Now will I prove by an other example, as this: 40 labourers (after 6 d the day for each man) have wrought 28 days: I would know what their wages doth amount unto. Where in the first place is the Multiplier (that is 1 days wages for one man) & in the second space is set the number of the warkmen to be multiplied. So appeareth the whole days wages to be 240 d that is 20 s. So is the whole wages of 40 workmen for 28 days (after 6 d each day for a man) 6720 d that is 560 s or 28 pound. Ma. Now if you would prove Multiplication, the surest way is by Division: therefore will I overpass it, till I have taught you the art of Division, which, you shall work thus. DIVISION. FIrst set down the divisor, for fear of forgetting, and then set the number that shall be divided, at the right side, so far from the Divisor, that the quotient may be set between them: as for example. If 225 sheep cost 45 lb. what did every sheep cost? To know this, I should divide the whole sum that is 45 lb, by 225, but that cannot be: therefore must I first reduce that 45 lb into a lesser denomination, as into shillings, than I multiply 45 by 20, and it is 900: that sum shall I divide by the number of sheep, which is 225, these two numbers therefore I set thus. Then begin I at the highest line of the divident, and seek how often I may have the divisor therein, and that may I do four times: then say I, four times 2 are 8, whithe if I take from 9, there resteth but 1, thus. And because I found the divisor 4 times in the divident, I have set as you see, 4 in the middle room, which is the place of the quotient: but now must I take the rest of the divisor as often out of the remayner, therefore come I to the second line of the divisor, saying: 2 four times make 8, take 8 from 10, and there resteth 2, thus. Then come I to the lowest number which is 5, and multiply it 4 times, so is it 20, that take I from 20, & there remaineth nothing, ☜ so that I see my quotient to be 4, which are in value shillings, for so was the divident: and thereby I know that if 225 Sheep did cost 45 lb, every sheep cost 4 s. Sch. This can I do, Example of wages as you shall perceive by this example. If 100 soldiers do spend every month 68 lb, what spendeth each man? First because I cannot divide the 68, by 160, therefore I will turn the lb into pennies by multiplication, so shall there be 16320d. Now must I divide this sum by the number of soldiers, therefore I set them in order thus. Then begin I at the highest place of the dividend, seeking my Divisor there, which I find once, therefore set I 1 in the neither line. Ma. Not in the neither line of the whole sum, but in the neither line of that work which is the third line. Sc. So standeth it with reason. Ma. Then thus do they stand. Then seek I again the rest, how often I may find my divisor: and I see that in the 300 I might find 100 three times, but then the 60 will not be so often found in ●0, therefore I take 2 for my quotient: them take I 100 twice from 300, and there resteth 100, out of which with the 20 (that maketh 120) I may take 60 also twice, and then standeth the numbers thus. Where I have set the quotient 2 in the lowest line: So is every Soldiers portion 102 d that is 8 s, 6 d. Ma. But yet because you shall justly perceive the reason of Division, it shall be good that you do set your divisor still against those numbers from which you do take it, as by this example I will declare. If the purchase of 20 acres of ground did cost 290 pound, what did one acre cost? Example of purchase. First will I turn the pounds into pennies, so will there be 69600 pence. Then in setting down these numbers, I shall do thus. First set the dividend on the right hand as it ought, and then the divisor on the left hand against those numbers from which I intent to take him first as hear you see, where I have set the divisor two lines higher than is his own place. Scho. This is like the order of Division by the pen. Ma. Truth you say, and now must I set the quotient of this work in the third line, for that is the line of units in respect to the divisor in this work. Then I seek how often the divisor may be found in the divident, and that I find 3 times, than set I 3 in the third line for the quotient and take away that 60000 from the dividend, and farther I do set the divisor one line lower, as you see here. And then seek I how often the divisor will be taken from the number against it, which will be 4 times and 1 remaining. Scho. But what if it chance that when the divisor is so removed, it cannot be once taken out of the divident against it? Ma. Then must the divisor be set in an other line lower. Sc. So was it in division by the pen, and therefore was there a cipher set in the quotient: but how shall that be noted here? Ma. Here needeth no token, for the lines do represent the places: only look that you set your quotient in that place which standeth for units in respect of the divisor: but now to return to the example. I find the divisor 4 times in the divident, and 1 remaining, for 4 times 2 make 8, which I take from 9, & there resteth 1, as this figure following showeth; and in the middle space for the quotient I set 4 in the second line 〈◊〉 he is in this work the place of units. Where you may see, that the whole quotient is 348 d, that is 29 s, whereby I know that so much cost the purchase of one acre. Sc. Now resteth the proves of Multiplication, and also of Division. Ma. Their best proves are each one by the other: for multiplication is proved by Division, and Division by Multiplication, as in the work by the pen you learned. Sc. If that be all, you shall not need to repeat again that that was sufficiently taught already: and except you will teach me any other feat, here may you make an end of this art, I suppose. Ma. So will I do as touching whole number: and as for broken number, I will not trouble your wit with it, till you have practised this so well, that you be full perfect, so that you need not to doubt in any point that I have taught you, and then may I boldly instruct you in the art of Fractions or Broken number: wherein I will also show you the reasons of all that you have now learned. But yet before I make an end, I will show you the order of common casting, wherein are both pennies, shillings, and pounds, proceeding by no grounded reason, but only by a received form, and that diversly of diverse men: for the merchants use one form, and Auditors an other. merchants use. And further you may see, that the space between d and s may receive but one counter (as all other spaces likewise do) and that one standeth in that place for 6 d. Likewise between the shillings and the pounds, one counter standeth for 10 s. And between the pounds and 20 lb. one counter standeth for 10 lb. But beside those you may see at the left side of shillings, that one counter standeth alone, and betokeneth 5 s. So against the pounds, that one counter standeth for 5 lb. And against the 20 pounds, the one counter standeth for 5 score pounds, that is 100 pound, so that every side counter is 5 times so much as one of them against which he standeth. Auditors Account. Now for the account of Auditors, take this example. Where I have expressed the same sum 198 lb 19 s 11 d. Auditors account. But here you see the pence stand towards the right hand, and the other increasing orderly toward the left hand. Again you may see that Auditors will make 2 lines (yea and more) for pence, shillings, and all other values, if their sums extend thereto. Also you see that they set one counter at the right end of each row, which so set there, standeth for 5 of that room: and on the left corner of the row it standeth for 10 of the same row. But now if you would add other subtract after any of both those sorts, if you mark the order of the other feat which I taught you, you may easily do the same here without much teaching: for in Addition you must first set down one sum, and to the same set the other orderly, and in like manner if you have many: but in Subtraction you must set down first the greatest sum, and from it must you abate the other, every denomination from his due place. Sc. I do not doubt but with a little practice I shall attain these both: but how shall I multiply and divide after these forms? M. You can not duly do any of both by these sorts, therefore in such case you must resort to your other arts. S. Sir, yet I see not by these sorts how to express hundreds, if they exceed one hundred, neither yet thousands. Master. And if you desire the same sum after Auditors manner: Lo here it is. But in this thing you shall take this for sufficient, and the rest you shall observe as you may see by the working of each sort: for the diverse wits of men have invented diverse and sundry ways, almost unnumerable. THE ART OF NVMbring on the hand. But one feat I shall teach you, which not only for the strangeness and secretness is much pleasant, but also for the good commodity of it, right worthy to be well marked. This feat hath been used above 2000 years at the least, and yet was it never commonly known, especially in English it was never taught yet. This is the art of numbering on the hand, with diverse gestures of the fingers, expressing any sum conceived in the mind. And first to begin. If you will express any sum under 100, you shall express it with your left hand and from 100 unto 1000, you shall express it with your right hand, as here orderly by this Table following you may perceive. Here followeth the Table of the Art of the hand. hand gesture 1 hand gesture 2 hand gesture 3 hand gesture 4 hand gesture 5 hand gesture 6 hand gesture 7 hand gesture 8 hand gesture 9 hand gesture 10 hand gesture 20 hand gesture 30 hand gesture 40 hand gesture 50 hand gesture 60 hand gesture 70 hand gesture 80 hand gesture 90 hand gesture 100 hand gesture 200 hand gesture 300 hand gesture 400 hand gesture 500 hand gesture 600 hand gesture 700 hand gesture 800 hand gesture 900 hand gesture 1000 hand gesture 2000 hand gesture 3000 hand gesture 4000 hand gesture 5000 hand gesture 6000 hand gesture 7000 hand gesture 8000 hand gesture 9000 1 In which (as you may see) 1 is expressed by the little finger of the left hand, closely and hard crooked. 2 Is declared by like bowing of the wedding finger (which is the next to the little finger) together with the little finger. 3 Is signified by the middle finger, bowed in like manner with these two. 4 Is declared by the bowing of the middle finger, and the ring finger or wedding finger, with the other all stretched forth. 5 Is represented by the middle finger only bowed. And 6 by the wedding finger only crooked: and thus you may mark in these a certain order. But now 7, 8, and 9, are expressed with the bowing of the same fingers, as are 1, 2, & 3, but after another form. For 7 is declared by the bowing of the little finger as is 1, save that for 1 the finger is clasped in, hard and round, but for to express 7, you shall bow the middle joint of the little finger only, and hold the other joints strait. Sch. If you will give me leave to express it after my rude manner, thus I understand your meaning: that one is expressed by crooking in the little finger, like the head of a bishops bagle: and 7 is declared by the same finger bowed like a gibbet. M So I perceive you understand it. Then to express 8, you shall bow after the same manner both the little finger, and the ring finger. And if you bow likewise with them the middle finger, then doth it betoken 9 Now to express 10, you shall bow your forefinger round, and set the end of it on the highest joint of the thumb. And for to express 20, you must set your fingers strait, and the end of your thumb to the partition of the foremost & middle finger. 30 Is represented by the joining together of the heads of the foremost finger & the thumb. 40 Is declared by setting of the thumb crossewaies on the foremost finger. 50 Is signified by right stretching forth of the fingers jointly and applying of the thumbs end to the partition of the middle finger, and the ring finger or wedding finger. 60 Is form by bending of the thumb crooked, and crossing it with the forefinger. 70 Is expressed by the bowing of the foremost finger and setting the end of the thumb between the 2 foremost or highest joints of it. 80 Is expressed by setting of the foremost finger crossewayes on the thumb, so that 80 differeth thus from 40: for that 80, the forefinger is set crossewayes on the thumb and for 40 the thumb is set cross over the forefinger. 90 Is signified by bending the forefinger, and setting the end of it in the innermost joint of the thumb, that is even at the foot of it. And thus are all the numbers ended under 100L. Sc. In deed these be all he numbers from 1 to 10, & then all the tenths within 100, but this teacheth me not how to express 11, 12, 13, etc. 21, 22, 23, etc. and such like. Ma. You can little understand, if you can not do that without teaching. What is 11? is it not 10 and 1? then express 10 as you were taught and 1 also, that is 11: and for 12 express 10 and 2: for 23 set 20 and ●: and so for 68, you must make 6, and thereto 8: and so of all other sorts. But now if you would represent 100, either any number above it, you must do that with the right hand, after this manner. You must express 100 in the right hand with the little finger, so bowed as you did express 1 in the left hand. And as you expressed 2 in the left hand, the same fashion in the right hand doth declare 200. The form of 3 in the right hand standeth for 300. The form of 4 for 400. Likewise the form of 5, for 500 The form of 6, for 600. And to be short: look how you did express single unities and tenths in the left hand, so must you express unities and tenths of hundreds, in the right hand. Scholar. I understand you thus: that if I would represent 900, I must so form the fingers of my right hand to express 9 And as in my left hand I express 10, so in my right hand must I express a 1000 And so the form of every tenth in the left hand, serveth to express the number of thousands, so the sum of 40 standeth for 4000 The sum of 80, for 8000. And the form of 90 (which is the greatest) for 9000, and above that I can not express any number. Master. No, not with one finger, how be it, with diverse fingers you may express 9999. and all at one time, & that lacketh but 1 of 10000 So that under ten thousand you may by your fingers express any sum. And this shall suffice for Numeration on the fingers. And as for Addition, Subtraction, Multiplication, and Division (which yet were never taught by any man as far as I do know) I will instruct you after the treatise of Fractions: and now for this time far well, and look that you cease not to practise that you have learned. Sc. Sir, with most hearty mind I thank you, both for your good learning and also your good counsel, which (God willing) I trust to follow. FINIS. ¶ THE second part of the Arithmetic touching Fractions, briefly set forth. Scholar. ALbeit I perceive your manifold business doth so occupy, or rather oppress you, that you can not as yet completelie end that treatise of Fractions Arithmetical, which you have prepared, wherein not only sundry works of Geometry, Music, and Astronomy be largely set forth, but also divers conclusions and natural works, touching mixtures of metals, and compositions of medicines, with other strange examples, yet in the mean season. I can not stay my earnest desire, but importunelie crave of you some brief preparation, toward the use of Fractions, whereby at the least I may be able to understand the common works of them, and the vulgar use of those rules, which without them can not well be wrought. Master. If my leisure were as great as my will is good, you should not need to use any importunate craving, for the attaining of that thing, whereby I may be persuaded that I shall any ways profit the common wealth, or help the honest studies of any good members in the same: wherefore, while mine attendance will permit me to walk and talk, I am well willing to help you as I may. What a Fraction is. Therefore first to begin with explication of this name Fraction, what take you if to be? Scholar. Marry sir, I think a Fraction (as I have heard it often named) to be a broken number, that is to say, to be no whole number, but a part of a number. Master A Fraction in deed is a broken number, and so consequently, the part of another number: but that must be understanded of such an other number, as can not be divided into any other parts than Fractions: for although I may take the third part of 60, or the fourth part of it, and so of other parts diversly, yet these parts be not properly, nor ought not to be called Fractions, because they may be expressed by whole numbers: for the third part of it is 20: the fourth part is 15: the twelfth part is 5, and so forth of other parts, which all be whole numbers. Wherefore properly a Fraction expresseth the parts or part only of an unit, What a Fraction in properly. that is to say, that the number which is the whole or entire sum of any Fraction, may not be greater than one: and therefore it followeth, that no one Fraction alone can be so great, that it shall make 1, as by examples I will declare as soon as I have taught you to know the form how a Fraction is expressed or represented in writing. NUMERATION. But first to begin with the expressing of a Fraction, which is the numeration of it, you must understand that a Fraction is represented by 2 numbers, set one over the other, and a line drawn between them as thus, ⅓. ●/4. ⅘ 10/17 which four Fractions you must pronounce thus: ●/3 one third part: ¾▪ three quarters: ⅖ two fift parts: 10/17. ten seventeen parts. Scho. I understand the form of their expression and pronunciation, but their meaning or valuation seemeth more obscure: yet I think that by the two first Fractions I understand the valuation of the two later Fractions, and so consequently of other. M. Value them then, that I may perceive your taking of them Scholar. ⅖ betokeneth two fift parts, that is to say, if one be divided into 5 parts, that Fraction doth express ij. of those fifth parts: 10/17 doth signify, that if one be divided into xvij. parts, I must take ten of them. And this I gather of the two first examples: for ⅓. that is one third part, doth easily declare, that if any one thing be divided into three parts, I must take but one of them: so ¾ that is three quarters, doth declare that one being divided into four quarters, I must take (for this Fraction) three of those quarters. If there be no more difficulty in their Numeration, than I pray you go forward to their Addition and Subtraction, ☜ and so to the other kinds of works for I understand that the same kinds of works be in Fractions, that be in whole numbers. Master. There are the same kinds of works in both, albeit the order of them is diverse, as I will anon declare: but yet more in Numeration before we leave it. You must understand, that those two numbers which express a Fraction, have several names. Numerator, and Denominator. The overmost which is above the line, is called the Numerator, and the other beneath the line, is called the Denominator. Scholar. And what is the reason of their diverse names? For in mine opinion both be Numerators, seeing both they do express the numeration of the Fraction. Master. You are deceived: for one only (which is the overmost) doth express the Numeration: and the denominatour doth declare the number of parts into which the unit is divided, as in this example, when I say: Divide a pound weight of Gold between four men, so that the first man shall have 2/15 the second 2/15 the third 4/15 and the fourth 6/15. Now do you perceive the by the denominator (which is one in all four Fractions) it is intended, that the pound weight should be divided into so many parts I mean 15, and by the four several numerators is limited the diverse portion that each man should have, that is, that when the whole is parted into 15, the first man shall have 2 of those 15 parts: the second man three of them: the third man 4: and the fourth man 6. And so may you see the several offices (as it were) of those two numbers, I mean of the Numerator and the denominator. And hereby you perceive, that a man can have no more parts of any thing than it was divided into, neither yet aptly so many: so that it were unaptly said: You shall have 15/15. that is xv fifteen parts of any thing, seeing it were better said: You shall have the whole thing. Sc. So doth it appear reasonably: for the labour is vain, to divide any thing, and than to apply the Division to no use. And much less reasonable were it to say 16/15: for if the whole be divided into 15 parts only, it is not possible to take 16 of them, that is to say, more than altogether. Master. This is true touching the proper and apt use of the name of a Fraction: ☜ yet improperly, and after a vulgar acceptation (for easiness in work) both those forms be called Fractions, because they be written like fractions, although they be none in deed for 15/15, and generally all such other: where the Numerator and Denominator be equal, are not Fractions: but the whole thing with all his parts. And so 16/12 is not to be called a fraction, but a mixed number, of a whole number and a Fraction: for it is as much, as 1 4/12, that is one whole one, and 4 twelve parts, as shall be declared in Reduction. Therefore they do abuse the names, that call them Fractions, where the Numerator is either equal or greater than the Denominator. Sc. But is there any needful cause why they should so abuse the name? Mai. There is cause why they shall sometimes, for easiness in work, writ some numbers after that sort, like fractions: but they needed not to call them fractions, but as they be whole numbers or mixed numbers (that is whole numbers with Fractions) expressed like fractions. Now must you understand, that as no fraction properly can be greater than 1, so in smallness under one the nature of Fractions doth extend infinitely: as the nature of whole numbers is to increase above one infinitely, so that not only one, may be divided into infinite Fractions or parts, but also every Fraction may be divided into infinite Fractions or parts, which commonly be called Fractions of Fractions, and they be expressed diversly: As for example, 3/●. ⅔. ½, that is three quarters of two third parts, of one half part. Whereby is signified, that if one be divided into two halves, and the one half into three parts, and two of those three parts, be divided idyntlye into four quarters, this Fraction of Fractions doth represent three of those quarters. Scholar. I pray you let me prove by an example in common money, whether I do rightly understand you or no. One Crown, which I take for an unit, doth contain 60 pennies, therefore the half of it is thirty pence: ⅔ of that half is 20 pence, whereof 3/● is fifteen pence, so then 15 pence is 3/●. 2/●. ½ of a Crown. And so 3 pence is ¾. ⅔. ½. of a shilling. Master. You perceive this well enough, but how happened that you found no doubt in the form of writing these Fractions, seeing the two latter Fractions have no line between their numbers, as the first hath? Sc. Because I had forgotten (as Scholars oft times do) that that was told me before: but I pray you, express the reason thereof. Mai. This form is but voluntary, and therefore hath none other reason than the will of the diviser, which form many do follow. Some other do make lines between every Fraction, and add words of distinction, after this sort, ¼ of ⅔, of ½, which form is good also. Some other express them thus in slope form, to distinct them from several Fractions of one whole number, for if they were set in one right line thus, ☞ ¾ ⅔ ½. then ought it to be pronounced, three quarters, and two third parts and an half, which maketh almost two whole units, lacking but one xii. part. And so is it nothing agreeable with the other Fraction of Fractions, wherefore it is a great oversight in certain learned men, which do express them so confusedly with such several Fractions, that a man can not know the one from the other. Therefore some men (as Stifelius) do express without a line numbers of proportion, being applied to Addition or Subtraction: because they must be taken as two, where the line in Fractions maketh them to be taken for one: for of the Numeratour and Denominatour is made one number. Scholar. Three several varieties. Then I perceive there be three several varieties in Fractions: First when one only Fraction is set for one number, as 4/●, that is four fifth parts. The second, is when there be set two or more several Fractions of one number, as ⅘ 2/5, that is iiij. ninth parts, and two fift parts. The third sort is Fractions of Fractions, as 4/9 2/5, that is 4 ninth parts of two fifth parts. Ma. You have said well, if you understand well your own words. Scho. If it shall please you, I will by an example in the parts of an old English Angel express my meaning. Mai. Let me hear you. Scholar. The old English Angel did contain 7 shillings 6 d, that is 90d. Now ⅘ of it, is 72d. And of the same 90 pence, if I take ⅘ and ⅖, that is four ninth parts, and 2 fifth parts, 4/9 is 40, and ⅖ is 36, which both make 76: but if I take 4/9 of ⅖, that is four ninth parts of two fift parts, seeing ⅖ is but 36, than 4/9 of 36 will yield but 16: for 1/9 of 36, is but 4, and that taken four times maketh 16. Master. This is plainly expressed, and truly and hereby (I doubt not) but you do perceive, that as great a difference as is between 16 & 76, so much difference is between these two Fractions 4/9 and ⅖: and 4/9 of ⅖. And now that you understand these varieties, I will proceed to the rest of the works: first admonishing you, that there is an other order to be followed in Fractions than there was in whole numbers, for in whole numbers this was the order: Numeration, Addition, Subtraction, Multiplication, Division, and Reduction, but in Fractions (to follow the same aptness in proceeding from the easiest works to the harder) we must use this order of the works: Numeration, Multiplication, Division, Reduction, Addition, and Subtraction. Scho. That Multiplication and Division should go together, and Subtraction to follow Addition, natural order doth persuade: but why Multiplication should be first in order here next to Numeration, and Reduction in the middle, I desire to understand the reason May. As in the Art of whole numbers order would reasonably begin with the easiest, and so go forward by degrees to the hardest, even so reason teacheth in Fractions the like order. And considering that Addition or Subtraction of Fractions can very seldom be wrought without multiplication and Reduction: and contrariwayes, Multiplication and Reduction may be wrought without this form of Addition or Subtraction. Therefore was it orderly required, that Multiplication and Reduction should go before Addition and Subtraction. And the same reason serveth for the placing of Multiplication before Reduction. Sch. Then if Multiplication be the easiest, I pray you declare the form of it first by rule, and then by example. May. Your example is good. MULTIPLICATION. THerefore when any two Fractions be proponed to be multiplied together, the Numerator of the one must be multiplied by the Numerator of the other: and the sum that amounteth thereof, must be set for a new numerator: likewise the Denominatour of the one must be multiplied by the Denominator of the other, and that that amounteth, shall be set for the common Denominator: & this new third Fraction expresseth the Product of the multiplication of the two first fractions proponed, whereof take this example, ⅗ multiplied by 5/12, doth make 15/6●. Scholar. I perceive then, that 3 being the Numeratour of the first Fraction, is multiplied by 5, being the Numeratour of the second Fraction, whereof amounteth 15, the Numeratour of the third Fraction. And so likewise, 5 being Denominatour of the first Fraction, is multiplied by 12 the Denominator of the second fraction, whereof amounteth 60 the new Denominator: so that I perceive how the work is done, but I do not perceive how 15/60 is greater than ⅖: For if I shall use my former manner of examination by the parts of some Coin, I see that ⅗ of a Crown is 36 d, and 5/12 of a Crown, is 25 d, whereof the one multiplied by the other, doth make 900 d, which is 15 Crowns: but by your multiplication there amounteth ●●/60, which is but 15 d, and that is much less than any of both the first Fractions. Mayst. That difference is between multiplication in whole numbers, and multiplication in broken numbers that in whole numbers the sum that amounteth, is greater than both the other whereof it came: but in Fractions it is contraryways: for the sum that amounteth is lesser than any of the other two fractions, whereof it came. Sc. I desire much to understand the reason thereof. May. Although I purposed to reserve the reasons of works Arithmetical for the perfect Book of Arithmetic, yet I will show you this, because of the strangeness of the work. You see in whole numbers, that of two numbers being multiplied together, is made the third number: which third number doth bear the same proportion to the number multiplied, that the multiplier doth bear to an unit. And so in Fractions, the third number which amounteth of multiplication, beareth the same proportion to each of the two first fractions, that the other of those two fractions doth bear to an unit. Scholar. Sir I understand your words thus: when 40 is multiplied by 12, there doth amount 480, which 480 doth contain 40 so many times in it, as twelve doth contain units, that is to say: twelve times. And so it appeareth, that 480 doth contain twelve so many times also, as 40 doth contain Unites, that is 40 times. But now I see not how the third number in this example of Fractions can contain any of the two former (as it happened in whole numbers) seeing it is lesser than either of them. Mai. No marvel if you cannot see that thing which is not possible to be seen of any man, how the third number in multiplication of Fractions should be greater than any of the two former Fractions, but yet this may you see (which I said) that the third number in Fractions so multiplied, doth bear the same proportion to any of the two former fractions, that the other of those 2 fractions doth bear to an unite, as in your example ⅗ being multiplied by 5/12, doth make 15/60. Now say I, that 15/60 doth bear the same proportion to ⅗, that 5/12 doth bear to an unit, as you may in your own form of examination by coin try it. For in an old Angel are 180 half pence, which I set for the entire unit whose parts (according to the Fractions aforesaid) are these, for 15/60 set 45 ob. for ⅖ take 160 ob and for 5/12 put 75 ob Now doth 45 bear the same proportion to 108 that 75 doth bear to 180: for 45 is 5/11 of 108, and so is 75 also 5/12 of 180. And for easier applying of each comparison, consider this form of setting all these numbers before your eyes, where the second demonstration towards your right hand is answerable to the first in every proponed part, where for ⅗ (of 180) stands his value 108: for ●/● stands 75: and for 15/60 is 45. But these reasons may be better reserved till another time, when the knowledge of proportions in due order shall be taught. Yet in the mean season I will show you how it cometh to pass that in Fractions the third sum must needs be lesser than any of the other two. Consider thus, that when a Fraction is proponed, as in the former example ⅗, if it be multiplied by more than 1, it will make more than one entire number. As if I multiply by 5, that is to say if I take it 5 times, it will make three entire units: example in a Crown, ⅗ of it maketh 3 s, which if I take five times, it will amount to 15 shillings, that is three entire Crowns: so if I take the same ⅗ but twice, it will yield 6 s, that is one entire Crown and ⅕. Now if I take it but once it cannot be more than it was before, that is, ☞ 3 s. And if I take it less than once, it cannot be so much as it was before. Then seeing that a Fraction is less than one, if I multiply a Fraction by another Fraction, it followeth that I do take that first Fraction less than once and therefore the sum that amounteth, must needs be less than the first fraction. Sc. Sir, I thank you much for this reason, And I trust I do perceive the thing, as by example of this same Fraction ⅗ I will express If I take ⅗ of a Crown once, that is to say, if I multiply ⅗ by 1, it will be as it was before, but 3 s: so if I do multiply it by ½, that is, if I take it but half one time, then will it be but half so much: likewise if I multiply it by ½, that is, if I take but the third part of once, it will yield but 12 pence, that is the third part of the first Fraction. And so to make an end. If I take it but the twelfth part of once, that is, if I do multiply it by ●/●●, it will yield but the twelfth part of the first Fraction, which is but three pence. And it followeth that if 1/●2 make 3 pence, than 5/12 must needs make five times so much, that is 15 pence, which was the sum that hath given the occasion of all this doubt. Master. Then I perceive you have sufficient understanding in this sort of multiplication for this time, wherefore I will omit that I might say more of Multiplication, till we come to reduction, and will pass to the other works, and first to Division, whose place followeth Multiplication, both by natural order, and also in eansinesse of work. DIVISION. WHen so ever two fractions be proponed, that the one should be divided by the other, I must set down first the Fraction that shall be divided (which is called the Dividend) and then after it the other, which is the Divisor. Then shall I multiply the numerator of the dividend by the denominator of the divisor, and that which amositeth, I must put for a new numerator. Again, I shall multiply the denominator of the dividend by the numerator of the divisor, and the number that amounteth thereof, I must put for the new denominator. And this third fraction is the quotient of the said division. Scholar. This seemeth easy in form, as by example, thus: If I would divide ⅝ by 2/6, first I must multiply 5 (being the numerator of the dividend) by 6, which is the denominator of the Divisor, and thereof riseth 30: then I multiply 8 (being the denominator of the dividend) by 2, being Numerator in the divisor, and so riseth 16, the which I must make in a third Fraction, thus 30/16. Ma. Me seemeth you are quicker in understanding now, than you were when I taught you the art of whole numbers: but that is no marvel, for the more knowledge that any man getteth, the readier shall he find his wit, and quicker in understanding: but yet of 2 things I will admonish you, which you might have observed here for ease of work and lightness of understanding the nature of the Quotient. Whensover you divide one Fraction by an other, either they be both equal together either else the one is greater than the other: if they be equal, their quotient shall be such, that the numerator and the denominator of it shall be equal also. And if the 2 first fractions be unequal, their quotient shall declare the same by the unequality of the numeratour and denominatour, as in these examples following shall appear. First of equal Fractions: 4/9 and 12/2● be equal together: and if the one be divided by the other, the quotient will be 108/108, as you may perceive by that rule aforesaid. Now in the unequal Fractions, as 4/9 and 3/10 the quotient will be 40/27: where the Numerator is greater than the denominator. Sch. I see it is so, but I see not the reason why it should be so. Master. Note how 〈…〉 2 numbers. The reason is this, when any Fraction is divided by an other, the quotient declareth what proportion the dividend beareth to the divisor. So ½ divided by ¼, maketh 2, which must be sounded, not two, but twice: declaring that ¼ is contained twice in ½. And note this, that the Numerator in the Quotient, representeth the Dividend, & the Denominator representeth the divisor▪ And this is always true, ☞ whether, the greater fraction be divided by the lesser, or the lesser by the greater. But this proportion will not be exactly known, till you have learned the art of proportions: notwithstanding somewhat of it will I declare in the next rule of Reduction. But now for the easy remembrance of the Quotient in division, as soon as you have set down your two Fractions, the one against the other, then make a straight line for the quotient: and as soon as you have multiplied the Numerator of the dividend, by the Denominator of the divisor set the number that amounteth, over the said line, and then multiply the other two numbers, & set their total under the same line. Scholar. I perceive you would not have me trust to memory till I were better expert, lest oftentimes I happen by miss remembrance to be abused. This example I take for that declaration. If I would divide ⅔ by ¾ I must set the numbers one against the other, (as here doth appear) & then make an other line for the Quotient in some good distance, where I may set the numbers of the Quotient, as soon as any of them is multiplied: So then as soon as I have multiplied 2 by 4, which maketh 8, I shall set that over that line thus. And then multiply ● by ●, which yieldeth 9: and that 9 must I set under the same line, and then will the whole quotient appear thus 8/9. Note. Whereby appeareth (as I remember your words) that ⅔ is in proportion to ¼, as 8 is to 9: but how may I perceive that? Master. Although you shall better perceive it by the rule of Reduction, yet this example may be declared in common coin, as in a common shilling of xii. pence, of which ●/2 maketh 8 d, and ¾ doth make 9 pence, and so you may easily see that their proportions do agree. And if you had taken this example before, when you took the example of▪ and ●/6, your Quotient would appear (as this doth) more easier to understand, whereas that Quotient being 10/16, is not an easy proportion for you to perceive, being yet little acquainted with proportions: whereof to give you some taste, I will enter to the rule of Reduction: in which also I will declare other works, both of Multiplication and also of Division, which now I must for a time omit, as things that do need the help of Reduction. REDUCTION. THerefore will I now declare the diversities of Reduction of Fractions, Five varieties of Reduction. which commonly have five varieties. 1. First, when there be, sundry Fractions of one entire Unity, they must be reduced to one denomination, and also into one Fraction. 2 Secondarily, when there be proponed fractions of fractions, they must be reduced likewise into one Fraction, for other ways they can not be brought into one Denomination. 3 Thirdly, when an Improper fraction is proponed, that is to say, a fraction in form, which indeed is greater than an unity, it must be reduced into apt form expressing the Unity or Unities of it, & the proper fraction distinctly. And some times also it shall be needful to convert such a mixed number of Unities, with Fractions into the form of a Fraction, that is into an Improper Fraction, which 2 forms I esteem but as one, because they work on one kind of number. 4 fourthly, there happeneth sometimes Fractions to be written in great numbers, which might be written in lesser numbers, therefore is there a mean to reduce such great numbers into their smallest terms. 5 Fiftelie, when any Fraction betokeneth the parts of a whole thing, which hath by common partition certain parts, but none of like Denomination with that Fraction, then may you reduce the said Fraction into an other, whose Denomination shall express the common parts of that whole thing. Scholar. This distinction in doctrine delighteth me much, but more with hope than present fruit, for as yet I do not understand scarcely the varieties, and much loss the practice and use of their works. Master. Reduction is an orderly alteration of numbers out of one form into an other, which is never done orderly but for some needful use, as in every of the said 5 several varieties I will distinctly declare. first therefore, when two or more several Fractions of any Unite be proponed, The first sort of Reduction. as for example, ●3/1 and 4/6: because it is hard to tell what proportion of the entire number those two Fractions do express, therefore was Reduction devised, to be a mean whereby these several Fractions might be brought into one Denomination and Fraction. And in these Fractions this is the art for bringing them to one denomination. Multiply first the Denominators together, How to reduce fractions of diverse denominations into one Denomination. and the total thereof you shall set twice down under two several lines for two new Denominators, or rather for one common Denominator: Then multiply the Numeratour of the first Fraction, by the Denominator of the second, and set the total thereof for the Numeratour over the first line. Likewise multiply the Numerator of the second Fraction by the Denominator of the first, and set that total over the second line for the Numerator of that Fraction, and so are those two first Fractions of several denominations, brought to one Denomination. Sch. If I understand you, as I think I do, my example shall declare the same. The Fractions which you proponed were these, ●/16 and 4/6▪ whose Denominators (being 16 and 6) I multiply together, and there amounteth 96, which I set under lines, thus. Then I multiply the Numerator of the first Fraction by the Denominator of the second, saying: 3 into 6 maketh 18, that set I over the first line for a new Numeratour, and it will be thus. Likewaies' I multiply the Numeratour of the second Fraction by the Denominatour of the first: saying: 4 times 16 maketh 64, that I set for the second Numeratour, and the Fraction will appear thus. So that both Fractions brought to one Denomination, must stand thus: Ma. You have done well. Scholar. I beseech you, let me examine it after my accustomed form, by common parts of coin. Ma. Go to. Scholar. A new Angel accounted at eight shillings, containeth 96 pence, whereof ●▪ that is the xuj. part, is six pence, and 1/16 is 18 pence, that is 18/●6. Again ⅙ of the same Angel, is 16 pence, so that 4/6 maketh 64 d, that is 64/●●. And so I find the sums to agree with the other before. Master. So have you now the Art to bring such two Fractions into one Denomination. And if there be more than ij, Note the Reduction of three Fractions or more to one. then must you multiply all the Denominators together, and set the total thereof so many times down as there be Fractions, and then to get for each one a new Numerator. Multiply the Numerator of the first, by the Denominator of the second, and the total thereof multiply by the denominator of the third, and so forth if there be more. Likewise multiply the Numerator of the second, by the denominator of the first, and the total thereof by the Denominator of the third. And in the same sort multiply the Numerator of the third into the Denominator of the first: & the total thereof into the Denominator of the second, and so forth, if there were more. So these 3 Fractions 2/5 ●/4 2/● doth make by Reduction these other 3. Fractions of one Denomination 24/6● 45/6● 4●/6●. All which you may bring into one Fraction by adding the Numeratours together, and putting that total for the common Numerator, reserving still that same common Denominator, And those 3 Fractions make one Improper Fraction thus. Scho. All this I perceive, and also that this last Fraction is more than an Unity, and therefore you did call it an Improper Fraction. M There be certain other forms of working in this reduction, which I will briefly touch also, to give you an occasion to exercise your wit therein. The first variety of this Reduction. The first variety is this. When you have made and written down your common Denominator (as I have taught before) then to get a Numerator for the first, do thus. Divide the common Denominator by the Denominator of the first Fraction, and the quotient multiplied by the Numerator of the same, yieldeth a new Numerator for the first new Fraction. So likewise do with the second and the third, and with all the residue if there be more. Scholar. That will I prove in your last example of these 3 Fractions . When the Denominators be multiplied, they make 60 for 5 into 4 maketh 20, and 20 by 3 yieldeth 60, that I set down 3 times, thus. : then to have a Numerator for the first, I must divide 60 by 5, (the Denominator of the first) & the quotient is 12, which I must multiply by 2 (the Numerator of the first) & that maketh 24, and so have I for the first Fraction 24/60. Likewise for the second fraction: The second variety. I divide 60 by 4, and there cometh 15, which I multiply by 3, and so have I 45 and the second fraction 45/60. Then for the third in like sort will come 40/60. Master. another way is this. If it happen so that the lesser Denominator can by any multiplication make the greater, than note the multiplier, and by it multiply the Numeratour over that lesser Denominatour, and for the lesser Denominatour put the greater, as thus in these two Fractions, 2/●2 and ⅔ three being the lesser Denominatour multiplied by 4, will make 12, which is the greater Denominatour: therefore by the same 4, I do multiply 2, which is the Numeratour over 3, and that maketh 8: under which I do put 12 being the greater Denominatour, which is also made by Multiplication of 4 into ●, & so have I these ij. Fractions 3/●2●, 8/●2: thus shortly reduced without altering the one Fraction. S This I understand. Ma. The third ●arietie. Then mark this third way: If the denominators do not happen so, that one by Multiplication may make the other, than look whether they both may be parts of any other one number, as in ●/1● and ●/18, although the lesser taken but twice be to great to make 18, yet they both may be parts unto 36: therefore look how many times 12 is in 36, and that quotient being multiplied by the numeratour over 12, the total shall be put in stead of the Numerator over 12, & for 12 put 36, thus, ●5/●6. So likewise look how often is 18 in ●6, and because it is twice, therefore by 2 multiply 7. which is over 18, and it will be 14, set that for the numerator, and in stead of 18 put 36, and then shall your Fractions reduced, stand thus, 1●/36 14/36 in stead of 5/●2 and 7/1●. Proof. And if you will prove whether you have wrought well or no, that may be proved by Reduction of them again to their former denominations, which art shall be taught in the fourth kind of Reduction, where greater terms of Fractions be reduced into smaller in number, but no smaller in proportion. ☜ And if in such Reduction the same terms or numbers come again that were before, then is the work good, else not. S Sir, I hear your words, but I do not understand many of them, which it may please you to declare. Ma. With a good will, when convenient place serveth, but that must be in the said iiij. kind of Reduction. In the mean season I will declare the second form of Reduction, which teacheth how to reduce Fractions of Fractions into one Fraction, and so to one Denomination. When fractions of fractions be proponed, Reduction of fractions of fractions into one fraction and denomination. you shall multiply the Numeratours of each into other, and set the total for the new Numerator, and then multiply all the Denominatours likewaies, and take their total for the new denominator, and so are they speedily reduced. Sc. If that be all than I understand it already, as by this example I will declare. These be the Fractions, ¼ of ⅔ of 6/7 of 7/9 which I would reduce to one denomination. Therefore begin I with the Numeratours, and multiple them all together, saying: 3 into 2 maketh 6, and 6 by 6 maketh 36, which multiplied by 7, yieldeth 352, that I set over a line for the Numerator, thus: Then I multiply the denominatours, 4 by 3 maketh 12, & that by 7 bringeth 84, which multiplied by 9, yieldeth 756, the new denominatour. And so the whole reduced fraction is this, which is to hard a Fraction for me to understand yet. Mai You think so, and no marvel, but anon you shall learn to judge it easily, for this Fraction is no more in deed then ⅓ although it be in greater terms, and therefore more stranger and more obscure. And this sufficeth for this Reduction, save that I will show you by a figure of measure, the just rate and reason of this kind of Fractions, and also the due understanding of the Reduction. The entire measure parted into 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 7/9 .1. .2. .3. 6/7 1 2 3 4 ⅔ 1 2 3 ¾ Hear you see the longest measure, (which standeth for the whole and entire quantity) first parted into 9 Divisions, whereof 7 are severed by the second measure: and thereof again are parted out 6. And that 6 being distinct into 3 parts, 2 of them are parted by the fourth measure, of which fourth measure, being divided into 4 parts, the lowest measure doth contain ¾, so that the same ¾ must be named, not 3/4 of the whole measure, but in deed is ¾ of ⅔ of 6/7 of 7/9 or as I would rather express it, 3/● 1/● 6/● 7/●. Scholar. This example is so sensible, that I can not choose but see it. And furthermore, I see also, that the same fraction is equal to 3/9 of the entire measure, as the lines which run up and down do expressly set forth. Also I see here, that ⅔. 6/7, 7/9 is equal to 4/9 And further yet that 6/7. ●/9 is equal to 6/5. Ma. I am glad that you see it so well, not doubting but you will gather greater light of knowledge hereby. The third form of Reduction. But now it is time that we come to the third form of Reduction, which teacheth of Improper Fractions, that is to say, Mixed numbers of Unites and Fractions, although they appear like Fractions, as this 2/● 6/5, which doth include 5 Unites wholly, and ⅕ over. Wherefore first you shall know them, by that the Numeratour is greater than the Denominatour. Scho. In deed sir that appeareth reasonable, that if the Numerator do express more parts to be taken of any unity than the Denominator doth signify that unity to be divided into, it must needs follow that such a fraction importeth more than the whole, that is to say, the whole with certain parts over. But what Reduction is there in it? Ma. There be two several kinds of Reduction, concerning such Fractions. Sometimes it shall be needful to convert those fractions into the Unites and the proper fraction that will remain And sometimes contrary ways it shall be meet to reduce mixed numbers, that is unities, written with fractions into the form of one simple fraction, & so be there two ways. Sch. What is the mean of the first way to turn improper fractions into Unities with their proper Fractions? M. That is thus. Reduction of improper frictions into units with their proper Fractions. Your Numerator being greater than the Denominator, must be divided by the same denominator, & the quotient thereof expresseth the unities, the remainder shall be put for the Numerator of the fraction that resteth, and the denominator must be the same that was before. Sc. For example, I take 17/5. And dividing 17 by 5, the quotient will be 2, and there will remain 2. Ma. That must you write thus, 3 ⅖ where (you see) I have written, without any line, as entire numbers ought to be written, and the 2 that remained, I have set over the former denominator with a line, as a proper fraction. And this number doth signify now 3 Vnits, and ⅖ of one. Scho Then if I would by Unites here understand crowns, so it were 3 crowns, and ⅖, that is 2 s. M. Even so, and therefore 17/5 did signify the same. But this happeneth sometimes, that when the Reduction is so wrought, there remaineth nothing. And then it is not a mixts number, but a simple entier number, represented like a Fraction. Scholar As ●●/● will make 3 just, and 15/3 will make even 6. This I will remember. But now, what is the second form of Reduction, that you spoke of for these sorts of Fractions? M. Reduction of whole numbers, either alone, or joined with fractions into ●ractions. Whensoever you have any of these two sorts of numbers, that is to say, whole numbers without Fractions, or whole numbers with fractions, and you would turn them into the form of a fraction you must multiply the whole number by that denominatour, which you will have to remain still, and to the total thereof add the numeratour which you have already, and all that shall you set for the new numerator, keeping still the former denominatour: as if you have 6 ¾ which you would convert into an Improper Fraction you must multiply 6 by 4, whereof cometh 24, and thereto add the numeratour which is 3, and so have you 27 for the numerator, and 4 still for the Denominator. Sc. Then is 27/4 equal to 6 ¾. Ma. Even just, and so backward (as appeareth by the former Reduction) 6 ¾ maketh 27/4 And thus one of these Reductions may be the proof of the others work. Sc. This I perceive, but now if you would turn whole numbers without fractions into any fraction, I see not how that may be done, because there is no denominatour to make the multiplication by. Ma. That was well marked: but this you know, that no man intendeth to turn any whole number into a Fraction, but he hath in his mind that denominator by which the Multiplication must be made: ☜ for the proof whereof I set down 7, which is a whole number. And if you will have this number converted into any certain fraction, will me to do it. Scholar. I pray you reduce 7 into a Fraction. Ma. Then you care not what the fraction be, so it be some fraction. Sc. No, I pass not for the sort of the Fraction. Ma. Then how can you think that you require me to do any thing certain, when you leave me to do as I list? and seeing you stand at that stay, whether think you that I must first intend in mind what fraction I will make of it, before I can do it indeed? Sc. Else you should do ignorantly. Mai. Then I will limit myself (seeing you will not) to turn it into quarters. And therefore I multiply 7 by 4 (which is the denomination of quarters) and there amounteth 28 to be set for the Numeratour, and the 4 must be set for the Denominator, and the fraction will be thus, 28/●. Sch. Indeed I perceive this to be reasonable, for without much trial I understand that 28/4 of any thing doth make 7. And so then if I would turn 8 into fift parts, it will make 40/5 which is all one with 8. for eight crowns turned into fift parts (that is, into shillings) will make 40 shillings, that is 40/5 of a crown. Ma. you understand now these three kinds of Reduction, I will declare unto you the fourth kind, that is when Fractions be written in greater Terms than they need, how they may be brought to lesser Terms. Scholar To write any thing in greater Terms than needeth, seemeth to be a fault, The fourth kind of Reduction. and so this Rule seemeth to amend that fault. Na. It were a fault to do any thing without need which after must be redressed: but in this case it is not so: neither did I say absolutely (as you do) that it needeth not to express those Fractions in so great terms, but that the fractions do not need, I mean for their value to be understanded: but yet it may be needful for the ease of those works whereto they be applied, as for example: In the first kind of Reduction this was your own example: 3/16 and 4/●, which when you would reduce, you were fain to turn them first into one Denomination, & so appeared they thus, 18/96, and 6●/96, where the Fractions (for their own understanding) needed not to be turned out of smaller terms into greater, but yet the easiness of working needed it. Sch. Sir, I understand now, not only the difference of this need (for the Fractions might better be understanded as Fractions several, each in his value, Terms of Fractions. when they were in lesser terms, although they could not so well be reduced) but also I understand what you mean by greater Terms and lesser Terms, whereof before I was in doubt, for I see you call the Numeratour and denominatour, the Terms of the Fraction. Master. I am glad you understand it so well. Now then when you would value any fractions (because that may best be done when the Terms are smallest) you shall reduce them to the smallest that you can, which thing you may do thus: Reduction of Fractions into their smallest terms. Divide the greatest of any such two Terms by the lesser, and if any thing remain by that remainder, divide the last divisor: and if any thing remain now, by that divide the last divisor (which was before the remainder of the first Division) and so continue still, till nothing do remain in the Division: and then mark your last divisor, for it is the number that will easily reduce your Fraction, if you divide both the Numeratour and the Denominatour by the same number, and put for the Numeratour the quotient of his Division, and for the Denominatour also his quotient, that riseth by his Division. Scho. I take for example 18/96, and because 96 is the greatest number, I divide it by 18, and the quotient is 5, and there resteth 6: what shall I do with this quotient? Master. Nothing in this work, but now seeing there remaineth somewhat, by that remainder must you divide the last divisor. Sc. If I shall divide 18 (which was the last divisor by 6, that was the remayner, so is the quotient ●, and nothing resteth Ma. As for the quotient I omit him yet: but because there doth remain nothing therefore is 6 (which was your last divisor) that number by which you may reduce the fraction proponed, Sch. Then as you taught me. I must divide the Numerator 18 by 6 and the quotient is 3, which I must put for the numerator over a line, thus. and then by the said 6, must I divide also the denominator 96, and the quotient will be 16, which I must take for the Denominator, and so is the fraction 3/16. And so me thinketh this Rule doth prove the work of the first Reduction. Maist. That is true, if the first Reduction were made of Fractions in their least terms, and else not, without some help, as the second number in that place will declare. Sc. The second number was ●/6, which was turned into 64/56, by that rule. Now if I shall by this Rule reduce it again into the least terms, I must divide 96 by 64, and there resteth 32, by which 32 I divide 64, & there remaineth nothing, wherefore I must take that 32 for the divisor, to reduce the said Fraction. Then do I divide 64 by 32, and the Quotient is 2, which I set for my Numeratour. Again, I divide 96 by 32, and the Quotient will be 3, and so have I but ⅔ Ma. Muse not at the matter, for you have done well enough: but you think you have not the fraction that you looked for, that is 4/6, yet have you one equal to it, as by the parts of a shilling you may prove. Sc. Truth it is, for each of them will bring forth 8 pence, so that, 8/●●, and 4/6 and 2/7, be all three equal. And now I perceive that because 4/6 was not written in the least Terms that it might be, therefore this Reduction brought forth not it, but that other which is written in the least terms. Now understand I this rule well. But is there any other way to work this Reduction. Master. Yes, but first note this, another way for to work Reduction. that if you find no such divisor, to reduce the Fraction till you come to 1, because 1 doth make no division, therefore that Fraction is already in his least terms, as by 71/100 you may prove and so of 81/98 and many other like. But now for your better aid to find the due proportion in least terms, with more ease for a young learner, you shall mediate or take the half of the Numerator, and also the denominator as long as you may upon a line always parting them with a dash of your pen as you work, which may easily be done, if the numbers be even: as 2, 4, 6, 8, or 10: But if they be odd (though it be but in one of them) then must you abbreviate them by ●, by 5, by 7, etc. And because Examples do most instruct, I have in the Page following, set down the manner of 2 or 3, whose last number at the end of the line, showeth the least term or valuation of that Fraction. Abbreviated first by 5, then by 293. Sc. Sir I thank you much, this is very easy, and good for a young learner. Ma. So it is, but yet notwithstanding, if you can without that division by memory espy the greatest number that may divide exactly both Terms of your Fraction proponed, then need you not to use that division, as in this fraction 6●/96 I see that 12 is the greatest number that can divide them both: and therefore without any work, by memory only, I turn that into ⅝, but this ability in knowledge is gotten by exercise. Yet one other way of easy Reduction in this kind there is, when your Fraction hath any Cipers in the first places of both terms, then may you by casting away the ciphers, make a brief Reduction, as thus 300/400, here take away the Ciphers, and it will be ¾, which is the same in value with 300/400. Scholar. And so if I have 400/650 it will be 4/65. Ma. You are deceived, for you take away more ciphers from the Numerator, ☜ than you do take from the denominator, which you may not do. Sc. I confess my fault, which came of too much haste, I was more gladder of the rule than wise in using it: but now I understand it I trust. Ma. Then may I go in hand with the fift or last kind of Reduction●, The fift kind of Reduction which teacheth how to turn any fraction proponed into any other denomination that you list: or into any parts of common coins, weights, or measures, or such like. For declaration whereof, first you shall mark whether your fraction be a simple fraction, either else a fraction of sundry parts, I mean of more terms than 2. And if your fraction be a fraction of Fractions, or otherways compound, you must reduce it to one simple fraction. And then mark well the denomination of that other fraction, into which you would turn this, for by that denominatour you must multiply the Numeratour of your first fraction, and the total Producte thereof shall you divide by the denominator of your first fraction, and that quotient shall be the numeratour to the denominatour proponed: as for example I have this fraction ⅕, which I would turn into tenth parts, therefore I multiply this 10 by 3, that is the Numeratour of my fraction, and there riseth 30, which I divide by 5, and the Quotient is 6, which must be the Numerator to 10, and so ⅗ will be 6/10. Sc. This is easy enough to do. Ma. Then shall you see an other example of the same fraction that is not so easy: as if I would turn ⅗ into viii. parts, prove you that work. Sc. I must multiply 8 by 3, and there amounteth 24, which I divide by 5, and the quotient is 4, then is the new fraction 4/8. Ma. And see you nothing doubtful in this work? Sc. I see that when 24 was divided by 5, there remained 4, which I did not pass of, because ye speak nothing of any remainer, but only of the quotient. Ma. By likelihood you remember what I said to you in division of whole numbers, that you should not pass of the remainer there, but only note it as a sum that could not be divided without knowledge of Fractions. Wherefore now mark this, that in all Divisions of whole numbers, when there is any remainder, you shall set it over a line as a Numeratour, and set the divisor for the Denominatour, and that Fraction doth make the Division complete, and is part of the quotient: as if I would divide 48 by 5, the quotient will be 9 ⅗: so in your former work when 24 was divided by 5, the quotient should be 4 ⅘, and so the new Fraction should be thus: that is 4/8 of the entire number, and ⅘ of 4/8: which you may prove be example of some coin. Sc. Then I take a Crown, whose ⅗ is 3 s. Now if I would prove whether that 3 s be 4 and ⅘ of 4/8 I shall have a cumbrous work to do. Master. In deed for whole pennies your example is troublesome: yet turning the Crown into half pennies, it is easy enough. Sc. Now will I do it. Ma. First let me tell you an easy way how to find any number that will easily be divided into such parts as you desire, which way is this. Set down the parts that you desire, and then by one of them multiply all the other, the total whereof shall contain all the parts proponed, as if I would have a number that may be divided into 4, 5, 6, and 7 parts, by 4 multiply 5, and there riseth 20: then multiply 20 by 6, & it will make 120: which multiplied by 7, will yield 840: and so of any other numbers. Sc. Then in our former example where is mention but of 5 parts and 8 parts, I shall only multiply 5 by 8, which maketh 40, and that number will serve. Ma. So will it. Sc. Then what is ⅕ of 40? Mai. Prove by the same rule which you confess easy enough: 3 times 40, is 120, which being divided by 5, maketh 24 and that is just. Now to know whether it be equal to 24, first I see by the same rule, that 4/8 is 20, and 2/8 is 5, of which 5, I must take ⅘: and that by the same Rule is 4. So that I see now, that is equal to ⅗. Ma. And by the way note this form of fraction how it is written, that is to say, both the Numerator and his Fraction above the line, although I know it may be written otherwise, as thus: 4/8 and ⅘ of 4/8, but I account the other way more apt a great deal. * Scho. Now I pray you sir let me prove how this 4/8 and ⅘ of 4/8 of a Crown be equal to ⅘ of a Crown our first proponed fraction, that hath brought forth these hard fractions to my thinking yet. First I see that ⅗ of a crown is ● s: which is 36 d, or 72 half pence. Now if I can find that this fraction 4/8 and ⅘ of 4/8 be equal unto 3 s: then am I fully answered. Because I cannot take 4/8 of a Crown, I turn the crown into half pence, as you willed me, which makes 120, which I divide by 8, my quotient is 15, which taken 4 times makes 60: now resteth me to have ●/5; of the ⅘, whereof ⅛ is 15, that 15 being parted into 5 parts, the quotient is 3, which taken 4 times makes 12, which with my 60 before, amounteth to 72: which are then equal to ⅕ my desire. Ma. And so may you express by an other way than is before mentioned, all Fractions of Fractions, as thus: That is ¾ of ⅝, and so of other, but I remit these forms to the arbitrement of every wise art's man, to use as he thinketh most apt and ready. But now one example more for this rule, and then shall we end it. If I have ●/15 of a Sovereign (accounting the Sovereign 20 shillings) how many shillings is that 7/15? Sc. I must multiply 7 by 20, and that maketh 140, which I shall divide by 15, and the Quotient will be 9 5/15: or else in lesser terms, ½. Ma. That is 9 s, and one third part of a shilling, that is 4 d, as by this same rule you may prove. And this for this time shall suffice for Reduction, save that I must now repeat a little touching Multiplication and Division, and so go forward. MULTIPLICATION. IN Multiplication it happeneth sometime that there be whole numbers to be multiplied with Fractions: And may be in two sorts, for either the whole number is several from the Fraction, and is the Multiplyer, or else, the whole number is joined with one, or both of the Fractions, Reduction of whole numbers into Fractions. and so maketh a mixed number thereof. If it be in the first sort, then needeth there no Reduction, but only multiply the Numerator of the Fraction by that whole number, and the total thereof set for the new Numerator. Scho. I understand you thus. If I have 6/2●, to be multiplied by 16, then must I multiply that 16 with 6, which is the Numerator, whereof cometh 96, and that must I set for the new Numerator, keeping still 23 for the Denominator, and so the Fraction will be 96/2● that is 4 4/23. Ma. And in this sort of work you may abridge the labour, thus. If it happen the denominator to be such a number, as may evenly be divided by the said whole number proposed, then divide it thereby, and set the quotient of that Division for the former denominator: but reserve still the Numerator, and so is the Multiplication ended. Scho. Then I fain this example, 7/20 to be multiplied by 5. And because 5 will justly divide 20, therefore I take the quotient of that division which is 4, and set in stead of 20, and so the Fraction will be 7/4, that is 1 ¼. Master. Which is all one with 35/●0, that would have followed of the other sort of work. Sch. I perceive it very well. Master. ●w to mul●ie mixed ●he●s. Now then for the other sort where the number is mixed, take this way: first to reduce the said whole number, and fraction into one fraction Improper (as I showed you in Reduction) and then multiply them together, as if they were proper fractions. Sc. 13 ●/5; being set to be multiplied by ●/9, first I must reduce the mixed number, as appeareth in the margin, by multiplying 13 by 5, and that maketh 65, whereto I must add the Numerator 3, and so the fraction will be 68/5, which now I shall multiply after the accustomed form, and it will be 340/40. Ma You have done well: and so may you see, that although most part of the forms of Multiplication may be wrought without Reduction, yet some can not, as namely mixed numbers. And yet one note more will I tell you of Multiplication, before we leave it: That is▪ when so ever you would multiply any Fraction by 2, which commonly is called Duplation, you may do it not only by doubling the Numeratour, but also by parting the Denominator into half, if he be even. Scholar. Then if I would double 5/●●, I may choose whether I will make it, 1●/1●, or else ⅚. And in deed I see that all is one, but that the dividing of the Denominatour seemeth the better way to make smaller terms of the Fraction, and so they shall need the less Reduction. M. It is so: and now I shall not need to tell you that Multiplication is proved by Division, and Division likewise by multiplication, but the like works that I showed you in Multiplication, will I show you in Division also. DIVISION. ●ision to ●de a ●ole nū● by a ●ion. WHen any whole number shall be divided by a Fraction, you must multiply the said whole number with the Denominator of the fraction, and set the total thereof for the new Numerator, and for the Denominator, let the Numerator of the Fraction. Scholar. Then 20 divided by will make 60/3, as here appeareth: Master. divide action whole ●er. Even so. But if you would divide the fraction by the whole number, them multiply the Denominatour by the same whole number, and set the total for the Denominatour, without changing the Numerator. Scholar. Then to divide 20/23 by 4, it will by 20/92. As here appeareth in this example. M. You say well. And by the same example you give me occasion to remember an other brief way to do the same: Another brief way for if you had divided the said Numerator by 4, and set the quotient for the Numerator, keeping still the old Denominator, it would have been not only as well done, but also in a fraction of lesser terms. Scholar. I guess it to be even so, by a like work that you taught me in Multiplication. And for proof thereof 20/23 being the dividend, and 4 the devisor, I divide the Numeratour 20 by 4, and the quotient is 5, which I set for 20 over 23, thus 5/23. And I see that it is all one with 20/62, as by dividing or abbreving both these terms by 4, and so reducing them to their least Denomination, I may easily prove: as appeareth in this example: M. You conceive it well. And if there be mixed numbers (either one or both) you must first reduce that mixed number into an improper Fraction. And then work as you have learned. S. That was sufficiently taught in Multiplication. Therefore I pray you go forward to some other thing. Ma. Then take this note yet for division. If the Denominatours be like, then divide the Numeratours as if it were in whole numbers, and the quotient whether it be Fraction, whole number, or mixed, is a good quotient for that Division. And generally if one of the Numerators may justly divide the other, by that quotient multiply the Denominator of the lesser Numerator, and set it that doth amount in the room of the same denominator, and then for a Numerator to it, set the denominator of the other fraction. Scho. Then if I would divide ●/4 by 12/1● I see that 3 will divide 12, and the quotient will be 4, by which I must multiply the other 4 that is the Denominator under 3, and then it is 16, which I set for the Denominator 4, and over it in stead of the ●, I must set 17, the other Denominator, and so is it thus, 17/16. Master. And so is 17/16 in stead of 11/48, which would have risen by the common work: as here appeareth: And now for Mediation (which is to divide by 2) mark this: If the Numeratour be even, set the half of it in his place without the divisor, and so have you done: and if the Numeratour be not even, then double the denominator. Sch. That is if I would mediate 6/11 I may make the quotient ●/11. And if I would mediate ●/11 I must make it 7/22. Master. Now trust I that you have sufficient knowledge in Reduction, Multiplication, and Division: and therefore will I go in hand with Addition and Subtraction, which now will appear easy enough. ADDITION. 〈◊〉 add ●●ctions of ●e denomination. Whensoever you have any fractions to be added, you must consider whether they be of one Denomination or not▪ And if they be of one denomination, then add the Numerators together, and set the that amounteth, for the Numerator over the common Denominator, and so you have done. The reason is, because that such differ little in Addition or Subtraction from the work of vulgar denominations, where the Denominators be no numbers: as 3 pence and 5 pence, make 8 pence, ●o add fra●●●ons of d●rse deno●●nations. where the denomination is not altered. But & if the fractions be not of one denomination, or any of them be mixed of whole numbers and fractions, then must you first reduce them to one denomination, and after add them. And if they be many, then add first two of them, and to the sum that doth amount of the Addition, add the third, and then the fourth, & so forth, if you have so many. Sc. This seemeth easy enough, now that I have already learned to multiply and to reduce, without which two, I could never have wrought this. And therefore now I see good reason, why you did place Multiplication and Reduction before Addition. Master. It is well considered, but yet refuse not to express your understanding of it, by an example. Sch. Then would I add first 7/18 with 5/18, and because the Denominators are like (and so needeth no Reduction) I add 7 to 5, which maketh 12, and then is my sum 12/18, that is in smaller numbers ⅔. And if I have many numbers to be added, as here 34/85 9/10, first I must reduce them (because they have diverse denominatours) into one denomination, and then will they be thus. or in less terms 15/4● 32/40 26/40, which by Addition do make 83/●● that is 2 3/●●. M Now may we go to subtraction. SUBTRACTION. SVbtraction hath the same precepts that addition had, for if the Denominatours be like, then must you subtract the one numeratour from the other, and the rest is to be set over the common denominator, and so your Subtraction is ended: but and if you have many Fractions to be subtracted out of many, then must you reduce them to one denomination, and into two several Fractions, that is, all that must be subtracted into one Fraction, and the residue into an other Fraction, and then work as I said before. Scholar For the first example I take 15/12 to be subtracted out of 17/12, and the rest will be ●2/●● or ⅙. For an other example I take 2/4 to be subtracted out of 7/3 which I must reduce, and it will be thus, 24/●2 and ●8/●2. Then do I subtract 24 out of 28, and there resteth 4, which I set over the common Denominator for a Remayner, thus, 4/32, that is ⅛. Now for the third example, I take ¾ and ⅚ to be subtracted from ⅞ and 9/10. And because their denominators be divers, I do reduce them thus, 1●●●/1920▪ 16●●/1720, ●●80/1920 17●8/1920. Then do I add the two first, & they make 3●●●/19●●. Also I add the two last, and they yield ●●08/162● Then do I subtract 3040 out of 3408, and there resteth 368, so is the remainder ●68/1920 that is in smaller Terms, 23/120. And thus have I done with Subtraction, except you have any more to teach me. Mai. Prove one example more of two Fractions of divers denominations. Scho I take these two Fractions, ⅞ and 7/24, which being reduced, will stand thus, 168/192. & 72/191 Now would I subtract 168 out of 72, but I can not. Ma. Then may you perceive that you mistook the Fractions: for you can never subtract the greater out of the lesser, although you may add, multiply or divide the greater with the lesser. And albeit that ⅞ hath both his terms lesser than 9/24, yet is 9/2● the lesser Fraction: for generally if you multiply the Numerators and denominators of two fractions cross ways, beatest ● fra● that Fraction is the greatest, of whose numerator cometh the greatest sum, as in this example: 7 multiplied by 24, maketh 168: and 9 being multiplied by 8, yieldeth but 72 therefore is the first Fraction ⅞ the greatest of these two, so can you not subtract it out of a lesser Fraction. But and you should subtract a fraction out of a whole number what would you do? Sc Marry I would reduce the whole number into a Fraction of the same Denomination that my Fraction is, and then work by Subtraction. M. So may you do, but it is easier much, if your fraction be a proper fraction, that is to say, less than an Unite, to take an Unite from the whole number, and then turn it into an Improper Fraction▪ and so work your subtraction. As if I would subtract ⅗ from 4, I may take one from 4, and turn it into 5/5, from which if I bate 7/6, there will remain 3 ⅖: And if the first fraction be an Improper Fraction, then may I take so many Unites from the whole number, that they may make an improper fraction greater than that first, and then work by Subtraction: As if there be proponed 10/3 to be subtracted from 6, because ●0/3 is more than 3, and not so much as 4. I must take 4 from 6, and turn them into thirds thus. 12/3 then abate 10/3, and there resteth ⅔ so the whole remainder is 2 ⅔. Or, else you may at your pleasure take 3 ⅓ which is 10/3 from 6 whole: Then set one under 6. as thus 6/1: And then to reduce those 2 fractions into one Denomination as here appeareth: Then 10/3 from 18/3 resteth 8/3: which maketh 2 ⅔ your desire: And thus will I make an end of the works of common Fractions for this time, not doubting, but you can apply them both unto the rules of Progression, and also unto the Golden rule, without any other teaching than you have learned before, which might seem tedious to repeat, save that in some special diversities, which be peculiar to Fractions, I can not overpass, but instruct you somewhat by the way. THE GOLDEN Rule. THerefore as touching the Golden rule for the placing of the 3 numbers proponed in the question, whereby to find the third, and for the form of their work, with other like notes, I refer you to that which you have already learned. But this easy form of working by fractions shall you note, that if your three numbers be fractions, for an apt work and certain, multiply the Numerator of the first number in the question, by the Denominator of the second: And all that again multiply by the denominator of the third number, and the total thereof shall you keep for to be the Divisor. Then multiply the Denominator of the first number by the Numeratour of the second, and the whole thereof by the Numerator of the third, and the total thereof shall be your dividend. Now divide this dividend by the divisor which you found out before, and that number shall be the fourth number of the question which you seek for: As in this example. If ¾ of a yard of Velvet cost ⅔ of a Sovereign (esteemed at 20 shillings) what shall ⅚ cost? Scholar. If it please you to let me make the answer, A question of Yeluet. I would first place these three numbers, as I learned in whole numbers thus. And then according to your new rule, I must multiply 3 being Numeratour in the first number, by three the Denominatour of the second, and thereof cometh 9, which I multiply again by 6, the Denominatour of the third number, and so have I 54. which I keep for the Divisor, then multiply I 4, the Denominator of the first, by 2, the Numeratour of the second, and there riseth 8, which again I multiply by 5, the Numeratour of the third, and it maketh 40: then must I divide 40, by 54, and it will be 40/●4 that is 20/27, in lesser terms, and then the figure will stand thus. But what that is in money, I can not tell except I shall work it by Reduction, as you taught me. Master. It forceth not now, you may reduce it when you list, but it were disorderly done here to mingle diverse works together, where we do not seek the value of the thing in common money, but in an apt number, which you have well done. And therefore will I yet show you an other like way of easiness in work, how you may change your 3 Fractions into 3 whole numbers, by which you shall work as if the question were proponed in whole numbers. The first number you shall find as I taught you: now to find the divisor of the second number, take the Numerator for the second fraction: and for the third number take that, that riseth of Multiplication of the Denominatour of the first, by the Numerator of the third, and then work your question. Scho. A question of silver. For example hereof, I put this question, If 11/12 of 1 lb weight of silver, be worth 11/4● of a Sovereign what is ½ of 1 lb. weight worth? For the answer, first I place the Fractions in order thus. Then to turn these fractions into whole numbers, I multiply 11 which is the Numerator of the first, by 4 (that denominator of the second) and there cometh 44, which I multiply by 2 the Denominator of the third, and so amounteth 88, which I set for the Divisor in the first place. Then in the second place I set 12, which is Numerator in the second Fraction, and in the third place I set the sum that amounteth of 12, being the Denominator in the first number, multiplied by 1, being numeratour in the third number, & so the figure will stand as here you see. Then to work it forth, I multiply 12 by 12, and there amounteth 144, which I divide by 88, and the quotient will be 1 ●6/88, or in lesser terms, 1 7/11 and then the figure will stand thus. Ma. These ij. forms now you understand well enough And as for any other, at this time I will not repeat, only this shall you mark for the proof of this rule, whether your work be well wrought or no. Multiply the first number by the fourth, and note what amounteth: The proof of the Golden rule. then multiply the second by the third, and mark what amounteth also Now if those two numbers so amounting be equal, then is your work well done, else you have erred. And this shall suffice for the former rule, The Backer rule. but in the barker rule, this shall you note for ease of work, that you multiply the Numerator of the first by the Numerator of the second, and the whole thereof by the Denominator of the third, and that amounteth thereof, shall be the Dividend. Then multiply the Denominator of the first by the Denominator of the second, and that whole by the Numerator of the third, & that riseth thereof shall be the divisor. Example of this: I did lend my friend ¼ of a Portugese seven. months, upon promise that he should do as much for me again: A question of lone. and when I should borrow of him he could lend me but 5/12 of a Portugese, now I demand how long time must I keep his money in just recompense of my loan, accounting 13 months in the year? Scholar The first number must be the first money borrowed, that is ●/4 of the Portugese: the second number the 7 months, that is 7/●3 of a year: and the third number the money that was lent in recompense, that is 5/12 of a Portugese: then I set the numbers thus. Then (as you taught me) I multiply three (being Numeratour in the firsts number) by 7 the Numerator of the second number, and it maketh 21, which I multiply by 12, the denominator of the 3 & so have I 252 for the divident: them I multiply 4 the denominator of the first, by 13 the denominator of the second, and it yieldeth 52. which I multiply again by 5, the Numeratour of the third, and it will make 260, that is the Divisor. Then must I divide 252, by 260, so it will be in the smallest fraction, 63/65 of a year. Ma. And this do you see some ease in working, better than to multiply and divide tediously so many Fractions. another question yet will I propose, to the intent you may see thereby the reason of the statute of assize of bread and ale, which in all Statute books in French, Latin and English, Statute of Assize of bread and ale. is much corrupted for want of knowledge in this art: for the right understanding whereof I propone this question. When the price of a quarter of Wheat is 2 s the farthing white loaf shall weigh 68 s. Question. than I demand, what shall such a Loaf weigh, when a quarter of Wheat is sold for 3 s? Scho. This Question must be wrought as it is proponed in whole numbers and not in Fractions. Mai. You seem to say reasonably, how be it, in that Statute of Assize, the rate is made by the proportion of parts in a pound weight Troy, else could it not be a Statute of any long continuance, seeing the shillings do change often, as all other moneys do: but this Statute being well understanded, is a continual rule for ever, as I will anon declare by a new table of Assize, converting the shillings into ounces and parts of ounces. Therefore here by a shilling you must understand 1/20 of a pound weight, and so by pence 1/2● of an ounce, wherefore although ye might work this question proponed by whole number well enough, for the time when the statute was made, yet to apply it to our time, and to make it to serve for all times generally it is best to work it by fractions, setting for 2 shilling 2/20: and for 68 shillings, 68/20: and so for three shillings 3/20, and then will the figure of the question stand thus. In which question because all the denominators be like, you shall work only with the numerators Sc. Then I shall multiply 68 by 2, whereof cometh 136, which if I divide by 3, the quotient will be 45 ⅓: but how shall I make a Fraction of that to stand with the other? Ma. Have you so soon forgotten what was taught you so lately? This is his form. Sc. I remember it now and then it signifieth 45 twenty parts, and the third deal of one twenty part. Ma. So is it, and that maketh in shillings, 45 s 4 d: whereby you may note one great error in the Statute books, which have constantly 48 s in that Assize. And by this rule, if you examine the Statute you shall find many sums false, wherefore for the true understanding of that statute and such like as I have made mention of it, and somewhat recognized it, so do I wish that all gentlemen and other students of the laws, would not neglect this art of Arithmetic as unnéedefull to their studies. Wherefore to encourage them thereto, and to gratify both them and all other in general, I will exhibit a Table of that part of the statute in two columns, and in a third column I will add the correction of those errors which have crept into it. Here followeth the Table. The price of a quarter of wheat. The weight of a farthing white loaf by the statute books. The correction by just Assize. s. d. li. s. d. li s. d. 1 0 6 16 0 6 16 1 6 4 10 8 4 10 8 2 0 3 8 0 3 8 0 2 6 2 14 4 1/● 2 14 4 ⅘ 3 0 2 8 0 2 5 4 3 6 2 2 0 1 18 10 2/● 4 0 1 16 0 1 14 0 4 6 1 10 0 1 0 2 2/● 5 0 1 8 2 ½ 1 7 2 2/● 5 6 1 4 8 ¼ 1 4 8 ●/●● 6 0 1 2 8 1 2 8 6 6 0 19 11 1 0 11 1/13 7 0 0 19 1 0 29 5 1/● 7 6 0 18 1 ½ 0 18 1 1/● 8 0 0 17 0 0 17 0 8 6 0 16 0 0 16 0 9 0 0 15 0 ¼ 0 15 1 ⅓ 9 6 0 14 4 ¼ 0 14 3 15/19 10 0 0 13 7 ½ 0 13 7 1/● 10 6 0 12 11 ¼ 0 12 11 3/7 21 0 0 12 4 ¼ 0 12 4 4/●● 11 6 0 11 10 0 11 9 2●/2● 12 0 0 11 4 0 11 4 In the common books there is no farther rate of assize made, than unto 1 2 s the quarter of wheat: but in an ancient copy of 200 years old (which I have) there is added the rate of assize unto 20 s the quarter, but yet was that assize also either wrong cast at the first penning, or else corrupt sith that time, for lack of just knowledge in the rule of proportion, which I will add here also, to gratify such as be desirous to understand truth exactly. The price of a quarter of wheat. The weight of the farthing white loaf by the statute books. The corection of the errors. s. d. s. d. s. d. 12 6 11 0 10 10 14/25 13 0 15 0 ½ 10 5 7/1● 13 6 10 1 ½ 10 0 8/● 14 0 9 7 9 8 4/● 14 6 9 2 ½ 9 4 16/20 15 0 9 1 ½ 0 0 ⅘ 15 6 9 1 1/● 8 9 9/3● 16 0 9 0 8 6 0 16 6 8 6 8 2 10/11 17 0 8 3 8 0 0 17 6 7 10 7 9 9/●5 18 0 7 6 7 6 ⅔ 18 6 7 3 7 4 8/●7 19 0 7 2 7 1 17/19 19 6 5 10 6 11 9/1● 20 0 5 6 6 9 ⅗ These 2 tables I have set several, because no man should think that I would either add or take away from any law those parts which might of right seem either superfluous either diminute, but yet I may not be so curious as to neglect manifest errors, which is not only my part, but every good subjects duty with sobriety to correct. And for avoiding of offence I have rather done it in this private book rather than in any book of the statutes self, trusting that all men will take it in good part. Sc. I would wish so, but I dare not hope so sith never good man that would reform error, could escape the venomous tongues of envious detractors, which because they either cannot or list not to do any good themselves, do delight to bark at that doings of other, but I beseek you to stay nothing for their perverse behaviour. Ma. I consider many things that some may object, whereunto I am not unprovided of just answers, but I will not seem so hasty to make the answers before I hear their objections, but as I trust that men are of a better nature, & more grateful now than some hath been in time passed, as I have done in the statute of Assize for bread in rate of s, so will I set forth the like table in pounds & ounces, & the parts thereof, that it may be easily applied to all times: but I mean not by this to alter any word of the statute (being so good an ordinance, & of so great continuance) but only to make it as a kind of exposition & declaration of the said statut, trusting that thereby the statute may be better understand, & consequently better put in execution. And here you shall note, A pound weight. that I have accounted the shillings after the rate of lx s, to the pound weight, because I esteem it the most apt rate for our time. Wherefore if in the first column you find the price of wheat, directly against it in the second column, you may find the weight of the farthing white loaf, in this our time: & if you double that number (as I have done in the third column) then have you the weight of the half penny white loaf & so in the fourth column is set the weight of the penny white loaf. It needeth not to tell you that, that the sight doth testify, how the every colunn is parted into 3 smaller pillars, whereof the first column hath these 3 titles, pounds, shillings, & pennies: the other 3 columns have each of them these 3 titles, pounds, ounces, & penny weights. And as in the first column xii d make a s, & 20 s make a pound, so in the other iii columns xx pence weight maketh an ounce. and xii ounces do make a pound. The price of a quarter of wheat. lb s d. 0 3 0 0 4 6 0 6 0 0 7 6 0 9 0 0 10 6 0 12 0 0 13 6 0 15 0 0 16 6 0 18 0 0 19 6 1 1 0 1 2 6 1 4 0 1 5 6 1 7 0 1 8 6 1 10 0 1 11 6 1 13 0 1 14 6 1 16 0 1 17 6 1 1● 0 The weight of the farthing white loaf. lb. unc. d w. 6 9 ½ 2 4 6 ¼ 3 3 4 ¾ 1 2 8 ½ 2 ⅘ 2 3 4 1 11 ¼ 1 2/7 1 8 ¼ 3 1 6 2 ⅔ 1 4 ¼ 1 ⅖ 1 2 ¼ 1 8/11 1 1 ½ 2 1 0 ½ 1 1/11 0 11 ½ 3 1/7 0 10 ¾ 2 ⅕ 0 10 4 0 9 ½ 2 0 9 1 ⅓ 0 8 ½ 1 15/1● 0 8 3 ⅕ 0 7 ¼ 0 3 0 7 ¼ 0 7 1 21/21 0 6 2/4 1 0 6 ½ 0 14/15 0 6 ½ 0 ●/● The weight of the half penny white loaf. ll. unc. d. w. 1 3 7 4 9 0 ¾ 1 6 9 ½ 2 5 5 1/● 0 ⅗ 4 6 ¼ 3 3 10 ½ 2 4/7 3 4 ¾ 1 3 0 1/● 0 ⅓ 2 8 ½ 2 ⅘ 2 5 ½ 3 5/11 2 3 4 2 1 2 2/13; 1 11 ¼ 1 2/7 1 9 ¾ 0 ⅕ 1 8 ¼ 3 1 7 4 1 6 2 ⅔ 1 5 3 11/19 1 4 ¼ 1 ⅖ 1 3 ½ 0 6/7 1 2 ¾ 1 ●/11 1 2 3 19/29 1 1 ½ 2 1 1 1 1/25 1 0 1 1/12 1 0 1 The weight of the penny white loaf. li. unc. d. w. 27 ¼ 3 18 1 ½ 2 13 7 4 10 10 ½ 1 ⅕ 9 0 ¼ 1 7 9 ¼ 0 1/7 6 9 ½ 2 6 0 ½ 0 ⅔ 5 5 1/● 0 3/5 4 11 ¼ 1 10/●1 4 6 ¼ 3 4 2 4 4/13 3 10 ½ 2 4/7 3 7 ½ 0 ⅗ 3 4 ●/4 1 3 2 ¼ 3 3 0 ¼ 0 1/3 2 10 ¼ 2 ●/19 2 8 ½ 2 ⅘ 2 7 1 ●/7 2 5 ½ 3 5/11 2 4 ¼ 2 15/23 2 3 4 2 2 2 6/25 2 1 2 2/13 2 ● ● Sch. Sir, I do thank you most heartily for this, not only in mine own name and in the name of all Students, but also in the name of the whole Commons, to whom the restitution of this Assize (I trust) shall bring restitution of the weight in bread, which long time hath been abused. And if you know any like things more, wherein you would vouchsafe to declare the errors and set forth the truth, you cannot but obtain great thanks of all good hearted men that love the common wealth. Mai. I have sundry things to declare but I have reserved them for a private book by itself, yet notwithstanding because the statute of the rate of measuring of ground is so common that it toucheth all men, and yet no more common than needful, but so much corrupt, that it is to far out of all good rate, not only in the English books of Statutes commonly printed, but also in the Latin books, and in the French also, for I have read of each sort, and conferred them diligently, I will give you a Table for the restitution of those errors, as may suffice for this present time. And first will I propose one question to you touching the use of that Statute, whereby you may perceive the order how to examine the whole Statute, A question of measure o● ground. and every parcel thereof, and the question is this. When the Acre of ground doth contain four perches in breadth, then must it contain 40 perches in length: then do I demand of you, how much shall the length of an Acre be, when there is in the breadth of it 13 perches? but before you shall answer to this question, I will declare unto you an other Statute, which is the ground of the former Statute. And that Statute is this. It is ordained that 3 Barley Corns, dry and round, shall make up the measure of an inch: 12 inches shall make a foot, A statute measures. and 3 foot shall make a yard (the common English books have an elne) five yards and a half shall make a perch, and 40 perches in length, and 4 in breadth, shall make an Acre. This is that statute: whereby you may perceive that the intent of the Statute is, An Acre. that one Acre should contain 160 square Perches. Now let me hear you answer to the question. Scholar. As I perceive by the words of that Statute, a perch to be 1/160 of an Acre, so could I make those numbers all in Frations, and so work the question: but seeing I may do it also in whole numbers I take that form for the most easy, therefore thus I set the question in form. Then do I multiply 40 by 4, and it maketh 160, which I divide by 13, and the quotient is 12 4/1●. Ma. Now turn that 4/1● into the common parts of a perch, as they be named in the former Statute: how be it, it shall be best to take one of the least parts in Denomination for avoiding of much labour, as feet, whereof the perch containeth 16 4/●. Sc. Then to turn 4/13; into feet, I Multiply 16 ½ by 4, and it maketh 66, which I must divide by 13, and the Quotiente is 5 1/13. Mayst. So I find that if the acre hold in breadth xiii perches it shall contain in length 12 perches, 5 foot, and 1/1● of a foot, which is not fully an inch, for the inch is 1/12 of a foot. Note this srour. But here all the Statute Books in Latin and English (that I have seen) do note it to be 13 perches, 5 foot and 1 inch: which maketh above 13 perches to many in the acre, so that I would have thought the error to have crept into the printed books by the great negligence that Printers in our time do use, save that in written Copies of great antiquity, I do find the same. Yet have I one french Copy, which hath 12 perches ●/●, and one foot, and that misseth very little of the truth. Sc. Then I see it is true that I have often heard say, that the truest copies of the statutes be the French copies. Ma. That is often true, but not generally, as I have by conference tried diversly: but in this statute the french book is most corrupt in all other places lightly. But now to perform my promise I will set forth the Table for measuring of an Acre of ground only by such parts as the Statute doth mention, because at this time I do of purpose writ it for the better understanding of the statute, and hereafter with other things I intend to set forth this same more at large. In this Table following, I have not done as in the other statute before compared by restitution with the faults crept into the Statute, but only have written that true measure, which the equity of the Statute doth pretends. For it were to vile to judge of so noble Princes and worthy Councillors, as have authorized & set forth this statute, that they would make one acre in any form greater than an other, but every one to be just and equal with each other, which is the ground also of my work, and hereby may all men perceive how needful Arithmetic is unto the Students of the law. But now I think best to make an end of these matters for this present time, sith the table hath in it none obscurity, that I should need to declare. The breadth The length of the acre. perdie. perdie. feet. 10 16 0 11 14 9 12 13 5 ½ 13 12 5 1/13 14 11 7 1/14 15 10 11 16 10 0 17 9 6 27/34 18 8 14 ⅔ 19 8 16 18/19 20 8 0 21 7 10 3/14 22 7 4 ½ 23 6 19 18/23 24 6 11 25 6 6 5/● 26 6 2 7/13 27 5 15 5/18 28 5 11 11/14 29 5 8 31/58 30 5 5 ½ 31 5 2 41/62 32 5 0 33 4 14 34 4 11 11/17 35 4 9 36 4 7 ⅓ 37 4 5 13/37 38 4 3 9/19 39 4 9/13 40 4 0 41 3 1 63/82 42 3 13 6/42 43 3 11 77/86 44 3 10 ½ 45 3 9 ●/● Sc. In deed Sir, I understand the Table (as I think) by those other which you set forth before. For in the first Column is set the perches of the breadth of any Acre, and then in the 2 columns following appeareth how many perches and how many foot the same Acre must have for his length Ma. You take it well: how be it to speak exactly of breadth & length, the first column doth sometime betoken the breadth: & sometime the length, ☞ for properly the longest side of any square doth limit his length, and the shorter side doth betoken the breadth, yet it is no great abuse in such tables, where a man cannot well change the title, to let the name remain, although the proportions of the numbers do change: for still by the first column, is expressed the measure of the one side, and by the two other pillars in one Column, is set forth the measure of the other side. And this shall be sufficient now for the use of the golden Rule. Now somewhat will I touch certain other rules, which for their several names may seem diverse rules & distinct from this, but in deed they are but branches of it: yet because they have not only several workings in appearance, but also pleasant in use, I will give you a taste of each of them. As for the rule of Fellowship, both single and double, with time and without time, I shall need to say little more than I have already said in teaching the works of whole numbers, yet an example or two will we have to refresh the remembrance of the same, and to declare certain proper uses and applications of it, as this for one. Four men get a booty or prize in time of war, the prize is in value of money 8190 lb, A question of unequal society. and because the men be not of like degree, therefore their shares may not be equal, but the chiefest person will have of the booty the third part, and the tenth part over: the second will have a quarter and the tenth part over: the third will have the sixth part: and so there is left for the fourth man a very small portion but such is his lot, (whether he be pleased or wroth) he must be content with one xx. part of the pray. Now I demand of you, what shall every man have to his share? Sch. You must be feign to answer to your own question, else is it not like to be answered at this time. Master. The form to understand the solution of this question, and all such like, is this: ☞ Reduce all the Denominators into one number by Multiplication, except that any of them be parts of some other of them, for all such parts you may overpass, and take for them all those numbers, whose parts they be: as in this example the shares be these ⅓ 1/10 ¼ 1/12 ⅙ 1/20 if I multiply all the Denominatours together, beginning with 3, and so go on unto 20, it will make 144000: but considering that 3 is a part of 6, I shall ommitte that 3, and likewise 10, which is a part of 20, I may overpass also, and then is there but 3 denominatours to multiply, that is 4, 6, and 20, which make 480, which sum I take for my work, because all the Denominatours will be found in it. Then I take such parts of it as the question importeth, that is for the first man, ⅛ 1/10, the ⅓ is 160 the 1/10 is 48: which I put in one sum for the first man's share, and it maketh 208. Then for the second man's share, I take ¼, which is 120, and 1/10. which is 48, and that maketh in the whole 168. Now for the third man which must have ⅙ I take 80 And for the fourth man there remaineth but 24, which is 1/20 of the whole sum: so that if the whole prey had been but 480 lb. than were the question answered: but because the sum was of greater value, by this means now shall I know the partition of it. I must set my numbers by the order of the Golden rule, putting in the first place the number that I found by multiplying the Denominatours, and in the second place the sum of the booty. The reason of this rule. And look what proportion is between the first number and that second, the same proportion shall be between the parts of that first number and the parts of the second, comparing each to his like. Therefore I must put in the third place, one of the parts or shares, and then work by the former rule of proportion or Golden rule. And because I have 4 several parts of the first number, by which I would find out 4 like parts of the second number, therefore must I make 4 several figures. Scholar. Now I trust I can answer to your question, as by your favour I will prove. And to try it, I set the 4 figures thus, marked with A, B, C, D, to show their order. And then in each of them I multiply the second number by the third, and divide their total by the first, and so amounteth the fourth sum which I seek for, for if I do multiply 8190 by 208, it maketh 1703520, which being divided by 480, maketh in the quotient 3549 for the first man's portion: And so working with the other three figures, I find for the second man 2866 ½, and for the third man 1365: and then for the fourth man 409 ½. And so is every man's share set forth in the figure here annexed. And thus I think I have done well. Ma. If you misdoubt your working and list to prove it, add all the shares together: and if they make the total, them seemeth it well done. Sc. I may set them thus: The proof by Addition and then by Addition the just sum doth amount, that is 8190, and therefore (as you say) it seemeth to be well wrought. But I beseech you, is there any doubt in this trial, that you use that word, The just proof. Seemeth? Master. You may easily conjecture, that if you did assign the first man's share to the last, and so change all the rest, that one had an others share, yet would the Addition appear all one, and therefore is not the proof exact. But if you will make a just proof for the first man's part take ●●/●● of the whole sum, and if it agree with the number in the figure, than it is well done. And so do for the second, third, and fourth sums, and this proof faileth not. Now will I propound certain other questions which have been set forth by certain learned men, albeit not without some oversight which questions I protest heartily I do not repeat to deprave those good men, whose labours and studies I much praise and greatelie delight in, but only according to my profession, to seek out truth in all things, and to remove all occasions of error, as much as in me lieth: and for that cause I will only name the questions without hurting the Authors name. The first question is this. A question of building Four men did build a house, which cost them 3000 crowns, their shares were such, that one man should pay 1/● of the sum, and 6 crowns over: the second should pay ⅓ and 12 crowns over: the third man must lay out ⅔, abating 8 crowns and the fourth man should pay ¼, and 20 crowns more, can you answer to this question? Sc No in good faith sir, and that you know best of any man, for I know no more than you have taught me. Master. Then I dare say you can not do it neither yet the best learned man that ever did propose it, for the question is impossible: An impossible question. for declaration whereof I will be bold to use first the representation of the numbers in their aptest form, (Although I have not yet taught you that manner of work) because it may appear plainly that the question is not possible, for here I have set the parts, and added them, and they make the whole sum and ¾ and 30 more. Now how is it possible to divide truly either gains either charges so, that the particulars shall be more than the total. Sc. It is against the form of proof by Addition of parts. Master. You say truth. And because you shall perceive it the better, I will try it after the vulgar form, as in this figure you see where the ½ with 6 over is 1506: for the total is as you heard before 3000: the ⅓ and the 12 more, is 1012: the ⅔ would be ●0●0, but then abating 8, it is but 1992, and then last of all, the ¼ is 750, and the 20 more maketh 770: which all being added in one sum, do make 5280, where the total sum should be but 3000, which sum if you divide by 4/3, so shall you have ¾ of it, that is 2250, and thereto add 30 more, then will those 3 sums make 5280: whereby you may see how this form as well as the other, doth declare that the particulars in that question would make more than the whole sum by ¼, and ●0 more: and therefore can that question not be accepted as a possible thing, but yet do certain learned men propound such questions, and answer to them. Therefore somewhat to say to their excuse rather of their good meaning than for their doing, I will anon declare what may be said for their defence: but in the mean season I will propound the question as it may be wrought by good possibility. As if four men build a house together, and it cost them 3000 crowns, and then for the partition they agree thus: that as often as the first man doth pay 6 crowns, so often the second shall pay four, the third man 8 and the fourth man three. Or else thus: that the first man shall pay double so much as the fourth, and the second man shall pay ⅔ of the first man's charge: the third man shall pay double so much as the second: (And these two ways are to one end) but further for their agreement it is appointed also, that the first man shall give 6 crowns overplus, and the second 12, and the fourth shall give 20, but the third man shall give no overplus, but shall have 8 crowns abated of his charge. Now is the question possible to be assoiled, and this is the way to do it. Mark the proportion of the several charges, ☜ and set out small numbers in that rate, by which you may reduce the work to the Golden rule, as here in the first form, the numbers are already named, 6, 4, 8, 3: and in the second form, although they be not plainly named, yet they may be the same numbers: for 6 is double to 3, and 4 is ⅔ of 6: and again 8 is double to 4. Now add these together and they make 21, which 21 must be set in the first number in the Golden rule: for if it with the overplus of each man's charge would make the total sum of the charges, them were those several sums the charges of each man, besides his overplus: but now it is not so. The rule. But yet this is true, that look what proportion each of these several sums doth bear to 21, the same proportion doth the just charges of every man (beside his overplus) bear to the total of the charges, the overplus being deducted: wherefore this may you note, that before you do apply the total of the charges to the Golden rule: you must deduct the overplus which is 6, 12, and 20, that is in the whole 38 but then 8 must be restored for the abatement of the third man, and then remaineth to be deducted 30, Take 30 therefore out of 3000, and there will rest 2970, which I must set in the Golden rule for the second sum: and for the third sum I must put each of the small numbers before mentioned, which although they be not the several charges, yet they represent them in proportion. And so making for every man's charge a several question, the figures will be 4, which I mark with four letters, a. b. c. d. thus. Where I have set for briefness the sum of every man's charge in the fourth place, presupposing that you can tell how to try out that fourth sum by so many examples as ye have had. Sch. As I trust that I understand this form, so I desire much to know what may be said for them that mistook this question. Master. You seem so desirous to know this error, that you have forgotten to examine whether this work be without Scholar. Me seemeth this work to be well done, because the Addition of the 4 several numbers doth make the total sum of 2970, which was to be divided into such four parts. Master. But then have you forgotten that the first man must pay 6 crowns more besides this share, and the second man twelve crowns more: the third man 8 crowns less: and the fourth man 20 crowns more, for without these, your first total of 3000 crowns will not be made. Scho. Then must I add to the first man's sum 6 more, and it will be 854 4/7: and to the second sum I must add 12, and it will be 577 5/7: from the third sum I must abate 8, and then will the sum be 1123 3/7: then adding unto the 4 sum 20, it will be 444 2/7: and these 4 sums will make 3000, which is the whole charge, as in this example it may appear, where first I gather the 14/7. that maketh ●, and so proceed I in the Addition to the end. Ma Now have you well done, and this work in the same sums is brought of other learned men for the true solution of the question as it was first proponed, which as (I said) was impossible: and now examine it by these several sums, and see whether it do agree with the sums in the question proponed. The first man must pay 1/● and 6 over of the total sum: how think you, is 845 4/7 the half and 6 more of ●000: Sc. No that it is not, for it would be 1506: and for the second man 101●: and for the third man 1992: and for the fourth man 770, whereof not one sum agreeth to this work. But I marvel that so wise men could be so much overseen. Ma. It is commonly seen, that when men will receive things from elder writers, and will not examine the thing they seem rather willing to err with their ancients for company, than to be bold to examine their works or writings, which scrupulosity hath engendered infinite errors in all kinds of knowledge, and in all civil administration, and in every kind of art: but these learned men did not mean any other thing by this question, than to find such numbers as should bear the same proportion together, as those numbers in the question proponed did bear one to an other: which thing you shall perceive more plainly by an other question of theirs, that is this. A question of a testament. A man lying upon his death bed, bequeatheth his goods (which were worth 3000 Crowns) in this sort. Because his wife was great with child, and he yet uncertain whether the child were a male or female, he made his bequest conditionally, that if his wife bore a daughter, then should the wife have half his goods, and the daughter ⅓, but if she were delivered of a son, than that son should have ½ of the goods, and his wife but ⅓. Now it chanced her to bring forth both a son and a daughter, the question is: How shall they part the goods agreeable to the testator his wil Scholar. If some cunning Lawyers had this matter in scanning, they would determine this Testament to be quite void, and so the man to die intestate, because the testament was made insufficient, sith this condition was not expressed in it, and also it might have chanced that she should have brought forth neither son nor daughter, as often hath been seen: so is the will unsufficient in that point also. Master. Such scanners should seem to cunning, and yet not so cunning as cruel: for the mind of the testator is to be taken favourably, for the aid of the legatories when there riseth such doubts. But let us try this work, not by force of law, but by proportion Geometrical, seeing the testator did mind to provide for each sort of them. Sc. If the son shall have ½ by force of the Testament, so must the mother have ⅓. Again because she hath a daughter also, therefore ought she to have ½ and the daughter 1/●: that is both ways ½ ⅓, and ½ ⅓, which cometh to the whole goods, and ⅔ more. Wherefore it seemeth also impossible. Master. In this matter the mind of the testator is so to be understand, that such proportion should be between the portion of the wife and the son, as is between ⅓ and ½ that is, the son must have ⅙ for 2/6 to his mother, so shall he have ● to 2, that is as much as his mother, and half as much more: and the mother must have the like rate in comparison to her daughter. Then must I find out 3 numbers in such proportion, that the first may be as much as the second, and half as much more (that is) in proportion sesquialtera, and the second to the third in the same proportion, such numbers be 6, 9, 4. Sc. I pray you sir, how shall I find out those numbers? M. That will I gladly tell you. To find three numbers in any proportion. Whatsoever the proportion be of any three numbers, multiply the Terms of that proportion together and the number that amounteth, shall be the middle number of the 3: them multiple that middle number by the lesser term, and divide that total by the greater, and the least number of the 3 will amount. So if you multiply that middle number by the greater extreme, and divide that total by the lesser extreme, then will the greatest number of that progression amount. Scholar To find the proportion between 2. numbers. Then in this example, to find the proportion of ½ to ⅓, I must divide (as you taught me in Division) ½ by ⅓, & the quotient will be ½, that is 1 ½, whereby I perceive that the proportion in this question is, as 3 to 2. Therefore (as you taught me even now) I multiply 3 by 2, and the sum is 6, which must be the middle number: then I multiply the middle number 6 by 2, which is the least term, and the sum is 12, that do I divide by 3, being the greater Term, and the quotient is 4, so is 4 the least number of the 3. Then I multiply 6 by 3, whereof cometh 18, and that I divide by 2 and so have I 9, which is the greatest number of the 3. Ma. another way yet may you find the third number in any progression, if you have 2 of them: for if the middle number be one of them which you have, then multiply it by itself (as in this example 6 by 6 maketh 36) and that total divide by the other number which you have, and the third number will be the quotient. Sc. Then if I divide 36 (which cometh of 6 multiplied by itself) by 4, the quotient will be 9, & if I divide 36 by 9, the quotient will be 4. But what if I know the first number and the third, and would have the middle number? May. Multiply the two numbers together, and in their total you must seek the root of that number, and it shall be the middle number: but because as yet you have not learned how to extract roots, therefore use the first form which I have taught you, till I teach you to extract roots. And now go forward with the answer to the same question. Scholar. I perceive then that the son must not have ½ of the goods, neither the mother ⅓, nor yet the daughter ⅓, but yet must the goods be divided in such proportion, that the son shall have 9 crowns for 6 to his mother: and the mother shall have 6 crowns for every 4 to the daughter. Then I apply it to the Golden rule in three examples thus: where the first number is the Addition of those three numbers 9, 6, 4 and the third is one of them severally: the second is the total of the goods in the testament: & then by the work of the Golden rule I find out the fourth number in every work that is for the son 1705 1/11: for the mother 1136 16/15: and for the daughter 757, 17/15, which three sums added together do make the sum of the whole goods, as may be seen by this example. And this (me thinketh) I do perceive, that because in this case there is a necessary remedy devised against an urgent inconvenience, therefore those learned men thought they might use the like liberty in that other question. Master Your guess is good, but they had so good reason for them in the one, as they have in the other: as in another example of theirs, it may better appear, that is this: A man left unto his iij. another question of ● Testament. sons 7851 crowns, to be parted in this sort, that the first son should have ½, the second son ⅓ and the third son ¼, which is not possible, for ½ ⅓ ¼ doth make 2/2 0/4: or 1/1 ●/2;, that is 1 ●/12, so is it more than the whole: but reduce these fractions into one denomination, the least that they will come to, and they will be 6/12, 4/12, 3/12, and so may you part the goods in such proportion as these 3 Numeratours bear together: that is, the first to have 6 for every 4 to the second: and the second to have 4 as often as the third hath 3: and so their portions will be for the first, 2623 7/13: for the second 2415 0/13: and for the third 1811 1●/13, and those 3 shares added together, will make the total sum of the whole goods, as you may easily see in this example Another question is there proponed thus: There is 450 Crowns to be divided bewéen 3 men, so that the first man must have ½ ⅓, the second man ⅓ ¼, the third man shall have ¼ ⅕. Scholar. I marvel that any man should be so overseen to propound that question as a thing possible, sith ½ ½▪ 1/● ¼▪ ¼ ⅕, do make 1 ⅓ ⅕, that is almost double the whole sum. But I perceive it might be thus proponed, that as often as the first man did receive 50 Crowns, so often the second man should receive 35, and the third man 27, for ½ ⅓, is equal to 50/60, and so is ⅓ ¼ equal to 1/6 5/●, and ¼ ⅕ is 2/6 7/●, and so working the question, the 3 figures will appear in this form: whereby the first man's portion is found to be 200 50/56: the second man's part is 140 15/56: the third man's share is 108 24/76: which in the whole doth make 450 crowns that was the whole sum to be divided between them. Master. And thus you are (I think) sufficiently instructed in the rule of Fellowship. The Rule of Alligation. NOw will I go in hand with the rule of Alligation, The rule of mixture. which hath his name, for that by it there are divers parcels of sundry prices, and sundry quantities alligate bound or mixed together, whereby also it might be well called the rule of mixture, and it hath great use in composition of medicines, and also in mixtures of metals, and some use it hath in medicines of wines, but I wish it were less used therein than it is now a days. The order of the rule is this: The reas●● of this ru● When any sums are proponed to be mixed, set them in order one over an other, and the common number whereunto you will reduce them, set on the left hand, then mark what sums be lesser than that common number, and which be greater and with a draft of your perme, evermore link 2 numbers together, so that one be lesser than the common number, and the other greater than he, for two greater or two smaller can not well be linked together, and the reason is this, that one greater & one smaller may be so mixed, that they will make the mean or common number very well, but 2 less can never make so many as the common number, being taken orderly: no more can two sums greater than the mean, never make the mean in due order as it shall appear better to you hereafter. And as it is of necessity to link every smaller (once at the least) with one greater, and every greater with one smaller: so it is at liberty to link them oftener than once, and so may there be to one question many solutions. When you have so linked them, then mark how much each of the lesser numbers is smaller than the mean or common number, and that difference set against the greater numbers which be linked with those smaller each with his match still on the right hand, & likewise the excess of the greater numbers above the mean, you shall set before the lesser numbers which be combined with them. ☞ Then shall you by Addition bring all these differences into one sum, which shall be the first number in the Golden rule: and the second number shall be the whole mass that you will have of all those particulars: the third sum shall be each difference by itself and then by them shall be found the fourth number, declaring the just portion of every particular in that mixture. As now by these examples I will make it plain. There is four sorts of wine of several prices, A question of mixing of wine. one of 6 d a gallon, an other of 8 d the third of 11 d, and the fourth if 15 pens the galon, of all these wines would I have a mixture made to the sum of fifty galons, and so that the price of each galon may be 9 pens. Now demand I how much must be taken of every sort of wine? Scholar. If it shall please you to work the first example, that I may mark the applying of it to the rule, than I trust I shall be able not only to do the like, but also to see reason in the order of the work. Master. Mark then this form and the placing of every kind of number in it. The common price. ☞ Here (you see) I have set down the several prices which be 6, 8, 11, 15, and have linked together 6 with 15, and 8 with 11. The common price 9, I have set on the left side: And the difference between it, and every particular price, I have set on the right hand not against the sum, whose difference it is, but against the sum that it is linked withal: so the difference of 15 above 9, is 6, which I have set not against 15, but against 6, that is linked with 15, and the difference between 6 and 9 (that is 3) I have set against 15. So likewise the difference between 8 and 9 is but 1, that have I set against 11, and the difference of 11 above 9 (which is 2) I have set against 8. Then add I all those 4 difference, and they make 12, which I set for the first number in the Golden rule: the second number I make 50, which is the sum of gallons that I would have, and the third sum is every particular difference. Now if you work by the Golden rule, you shall find the number of Gallons that shall be taken of each sort of wine: For the better distinction whereof, I have set these letters abcd both against the numbers for which the works do serve, and over the works also, which severally serve for each of them. And now if you list to examine the truth of these works, add those four sums together, The proo of 〈…〉. and they will make fifty, that is the total which I would have, as by this example you may easily perceive. And for to prove how the prices do agree, do this. Multiply this total sum 50, by the common price 9, and it will make 450: then keep that sum by itself, and afterward Multiply every several sum of Gallons, by the price belonging to the same Gallons, and if that sum do agree with this, which you have kept first, then is your work well done. As here, 25 is the number of gallons of 6 d price, multiply then 25 by 6, and it maketh 150, which you shall set down: then multiply 8 2/6 by 88 which is the price for the number of Gallons, and it will make 66 4/6: so again 4 ¼ multiplied by 11 doth make 45 5/6. And last of all 12 3/6 multiplied by 15, maketh 187 ⅙. And these added together do make 450, as in the example annexed you may see: wherefore seeing it doth agree with the former sum of 50, multiplied by 9, I may justly affirm this work to be good and well done. The variation of this question. And now to prove how you can do the like, I propound the same question, only willing you to use some other form of combining or linking the sums. Scho. That shall I prove with your favour, and therefore I combine 8 with 15, and 6 with 11, and then the form will be as followeth: whereby amounteth the same sum in total of the differences, as did before: and yet now the differences be altered, as the combination is changed, whereof I understand the reason by your former work. And therefore here appeareth no strange thing, but that now I must have 8 2/6 gallons, of six pence and 25 gallons of 8 d and 12 gallons and ⅙ of 11 d, and so consequently 4 gallons and ⅙ of 15 d, so the multiplying 8 2/6 by 6, it maketh 50, and then 25 multiplied by 8, maketh 100: likewise 12 3/6 multiplied by 11, yielded 137 3/6, and 4 ⅙ multiplied by 15, maketh 62 3/6, which four sums added in one, will yield in the total 450, which agreeth with the Multiplication of 50 (being the total sum of gallons) by 9 the common or mean price. Mai. Seeing you conceive this work so well, I will propound an other example unto you of more variety in the Allegations or combines: as thus. A question ●f spices. A Merchant being minded to make a bargain for spices in a mixed mass, that is to say, of Cloves, Nutmegs, Saffron, Pepper, Ginger, and Almonds, the Cloves being at 6 s apounde, the Nutmegs at 8 s. Saffron at 10 s. Pepper at 3 s. Ginger at 2 s. and Almonds at 1 s. Now would he have of each sort some, to the value of 300 lb, in the whole, and each pound one with an other to bear in price 5 s. how much shall he have of each sort? Sch. That will I try thus. First I set down those six several prices, and at the left hand I set the common price 5 s. Then I link them thus, 1. with 10, ●, with 6, and 3 with 8. As in the example following. Master. I had minded to have combined them in more variety, but I am content to see your own work first, and then more varieties in combination may follow anon. Scholar. Then to continue as I began, I seek the difference between 1 and 5 (which is 4) and that I set against 10: then against 1 I set 5, which is the excess of 10 above 5: so I gather the difference between 2 and 5, which is 3, and that I set against 6, because it is combined with 2: and likewise the difference of 6 above 5 (which is ●) I set against 2. Then take I the difference of ● from 5, which is 2, and that I set against 8, and before that 3 I set the difference of 8 above 5, which is 3. Then gather I all these differences by Addition, and they make 18, which I set for my first number in the Golden rule, & so appeareth by those works, that of Almonds I must take 83 lb ⅓, of Ginger 16 lb ⅔, of pepper 50 lb, of Cloves 50 pound, of Nutmegs 33 pound ⅓ and of Saffron 66 pound ⅔. Then for trial hereof, I multiply every parcel by his several price, as 83 1/● which is the sum of Almonds, I multiply by 1, which is their price. Also 16 ⅔ the sum of Ginger I multiply by 2, which is the price of it. And so each other in his kind, as this table annexed doth represent: and then adding them altogether, I find the total to be 1500, which also will amount by the multiplication of the gross mass of 300 by the common price 5, wherefore it appeareth well wrought. Ma. Now will I make the Alligation to prove your cunning somewhat better: but because you shall not think yourself pressed so much, I will also note the differences, as in this example you may see. where I have alligate 1 with 6 and 8: and therefore have I set against 1, both their differences: that is 1 and 3. Likewise because 2 is combined with 8 and 10, I set before him their differences, 3 and 5. Against 3 I have set only 5, which is the difference of 10, with whom 3 is combined only: likewise 6 is only Alligate to 1, and therefore is the difference of 1 only set against it: 8 is linked with 1 and 2 and therefore hath he against him both their differences, 4 and 3: and 10 is joined with 2 and 3, therefore hath be their differences 3 and 2. And because of ease for you, in an other column I have set the differences reduced into one number, for every several sort, and have also added them together, whereby appeareth that they make 33, and so consequently you see the works of the golden rule set forth for the six several drugs: I have added letters a, b, c, etc. as before. But I would not wish you to clean still to these elementary aids, but accusiome memory to trust herself, so shall occasion of negligence be best avoided. And as for the proof, try it at more leisure, because that time now is short, and you sufficiently instructed in that proof. And there resteth diverse things behind yet, of which I would gladly give you some taste before our departure. Scholar. But if it may please you to let me see all the variations of this question, before you go from it, for me thinketh I could vary it two or three ways more yet. Ma. I am content to see you make two or three variations, but I would be loath to stay to see all the variations, for it may be varied above ●00 ways although many of them would not well serve to this purpose. Sc. I thought it impossible to make so many variations. Ma. Marvel not thereat, for some questions of this rule may be varied above 1000 ways, but I would have you forget such fantasies, till a time of more leisure. And now go forward with some variation of this question. Scholar. For the first variation, I link the first number 1 with 8 and 10, and 2 I combine with 6 and 10, then join I 3 with 6, 8, and 10, as in this form. And so doth there appear the portion of weight for every kind of drug in this mixture. Now for the trial. Master. Nay stay there, you shall not need to make trial in one example so often, or if you list to do it by yourself, I am content. But now set forth (for declaration that you conceive the rule) two or three examples of several combinations, and then will we and then will we pass to some other example and so end this rule. Scholar. As it pleaseth you so will I do. And these be the varieties in which as the combinations are several, so doth it plainly appear, that the differences by which the proportion of each several kind is taken, are also several. And yet I see in the three first of these five varieties, and in one other, before, the total sum of the differences to be one, that is to say 18, whereby I perceive that the variety of their mixture doth depend of the variety of their differences several, and not of the variety of their total sum. Master. So is it. And seeing you conceive it so well, I will make an end of this rule, only exhibiting to you one question or two of the mixture of metals, that by it you may devise other like, and exercise yourself therein also, because the use of it serveth often in business of charge, not so much for Goldsmiths, as for coinage in mints. first I demand of you this question. If a Mintemaister have gold of 22 karectes, and some of 23 karectes, some of 24: Again, some of 15, some of 16, and some of 18 karectes, and would mix them so, that he might have 100 ounces of 20 karects, how much shall he take of every sort? Scholar. To know that, I answer in order thus. Master. You have wrought the question well, but how chanced you made no doubt of that new name, Karecte? Scholar. Because I thought it out of time to demand such questions now, seeing you make so much haste to end: and again in this case the proportion of the numbers is sufficient for my purpose in this work, trusting, that an other time you will instruct me as well of this, as of sundry other things, which I have heard you talken of, so I have a great desire to know them. Master. Your answer is reasonable: and your request and trust with Gods help I intend to satisfy. And to go forward with this matter, let me see your examination of this last work. Scholar. First for the one part I add together all the particular sums as they appear in the work, and they make 100, as here by their Addition it doth appear. And so it seemeth, that the sums are well gathered, but for the farther trial of them, I multiply first 20. which is the common or mean sum of the karectes by 100, which is the sum of the whole mass which I would have and it maketh 2000 Then I multiply every particular sum by the karects that it doth contain, as 10 by 15, and that maketh 150. Likewise I multiply 15 by 16, and it yieldeth 240: so 20 by 18 maketh 360. And 25 by 22 yieldeth 550: likewise 20 by 23 bringeth forth 460: and last of all 10 multiplied by 24, yieldeth 240: which sums all joined together make 2000, that doth agree with the like sum before: wherefore I may well say, that the work is good. And now if it please you I would set forth some varieties of this question, to prove my wit. Master. Go to, let me see. Scholar. Here be four varieties. And more yet I could make, but not like is the number that you spoke of in the variation of the other question. Master. That will I teach you at more leisure, seeing it is a thing rather of pleasure, than of any necessity. But now for your exercise in this rule, one other question I will propose. A mint master hath 6 ingots of silver of sundry fineness, some of 4 ounces fine, and some of 5 ounces, some of 6, and other of 8, some of 11, and other of 12: and his desire is to mix 500 pound weight, so that in the whole mass every pound weight should bear 9 ounces of fine silver, how much shall he take (say you) of every sort of silver? Scholar. To find out that I set the numbers thus in order. And gathering the differences, it will appear, that of the first sort there must be 43 ½ ½ of the second like much: of the third sort, 65 5/23: and of the fourth sort as much: of the fifth sort 195 ½ 5/●: and of the sixth sort 86 2/2 ⅔, which in the whole will make 300 lb weight: and in ounces after 9 ounces fine 4500, that is of the first sort 173 2/2 ½: and of the second sort 217 9/23: of the third sort 391 7/23: of the fourth sort 521 ½ 7/3: of the fifth sort 2152 4/2 ●/3, and of the sixth sort 1043 ½ ⅓, which all together do make 4500 ounces, agreeable to the multiplication of 9 by 500 Master. This is well done of you, therefore now make three or four variaties, and so an end of this rule. Sc. These 4 varieties I set for example. Master. And by these it appeareth, that you can find out more, with which I will not now meddle, save only for to show you an easy help in drawing the lines of Combination, I will set forth two varieties here. And this shall suffice now for the rule of Alligation or mixture, for by these examples may you easily conjecture such other as do appertain to it, as well for the due working, as for variety of drawing the lines of combination. Scholar. Sir, albeit it pleased you ere while, to put me from my musing at the manifold varieties, that may fall in these combinations, and termed them fantasies, yet my fantasy giveth me, that the consideration of this should in many other examples and cases of importance be very needful, and the knowledge of it most profitable. Therefore ye may well think, that at another time convenient I will request you to aid me herein. Master. Truth it is, that this consideration may fall in practice as well Politic, as Philosophical, and sundry ways in them be applied, therefore when time shall fall ● for the discussing of this consideration, you shall not want my helping hand. The rule of Falsehood. Now will I briefly also teach you somewhat of the rule of Falsehood, which beareth his name, The occasion of the name. not for that it teacheth any fraud or Falsehood, but for that by false numbers taken at all adventures, it teacheth how to find those true numbers that you seek for. Scholar. So might any other rule be called the rule of Falsehood, for they work by wrong numbers, and by them find out the right numbers, so doth the rule of Alligation, the rule of Fellowship, and the Golden rule partly. Master. In the Golden rule, the rule of Fellowship, & the rule of Alligation, although the numbers that you work by, be not the true numbers that you seek for, yet are they numbers in just proportion, and are found by orderly work: whereas in this rule, the numbers are not taken in any proportion, nor found by orderly work, but taken at all adventures. And therefore I sometimes being merry with my friends, and talking of such questions, have caused them that proponed such questions to call unto them such children or idiots, as happened to be in the place, and to take their answer, declaring, that I would make them solve those questions, that seemed so doubtful. And indeed I did answer to the question and work the trial thereof also, by those answers which they happened at all adventures to make: which numbers seeing they be taken as manifest false, therefore is this rule called the rule of false positions, and for briefness, the rule of Falsehood, which rule for readiness of remembrance, I have comprised in these few Verses following, in form of an obscure Riddle. Guess at this work as hap doth lead, By chance to truth you may proceed. And first work by the question, Although no truth therein be done. Such falsehood is so good a ground, That truth by it will soon be found. From many bate to many more, From to few take to few also. With too much join to few again. To too few add to many plain. In cross ways multiply contrary kind, All truth by falsehood for to find. The sense of these Verses, and the sum of this rule, is this: When any question is proponed appertaining to this rule, first imagine any number that you list, which you shall name the first position, and put it in steed of the true number, and then work with it as the question importeth: and if you have miss, then is the last number of that work, either too great or too little: that shall you note as hereafter shall be taught you, and you shall call it the first error. Then begin again, and take an other number which shall be called the second position, and work by the question: if you have miss again, note the excess or default as it is, and call that the second error. Then multiply cross ways the first position by the second error, and again the second position by the first error, and note their totals severally by the names of Totalles. Then mark whether the two errors were both like, that is to say, both too much, or both too little: or whether they be unlike, that is, the one too much, and the other too little, for if they be like, then shall you subtract the one total from the other (I mean the lesser from the greater) and the Remainder shall be your dividend, so must you abate the lesser error out of the greater, and the residue shall be the divisor. Now divide the dividend by that divisor, and the quotient will show you the true number that you seek for: But and if the errors be unlike then must you add both those totalles (which you noted) together and take that whole number for the dividend, so shall you add both the errors together, and that whole number shall be the divisor, and the quotient of that Division shall give you the true number that the question seeketh for: and this is the whole rule. Scholar. This rule seemeth so unlike any other, that without some example I shall not easily understand it Master. Therefore take this example: A Mason was bound to build a wall in 40 days, and it was covenanted so with him, that every day that he wrought, he should have for his wages 2 s, 1 penny, & every day that he wrought not, he should be amerced 2 s, 6 d, so that when the wall was made, and the reckoning taken of the days that he wrought, and of the other that he wrought not, the Mason had clearly but 5 s, 5 d, for his work. Now do I demand, how many days did he work of those 40, and how many did he not work? Scholar. I pray you express the order of the work, that I may partly by imitation, and partly by comparing it with the rule, be able again to do the like. Master. This order shall you keep in the work of this rule: first take some number (as you list) at adventure, as for example: I say he played 12 days, and wrought 28 days. Now cast you the wages of every day, and see whether it will agree with the sum of 5 s, 5 d. Scholar. The 28 days that he wrought after 25 pence the day, yieldeth 700d. Then the 12 days that he wrought not, at 30 pence each day, doth amount to 360 pence, which if I abate out of 700 pence, there resteth 340: but you say he had not so much. Master. He had but 65 pence, and by this supposition he should have had 340: therefore is this sum too much by 275 which sum I must set down after this sort, as you see here, where first I have made a cross (commonly called Saint Andrew's cross) and at the over corner on the left hand I have set the first position 12, and at the other corner under it, I have set 275 which is the first error, with this figure 4, which betokeneth too much, as this line,— plain without a cross line, betokeneth too little. On the right hand of the cross I have left two like rooms for the second position and his error. Therefore to prosecute the work, I suppose he played 16 days, and wrought 24. Scholar. I was a while in doubt why you named the days of his working, seeing they be not set in the figure: and I doubted how you knew them, or else whether that you did suppose them at all adventures, as you did the days that he played: but now I gather, that seeing 40 days is the whole time limited, than the days that he played being supposed the rest of 40 must needs be the days that he wrought, and therefore 28 followed 12 of necessity and 24 followeth 16 also of necessity: but yet I scarce perceive why you set not in the figure as well 28 as 12. Master. It forceth not which of them I take, ☞ so that in the second position I take the numbers of the same nature that is here both of working days, or both of idle: but now examine you this second position. Scholar. If he played 16 days, then abating 16 times 30 d the sum will be 480d. And for 24 days that he wrought, every day yielding 25 d, the total is 600 d: so that abating 480 out of 600, there resteth 120: and as you say it should be but 65, therefore it is too much by 55, that must be set on the right hand of the figure at the neither part, and over it on the same side 16 which is the second position, thus. And as I gather by your words, it were all one if I did set 28 in stead of 12, and 24 in stead of 16. M. So were it. But this shall you mark, ☜ Note: that of what nature so ever the two positions be, of the same nature is the quotient. Therefore when the positions in this question are 12 and 16, The proof of this rule. which both being numbers of the playing days, the quotient shall declare the true numbers of playing days, where as if the positions had been 28 and 24, which are supposed to be the working days, than would the quotient declare the true number of the working days, & not of playing days as it will do now. And therefore to continue the work of this question, and to find the true number of playing days, I must multiply cross ways the first position 12 by 55, that is the second error, and the total will be 660, than I multiply 275 by 16, and it yieldeth 4400. Now because the errors are like, that is to say, both too much, I must subtract 660, out of 4400, and so remaineth 3740, which is the dividend. Again I must subtract the lesser error 55 out of 275 that is the greater error, and there will remain 220, which shall be the divisor, then dividing 3740 by 220, the quotient will be 17. Wherefore I say now constantly, that 17 is the true number of days that the Mason played: and then it followeth, that he wrought 23 days, and so is the question answered. Now for the order of trial of this work there needeth none other proof but only this, to work with this number according to the question, and if it agree, then appeareth the number to be it that you would have. As here now seeing he wrought 23 days, and must have for every day 25 pence, the whole sum cometh to 575. Then again seeing he played 17 days, and must abate 30 pence for everiedaye, the whole sum of the abatement will be 510: therefore I subtract 510 out of 575, and there will remain 65, which maketh 5 s, 5d. the clear wages of the Mason for his work, according to the question. Scholar. Now I trust I understand the work and the rule so well (and the better by this proof) that I can be able to do the like. And for a proof I take the same question all save the last number, where I will suppose that he had 10 s, for his wages clear. And now to guess at the number of the days that he wrought, I suppose first that he wrought: 20 days, then say I, if he wrought 20 days his wages must be 500 d, then did he play other 20 days, for which must be abated 600 d, and then he loseth 100d. And so am I at a stay, for it is not like unto your former work. Ma You should have required of me some question, and not have taken a question of your own fantasying, until you were more expert in this art: ☜ for so might you as well happen on an impossible question as on a possible: but now to go forward, consider that this number is too little by 220, seeing he should gain by your supposition 120 d, and in this position he loseth 100, those both make 220, which you shall set down for the first error with this sign—, betokening to little, as here in this form following doth appear. And now for the rest go forward yourself once again, Scholar. As my error hath uttered my folly, so it hath procured me better understanding. Now therefore considering this position not to solve the question, I take an other, supposing that he wrought 30 days, then for his wages he must be allowed 750 d, and for the 10 days which he wrought not, he must abate ●0● d, and so remaineth clear 450 d: but it should be only 120 d, therefore is it to much by 330, which I set down in the figure with the former position, and his error and the figure appeareth thus. Now must I multiply in cross ways 220, by 30, and it will be 66000. Then again I multiply 330 by 20, and it will be also 6600. Wherefore if I shall subtract the one out of the other, there will remain nothing to be the Dividend. Ma. In this you forget yourself again: for in as much as the signs in the errors be unlike, therefore must you work by Addition, adding together those two totals to make the dividend, and also adding the two errors to make the divisor. And because you shall no more forget this part of the rule, take this brief remembrance: Unlike require Addition, And like desire Subtraction. Sch. You mean, that if the errors have like signs, then must the dividend and the divisor be made by Subtraction, as is taught before: And if those signs be unlike (as in this last example they be) then must I by Addition gather the Dividend and the divisor. Therefore must I add 6600 to 6600, and it will be 13200, which shall be the dividend. Then again I add 220 to 330, and it will be 5●0, which must be the divisor: wherefore dividing 13200 by 550, the quotient will be 24, whereby I know that the Mason wrought 24 days: and then it followeth that he played 16 days. M. Examine your work whether it be agreeable to the question or no. Sc. For 24 days work, the wages must be 600d. and for 16 days which the Mason wrought not, there must be abated 480, and then remaineth clear to the Mason 220 pence, as the question importeth, wherefore it is evident, that 24 is the true number of the days that he wrought. Master. Although you seem now to understand this work, yet to acquaint your mind the better with the new trade of this rule, I think it good to propone to you five or six examples more, before I make an end of it. Sch. Sir I thank you, that you do so consider my commodity and profit in knowledge, for undoubtedly it is practise & exercise that maketh men prompt & expert in every kind of knowledge. Master. You say well so that they follow some certain precepts to govern and rule their practice by, else may practise procure custom of error, and a repugnance to exactness of knowledge, namely as long as the error is not plainly known to the vulgar sort. But to return to our work. There is a servant that hath bought of the velvet and damask for his master 40 yards, the velvet at 20 s, a yard, and the Damask at 12 s, & when he cometh home, A question of wares. his master demandeth of him how much he hath bought of each sort: I can not tell (saith he) exactly, but this I know, that I paid for damask 48 s. more than I paid for velvet, now must you guess how many yards there is of each sort. Scho Although the guess seemeth difficult, yet I will prove what I can do: for I remember your saying, that it forceth not how fond or false the guess be, so it be somewhat to the question, and not an answer of a contrary matter. Therefore first I imagine that he bought 20 yards of Damask, for which he should pay after the former price 240 shillings: then must he needs have of velvet other 20 yards (to make up the 40 yards) and that would cost 400 s. So that the total of the price of the damask is less than the sum paid for velvet 160 s, and should be more by 48. therefore the first error is 208 too little. Then begin I again, and suppose he bought of Damask 30 yards that cost 360 s, than had he but 10 yards of Velvet, which cost 200 s: and now the price of Damask is greater than the price of the Velvet by 160 shillings, and should be but 48, therefore is the second error 112 too much, which I set in form of a figure as here doth appear. Then do I multiply in cross ways. 280 by 30 and the sum will be 6240. Also I multiply 112 by 20, and there will amount 2240. And in as much as the signs of the errors be unlike, I know I must work by Addition, therefore add I those two totals together, and they make 8480, which is the dividend: then add I also the two errors together. 208 and 112, and they make 320, which is the divisor. Wherefore dividing 8480, by 320, the quotient will be. 26 ½, which is the true sum of yards of Damask that he bought: and in Velvet 13 yards ½, and that appeared by examination thus: 26 ½ yards of Damask at 12 s. the yard maketh 318 s, then in Velvet he had but 13 yards and ½. that cost 270 s at 20 s. the yard. Now Subtracte 270 out of 318, and there will remain 48, which is the number of shillings that the Damask did cost more than the Velvet. M Now shall you have a question of an other kind. A question of debt. There are three men that do own money to me, and I have forgotten what the total sum is, and what the particulars be. Scho Why? then is it impossible to know the debt. Master. Peace ye are to hasty: there is more help in it than you yet see: I have three several notes, whereby it appeareth that I did confer their debts together, and found the debt of the first and the second to amount to 47 lb, the debt of the first man and the third did make 71 lb and the second man his debt with the third, did rise to 88 lb. Now can you tell what every man did owe, and what was the whole total? Scholar. Nay in good faith: but as I perceive that it must be found by conjecture, so will I guess at it, supposing that the first man did owe 20 lb, and the second man 30, and the third. M. Nay stay there for you are to far gone already, ☜ you may not suppose a several sum for every man, for it is enough to suppose one sum for the first man, and let the other rise as the question importeth. Therefore seeing you set the first man his debt to be 20 lb, the second man can not owe 30 lb, for the declaration is that their debts added together, did make 47 lb. so must the second man his debt be but 27 lb. Now this second debt with the third must make 88, therefore subtract 27 out of 88, and there will remain 61, as the third man his debt. Then saith the declaration, that the first and third men's debts do make 71: but by this supposition they make 8● that is 10 too much: which I must set for the 〈◊〉 error. Now work you the second position. Sc I suppose the first man's debt to be 24 lb, then must the second man's debt (by your declaration) be but 2● lb. seeing both they make but 47 lb. Also the second man his debt with the third, do make 88 lb, and the second man oweth but 2●, therefore the third man must d●●e 65 lb. Now the third man's debt with the first, should make by the declaration 71 lb, & they do make 89 lb: that is 18 lb too much: and that is the second error, which I set down with the first, and their position in this form and then do I multiply in cross ways 20 by 18, & it is 360. Also 10 by 24 maketh 240. And because the signs of the errors be like, I must work by subtraction: therefore I subtract 240 out of ●60 and there resteth 120, which is the dividend: then do I subtract 10 out of 18 by the same reason, and so is the divisor 8, which is found 15 times in 120, therefore I say that the first man did owe 15 lb, and then the second man must owe 32 lb, for those 2 do make 47 lb, and the third man his debt is 56 for so much remaineth if I bate 15 out of 71, or if I take 32 out of 88 M For the third example take this easy question for the variety in work. The third question. Two men having several sums which I know not, do thus talk together: the first saith to the 2, If you give me 2 s of your money, them shall I have 3 times so much money as you: the 2 answereth: It were more reason, that our sums were made equal, and so will it be, if you give me ● s of your money. Now guess what each of them had. Scholar. I imagine that the first had 9 s. Note: Master. Consider evermore in your imaginations that you take a likely sum, as in this question take such a sum that having 2 added unto it, may be divided into 3 parts even. Scholar. Why? I remember you said before, it forced not how fondly so ever I guessed. M As for the possibility of the solution it is truth, but for easiness in work, the aptest numbers are most convenient. Scholar. I thought no less, and therefore I took 9 as an apt number to be parted into: but I perceive I should have considered the aptness of that partition after the addition of 2 unto it, and then 7 had been more meeter. M That is truth, and then should the second man his sum be 5: for although he have now but the third part of 9 that is 3, yet you must remember that he lent the first man 2, and so had he 5. Scholar Then to go forward: if the second man had 3 of the first man, than should he have 8, and the first man but 4, so hath he double to the first man: yet he said in the question they should have equal: wherefore it appeareth that he hath 4 to much. Therefore I note that error with his supposition, and guess again that he hath 10 s: whereunto I add 2 shillings borrowed of the second man and then hath he 12 shillings, so the second man hath remaining but four, whereunto if I add the 2 that he lent to the first man, so had he but 6 s at the beginning. Then take 3 shillings from the first man, and give to the second, then hath the first man but 7, and the second hath 9, which are not equal, but there are 2 to many, wherefore I set down both the positions with their errors as here you see, and multiply a cross, so cometh there 40 and 14: and because the signs be like. I take 14 out of 40, and so resteth 26 to be the dividend, them likewise I take 2 out of 4, and there resteth 2 by which I divide 26, and the quotient will be 13, which is the sum that the first man had. And so appeareth that two being added thereto, the sum will be 15, so hath the second man now but 5, and before he had 7: then take three from the first, and put to his seven so have each of them 10, and that is equal, as the question would. Master. For the fourth example take this question. One man said to an other: The fourth example. I think you had this year two thousand lambs: so had I said the other: but what with paying the tith of them, and then three several losses they are much abated: for at one time I lost half as many as I have now left: and at an other time the third time of so many: and the third time ¼ so many. Now guess you how many are left. Sch. Because here is mention made of certain parts, I must take a number that may have all those parts: that is to say, ●/2, 1/●▪ and ●/●, which will be 24, howbeit 12 hath the same parts. Therefore first I take 12 to be the number that doth remain, so hath he lost 6, 4, and 3, that is 13: and in the whole 25, but it should be 2000 Master ye are deceived yet still: you have forgotten the 10 part, which must be defalked, that is 200, so there remaineth but 1800 and now go on again. Scho. Then to find the error, I take 25 out of 1800, and there remaineth 1775 to few, which I set for the first error. Then for the second position I take 24, whose half is 12, the third part 8, & the quarter 6, where by riseth 50, which is too little by 1750, therefore I set down both the positions with their errors thus. And multiply in cross ways 1775-1750- 1775 by 24, whereof cometh 42600. Also I multiply 1750 by 12, and there ariseth 21000. And because the signs are like, I do subtract the one from the other, and so remaineth the dividend, 21600: then do I subtract 1750 out of 1775, and there resteth 25: by which I divide 21600, and the quotient is 864, whereof the half is 432, and the third part is 288, the quarter is 216, which all being added together, will make 1800. And if you add thereto the tenth which was abated before, then will the whole sum be 2000 And now doth there come a question to my memory which was demanded of me, but I was not able to answer to it, and now me thinketh I could solve it M. Propone your question. Sch. There is supposed a Law made that (for further tillage) every man that doth keep sheep, A question● Of sheep and tillage. shall for every ●0 sheep ear and sow one acre of ground: and for his allowance in sheep pasture, there is appoints for every four sheep one acre of pasture Now is there a rich sheepemaister 〈◊〉 hath 7000 acres of ground, and would gladly keep as many sheep as he might by that Statute, I demand how many sheep shall he keep. M. Answer to the question yourself. Scho. First I suppose he may keep 500 sheep, and for them he shall have in pasture after the rate of 4 sheep to an acre, 125 acres, and in arable ground 50 acres, that is 175 in all: but this error is to little by 6825. Therefore I guess again, that he may keep 1000 sheep, that is in pasture 250 acres: and in tillage 100 acres, which maketh 350: that is too little by 6650. These both errors with their positions I set down as you see, and multiply in cross 6825 by 1000, & it maketh 6825000. Then I multiply 6650 by 500, and it doth amount to 3325000, which sum I do subtract out of the first and there remaineth 3500000 as the dividend. Also I do subtract the lesser error out of the greater, and so remaineth 175, by which I divide the said dividend, and the quotient will be 20000, so that I see, that by this rate he that hath 7000 acres of ground, may keep 20000 sheep: and thereby I conjecture, that many men may keep so many sheep, for many men (as the common talk) have so many acres of ground. Master. That talk is not likely, for so much ground is in compass above 48 ¾ miles, leave this talk and return to your questions, least your pointing be scarce well taken. Scholar. Indeed I do remember, that the Egyptians did grudge so much against shepherds till at length they smarted for it, and yet they were but small shéepemaisters to some men that be now, and the sheep are waxen so fierce now & so mighty, that none can withstand them but the Lion. Maist. I perceive you talk as you hear some other: another way of working. but to the work of your question: both this last question, and the next before might be wrought without the second position, by the rule of proportion, as this. When in this question ye found in the first error, that for 500 sheep, there must be 175 acres, than might you reduce it to the Golden rule, thus If 175 acres will admit in allowance 500 sheep than 7000 will have 20000. And so by one position with the help of the Golden rule may you answer that question. Likewise for the question of Lambs, when you had found that 12 came of 25, you might have set the figure thus as ye see & have said: If 25 do leave but 12, what shall 180 leave? and it would appear to be 864. Scholar. Sir, I thank you for this aid, for it doth much shorten the work of this rule. Master. another way yet. Yet again I will show you an other way, to answer to this last question without this rule of False position, and that by the rule of Fellowship, for it appeareth in the proponing of the question, that 10 sheep must have in pasture 2 acres and ½, and for them must there be eared but one acre: so it followeth, that for two acres eared, there must be five set to pasture. And if you put them both into one sum, they will make 7. Therefore look what proportion 7 being this total, doth bear to 5 and to 2, such proportion shall any total in this question bear to the pasture ground, and the eared ground. Scholar. This serveth wondrous aptly. Therefore to prove it, I demand this by the former supposition: If a man have ●00 acres, how much shall he leave in pasture, and how much shall he turn to tillage? You say that as 7 is to 5, so shall 300 be to the acres of pasture: and as 7 is to ●, so is 100L to the acres of tillage, whereof for both I have set examples here following, whereby appeareth that of pasture there shall be 214 2/7 acres, and of tillage 85 ●/7 which both sums added together, do make 300. Master. Now take an other example: A man hath three silver cups with one cover, the cover weigheth 18 ounces the second cup weigheth even half the weight of the first and the third. Now if the cover be put to the first cup, they weigh just as much as all the three cups do weigh: and if the cover be joined with the second cup, they weigh as much as the second twice, and the third: and if the cover be put to the third cup, they will make twice as much as the first and the second cup. Now try you what was the just weight of every cup. Scholar. I do set the weight of the first cup to be 9 ounces: then in as much as these two (that is to say, the cover and the first cup) do weigh the weight of the three cups, I see that the three cups must weigh 27 ounces, for so much is 18 and 9 Also because the first and the third do weigh double so much as the second, therefore is it the third part of that weight, that is 9, and then would it follow, that the third cup also should weigh 9 ounces, but then the question saith, that the cover being joined to the second cup, they weigh as much as the second twice, and the third once, that should be 27, and so it doth: then being joined with the third cup, they should weigh twice as much as the first and the second, that should be 36, and they weigh but 27, so is that error 9 too little. Then begin I again, and say, that the first cup doth weigh 12 ounces, which I join with the cover, and they make 30 ounces: then seeing the second is ⅓ of that weight, it must needs weigh 10 ounces, and the third must weigh 8 ounces, seeing the first and the third must weigh 20 ounces. Now put I the cover to the second cup, and they weigh 28 ounces, which should be even so: then join I the cover with the third Cup, and so should it weigh twice the first, and the second, that is 44 ounces, and they do weigh but 26, that is 18 too little: those errors with their positions I set down, and multiply in cross ways 9 by 12, whereof cometh 108. Also 9 by 18, and that yieldeth 162: and in as much as the signs be like, I abate the lesser out of the greater, and there doth remain 54. Then do I also abate the lesser error from the greater, and so remaineth 9, by which I divide 54, and the quotient is 6: which I take for the true weight of the first cup: which being joined with the cover must weigh as much as the three cups, so do they weigh but 24 ounces. Then seeing the second cup is the third part of that weight, for the other two cups (you say) must weigh double his weight, the weight of the second cup is 8 ounces, and so the weight of the the third must be 10 ounces. Now put the cover to the second cup, and it will make 26 ounces: that must be the weight of the second twice, and the third once, that is twice s, and once 10, and so is it. Again, put the cover to the third cup of 10 ounces, and they must weigh twice as much as the first and the second, that is 28: and so is all agreeable. Master. Then answer to this question. A question of water. There is a Cistern with four cocks, containing 72 barrels of water: and if the greatest cock be opened the water will avoid clean in six hours: at the second cock it will ask eight hours: at the third cock it will avoid in no less than nine hours: and at the smallest it will require twelve hours. Now I demand, in what spaes will it avoid, all the cocks being set open? Scholar. first I imagine that it will avoid in two hours. Master. Then must there avoid by the first cock 1/● of the water, that is ●4 barrels, and by the second cock ¼, that is 18, and by the third cock 2/● that is 16 barrels, and by the smallest cock ⅙, that is 12 barrels, all which sums put together do make 70, as by their addition it doth appear, but it should be 72, therefore the error is 2 too few. Scholar. Then I begin again by your favour, because I think I understand the work, and put three hours for the due time: so shall there run out at the greatest cock ½, that is 36 barrels, and at the second hole ●/8, that is 27, and at the third cock ●/3 that is 34, and at the smallest hole ¼, that is 18 barrels, which all together do make 105, and should be but 72, so is it too much by ●3, therefore do I set the errors in order of the figure with their positions, and work by multiplication, in cross, saying: 2 times 3 is 6 and 2 times 33 maketh 66: and because the signs are unlike, I must add those two totalles together, which make 72: also I add the two errors, and they make 35, by which I divide 72, and the quotient riseth 2 2/25, whereby I see that all the cocks being set open, the water will avoid in 2 hours, and ●2/35 of an hour. Master. This exercise maketh you to grow expert in the rule. Therefore I will enure you somewhat more with a question or two. There were two men that had been partners, and had in account between them 300 ducats: whereof the one should have for his part 180, and the other 120: but in the parting of them they fell at variannce, so that each of them catched as many as he could: yet afterward being reconciled, they agreed that he which had gotten most part of them, should lay down ¾ of them again, and he that had gotten least, should lay down ⅓ of those which he had taken, and then parting them unto two equal parts, each man to have half thereof, and so had they their just portions, as they ought: now I demand of you what each of them had gotten by the scambling? Scholar. I suppose he that had least, got 108 ducats, than the other had 192: wherefore in laying down again of the 192, there was put down ¾ that is, 144, and so had he left but 48. Also of the 108: there was laid down 36, that is ●/●, and so he had left 72. Then I put together 144, and 36, and it maketh 180 which I part into two parts even, and so cometh 90 to be given to each of them: which sum put to 72, maketh 162, and joined to 148, it maketh 238: and now I doubt how I shall go forward. Master. You need not to take but one of them which you list, the greater or the smaller, for all cometh to one purpose: and so may you compare it that you take to any of the other sums, remembering that you make comparison to the same in the second work: as for example of the first part, If you compare 138 with the lesser sum due, that is 120, so is it 18 too much: and if you compare it with the greater sum, then is it 42 too little. Again, if you compare 162 to the greater sum, the error will be 18 as it was in the other: but it will have a contrary sign: and if you compare it with the lesser sum it will be 42 too much: so that the error both ways is either 18 or 42: & as for the signs it little forceth, for in them is nothing considered here, but likeness and unlikeness, which in this case, doth neither further nor hinder. But now go on with the work. Scholar. If it be so, then am I out of my greatest doubt Than I join that 90 (which I found as the half of the latter partition) unto 48, which is left with the one man, and so hath he 138, which (I may say) is 18 too many for the least should be but 120: that error do I note, and then make a new position, supposing the one man to have 204, and the other to have 96, wherefore of the 204 there must be laid down 153, and so remaineth with him 51. Also of the 96 there must be laid down ⅛, that is 32, and so resteth with that man. 64. Now of the 153 and 32 I make one sum as 185, which I must divide into 2 equal parts. and so each man shall have 92 ½, whereunto if I add their former portions reserved, than the one shall have 156 ½ and the other hath 145 ½. Wherefore I take the lesser sum now again, as I did before, that is 143 ½, and find that he hath too many by 23 ½, for he should have but 120, so have I for my two positions two errors, which I set down, as here may be seen, each error under his position, and then by the rule I do multiply in cross ways 108 by 23 ½ and there riseth 2538 which I note then again I multiply 96 by 18, and thereof amounteth 1728. Now because the signs are both like, that is both too many, I must work be Subtraction, and so abating 1728 out of 2538, there will rest for the dividend 810: then for the divisor I subtract 18 out of 23 ½ and there remaineth 5 ½, by which I divide 810, and the quotient will be 147 ●/33●, which is the just portion of him that had the least sum. And if I do subtract it out of 300 being the total sum, then will there remain 152 9/1●, as the portion that the other did get. Master. For the proof of this work, you may choose whether you will examine those numbers according to the form of the question, or else work by other two positions for to find the second number: and if those positions bring the same numbers that did amount by the first two positions, then doth each work confirm other. Scholar. By your patience, I will prove both ways, not only to see their agreement, but also to accustom my mind to those works: for I perceive it is exercise that must be the chief engraver of these rules in my memory. Master. You consider it well: then go to. Scholar. first I will by two other positions try to find the portion of him which had most. Master. Although you may do it with any positions, yet to see the agreement of your work the better, take the same positions that you did before, comparing them now to the greater, as you did before unto the lesser. Scholar. Then I suppose, that he that had most, had 192, so had the other 108. Now if I take ¼ of 192, that will be 144, and there will rest to that man but 48. And from the second which had 108, if I take ⅓, that is 36, there will remain to him 72: then joining 144 with 36, it will make 180, the half whereof being 90. If I add to each of those two men's portions remaining with them, the one shall have 138, and the other 162, of which two I take the greater (that is 162) and see it to be 18 too few, for it should be 180, that error I note under his position. Then for the second position I take (as I did before) 204 for the one, and so resteth 96 for the other: then take I ¾ of ●04 and it will be 153, and there resteth to him 51. Also of the 96 I take ¼ that is 32, and there remaineth to him 64. Now put I that 32 to 15●, and it yieldeth 105: which being parted in equal values, maketh 92 ½. to be added to each man's remainder, and so the one hath 143 ½, and the other 156 ½: wherefore I take the greatest sum, and it is 23 ½ too little, that do I note also, and set both these errors under their positions, as in this example following doth appear. And then multiplying 192 by 23 ½, there doth arise 4512. Again, I multiply 204 by 18, & it maketh 3672, which I do subtract out of 4512, because the signs be like, & there resteth 840 for the dividend: then subtracting 18 out of 23 ½, there will remain 5 ½, which I must take for the Divisor. And so dividing 840 by 5 ½, the quotient will be 152 8/11, whereby I have found an agreeable sum to that which I found by the former positions, for him that had most, which if I do subtract out of 300, that is the total, there will rest 147 ●/11, which was the portion of him that had the least part. Master. So by divers positions you see, that one doth confirm the work of the other. Now examine those two numbers by the form of the question, and so shall you prove your work good also. Scholar. If that he which gate most, had 152 3/11, then must he lay down ¾ of his sum, that is 114 6/11▪ and so shall remain with him, but only 38 2/11 The other which had least, that is 147 3/11. must put down of his sum ½. that is 49 1/11, and so doth there remain with him yet 98 3/1●. Then do I add together 114 6/11 and 49 1/11, and it will make 163 ●/1●, which I must part into two equal parts, and that will be 81 9/11 to be given to each of them: so putting 81 9/11 unto 38 2/11, there doth amount 120 just, which is the true portion of him that should have the lesser sum: and adding 81 9/11 to 98 2/11, the total will be 180. the true portion of the other. And so is the work by this proof also tried to be good. And this I mark by the way, that in their scambling, he gate moste (as it chanceth often) that aught to have had least by just partition. Master. Let your study be to learn truth and just art of Proportion, and to distribute and part according thereunto, as often as occasion shall be ministered. And here would I make an end of this rule, save that I remember one pleasant question which I can not overpass, which I will declare somewhat largely, because you shall as well understand some reason in the pleasant invention, as apt proceeding in the witty working thereof. Hiero King of the Syracusanes in Sicilia, An example of mixture of Gold and Silver. had caused to be made a Crown of Gold of a wonderful weight, to be offered for his good success in wars: in making whereof, the Goldsmith fraudulently took out a certain portion of Gold, and put in silver for it, so that there was nothing abated of the full weight, although there was much of the value diminished. Which thing at length being uttered, (as no evil can always lie hid) the King was sore moved, and being desirous to try the truth without breaking of the Crown, proponed the doubt to Archimedes', unto whose wit nothing seemed unpossible, which although presently he could not answer unto, yet he had good hope to devise some policy for that invention. And so musing thereon, as he chanced to enter into a bane full of water to wash him, he observed that as his body entered into the bane, the water did run over the tub: whereby his ready wit of such small effects conjecturing greater works, conceived by and by a reason of solution to the King's question, & therefore rejoicing exceedingly more than if he had gotten the Crown itself, forgot that he was naked, and so ran home, crying as he ran, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, I have found, I have found. And there upon caused two massy pieces, one of gold, and an other of silver to be prepared of the same weight that the said Crown was of: and considering that gold is heavier of nature than silver, and therefore gold of like weight with silver, must needs occupy less room, by reason it is more compact & sound in substance, he was assured, that putting the mass of gold into a vessel brim full of water, there would not so much water run over, as when he should put in the silver mass of the like weight. Wherefore he tried both, & noted not only the quantities of the water at each time, but also the difference or excess of the one above the other, whereby he learned what proportion in quantity is between gold and silver of equal weight. And then putting the crown itself into the vessel of water brim full (as before) marked how much water did run out then, & comparing it with the water that run out when the gold was put in, noted how much it did exceed that: & likewise comparing it to the water that run out of the silver, marked how much it was less than that: & by those proportions found out the just quantity of gold that was taken out of the crown, & how much silver was put in steed of it. But seeing vitrvuius which writeth this history, doth not declare the particular work of this trial, it shall be no inconvenience to suppose an example for declarations sake, wherein although the true and just proportions be not expressed, yet the form of trial shall be truly set forth And for an example, I suppose the weight of the Crown to be 8 lb, and so of each of the other two Masses. And when the mass of Gold was put into the water, I imagine that there ran out 2 pound of water: and when the mass of silver was put in, I suppose there ran out 3. pound ½ Again when the crown was put in, there run out 2 pound ¼ Now to know what quantity of silver was in the Crown, work by the rule of false position, and imagine that there was 2 pound of silver: then must there be 6 pound of Gold▪ Then say thus by the rule of Proportion: If 8 pound of gold do expel 2 lb of water, what shall 6 lb. expel? and it will be 1 pound ½. Again for the silver: If 8 lb of silver expel ● lb ½ of water, what shall 2 lb of silver put out? it will be ⅞. Now add those two weights of water together, and they will make 2 lb 3/● and it should be by the supposition 2lb ●/●, so is it too much by ⅛ Sch. Now do I understand the work as I think, therefore I pray you let me work the rest of the question. And because this first supposition did err, I note that position, and his error, and take a new position, esteeming the silver to be but one pound, so must there be in Gold 7 pound, Then say I: If 8 lb of Gold yield 2 lb of water, what shall 7 lb yield? and it will be 1 lb. ¾. Again if 8 lb, of silver expel 3 lb. ½ of water what shall 1 lb expel? and it will be ●/2 7/6. Now must I add those two sums together, and they make 2 lb, ●/16 and they should make 2 lb ¼ so is it too little by 2/16 Therefore I set the positions with their errors in order, as here followeth. And then I multiply in crossewaies 2 by 6/11 and it maketh ⅛: likewise 1 multiplied by ⅛: maketh 2/8. And because the signs be unlike, I must add those two sum, which make ¼ and that is the dividend. Again I must add ⅛ to 1/16, and it will be ●/●●▪ that is the divisor. Now I shall divide 1/● by ●/●● and the quotient will be ●●/●● that is, 1 ½, whereby I know that there was put 1 lb and 1/● of silver into the Crown, and so much Gold taken out for it. M Prove it now by examination according to the question. Scho. If there were 1 pound ●/3 of Silver than was there of Gold 6 pound ⅔. Now say I by the rule of proportion: if eight pound of Gold expel two pound of water, what shall 6 pound ⅔ expel? It will be one pound ⅔. Again, if 8 lb of Silver expel 3 lb ½ of water, what shall 1 ½ expel? It will be ●/1 7/2. Now must I add together 1 lb ½. and ●/1 7/2 and they will make 2 lb, 9/●6, that is 2 lb ¼, according to the supposition of the question, whereby I perceive the work to be well done. And as I can not but much rejoice of this excellent invention, so my desire is kindled vehemently to be perfectly instructed in every part thereof, and namely in this point, whether the portion between water and gold be such, that for 8 lb of gold into a vessel full of water, there shall run out 2 lb of water: & for as much silver, whether 3 lb ½ of water would avoid? Ma. I perceive your meaning, and conjecture your imagination to be thus: that if you knew the exact proportion between Gold and Silver and Water both in their weight and quantities, than could you easily find out the mixtures of them, which thing I have reserved for an other work that entreateth such matters specially. And at this time you must consider, that you learn Arithmetic, which entreateth of the manner to solve doubtful questions touching number, without regard what matter is signified by that number, else were it necessary in Arithmetic to teach all arts, seeing in it may be moved questions of all arts. But seeing you are so desirous to know this thing, I will tell it you in such a sort, that you shall practise your art in finding it, and propound it in form of a question. Gold beareth greater proportion to water than silver doth, and their two proportions be in proportion together as 4●/25. But to help you somewhat in this riddle, you shall note that the proportion of quick silver unto water, is the just middle number proportional in Progression Geometrical, between the proportions of Gold and silver unto water. And his proportion is as 29●/●1. Now if you will know the just numbers of these 3 proportions than must you find out 3 numbers in Progression geometrical, whereof the middlemost must be 290/21, and the first must be unto the last, as 25 to 48. And thus I will leave you to find those numbers when you be at leisure. Scholar. Yet sir I thank you heartily for this much, for now I see the possibility to find them out. Howbeit, because this question seemeth strange, if it might please you to instruct me somewhat in the order of working for it, I should the more easily find the true working. M. You desire too much ease if you will study for nothing: therefore to occasion you to study the better, I will leave this doubt wholly to your own search. But as touching the generalty of the rule, Archimedes needed not to take two masses of gold and silver equal in weight with the crown, for the proportion might as well be found in any other weight yea although the mass of gold were of one weight and the mass of silver of another. As for example. If the crown were of 8 pound weight, as I did suppose, and I have not so much other fine gold, but only 1 lb, and trying that by water, & finding that it doth expel but ●/4 of an ounce of water, yet then by it may I infer, that 8 pound of gold would expel 6 ounces of water And likewise of the silver: whereof if I had but 2 pound, and find that it doth expel three ounces of water, than might I affirm that 8 pound would expel 12 ounces, that is 1 lb weight. And so is it, as good as if the 3 masses were all of one weight. And thus for this time I will make an end of this other part of Arithmetic. Sc. Although I can not sufficiently thank you for this, yet your promise made me to look for the art of extraction of roots, whereof hitherto I have learned nothing. Master. I will not break my promise, but intent (God willing) to perform it within these three or four months, if I perceive this my pains to be well taken in the mean season. And you shall not repent the tarrying for it: for it shall be increased by the tarrying. And in the mean time, you shall take this Addition, not for the second part of Arithmetic which I promised, but for an augmentation of the first part, unto which I would have annexed the extraction of Roots square and cubike, namely for examples of the Statute of Assize of wood, but that in the second part I must write of divers other roots, and thought it best to reserve those rules also with their examples unto the same second part. Scholar. Sir, although I can not recompense your goodness, yet I shall always do mine endeavour to occasion you not to repent your benefit on me thus employed. Master. That recompense is sufficient for your part. FINIS. The third part or Addition to this Book entreateth of brief rules, called rules of practise of Rare, Pleasant and commodious effect, abridged into a brieffer Method than hitherto hath been published: With divers other necessary Rules, Tables, and Questions not only profitable for Merchants, but also for Gentlemen, and all other occupiers whatsoever, as by the Contents of this Book may appear. Set forth by john Mellis Schoolmaster. The first Chapter of this Addition entreateth of brief Rules, called Rules of practise, with diverse necessary questions profitable, not only for merchants, but also for all other occupiers whatsoever. THe working of Multiplication in practise, is no other thing, than a certain manner of multiplying of one kind by another: whereupon is brought forth the product of the proponed number which is accomplished by the means of Division in taking the half, the third, the fourth, the fift, or such other parts of the sum which is to be multiplied. And for the better understanding of such conversions: you shall understand that in the manner and use of these Rules of practise, you ought first to know the even or aliquot parts of a shilling, which in this Table following doth appear. Item d 6 is the ½ of a s. 4 is the ⅓ of a s. 3 is the ¼ of a s. 2 is the ●/● of a s. 1 is the ●/12 of a s. Wherein as you see according to the order of these rules of Practice at 6 d the yard of any thing, you must take the ●/2 of your number which is to be multiplied, and the product, that cometh thereof shall be shillings, if any unity do remain it is 6 d. For 4 d take the ½ of the number that is to be multiplied and the product also produceth shillings if any unities do remain, each one shall be worth in value 4 pence. The like is to be understood of the other 3 etc. Example I. At 6 d the yard what 379 yards II. At 4 d the yard: what are 104 yards worth III. At 3 pence the yard : FOUR At 2 pence the yard V At 1 penny the yard Here you may see in the first example the 379 yards at 6 d the yard, are worth 189 s 6 d in taking the ½ of 379. And in the second example the 104 yards at 4 d the yard: are worth 34 shillings 8 d: in taking the ⅓ of 104. Likewise in the third example 5014 yards at three pence the yard bringeth forth 1253 s 6 d in taking the ¼ of 5014. Also in the fourth example at 2 pence the yard, maketh 88 shillings 8 d. And lastly in the fifth example: 409 yards at 1 d the yard, amounteth to 34 s and 1 d, in taking the 1/12 of 409: And so is to be done of all other questions the like, when the number of the pence is any of the even or aliquot parts of 12 d. Item to bring the productes of these shillings and all other the like in pounds is very easy in dividing of it into your mind by 20, for it is to be understood that as often as 20 is found in that product: So many pounds doth it contain, which with facility to perform, always strike of the figure toward your right hand, with a right down dash of your pen for the 0 that appertayto 20: And then begin at the left hand, in taking the ½ of the rest. And if at the last any unity do remain, the same shall be joined with the figure that is cut of, which shall represent the odd shillings contained in that work. As for example in your third question at 3 d the yard which amounteth to 1253 s. 6 d: the producte whereof maketh 62 lb 13 s 6 d: as here you may see is easily performed in the mergent. Item also for the working of 1 penny the yard, it is something harsh and hard to take the 1/12; part of some products: Therefore to ease that hard work you shall first bring your delivered sum into groats, by taking the ¼ part of the product. And if any unites remain of that ¼ part, as sometimes there may they are pence: and must be signified with a line from the groats with their title of pence: And because that 60 groats maketh a pound or twenty shillings, strike of the first figure toward your right hand for the 0 that appertaineth to 60 (as you did even now for the 0 that belongeth to 20:) then in taking the ⅙ of that product, if there do remain any unities the same shall you join with the figure that you cut of, esteeming them as groats: which keep in your mind. And by taking the ½ part of them, you shall turn into shillings: And so have you done as for example by a question or 2 hereafter proponed shall more plainly by the work appear. At 1 d the yard Here in taking the ⅙ part of 1359: in coming to the last work the ⅙ part of 39 being taken, the remainder is 3 which joined with the 2 that was cut off, maketh 32 groats: which converted into shillings by taking the ⅓ part: maketh as appeareth 10 s 8 d: Many other ways there are, but none more apt for a young learner to understand than this: wherefore this one way well impressed in memory is better than 20 ways doubtfully understood. At 1 d the yard, what 4533 yards At 1 d the yard what 64768 yards Now followeth also to be understood that if the number of pence be not an aliquot part of 12, 2. Rule. you must reduce them into some aliquot part of 12. And after the aforesaid manner, you shall make of them 2 or 3 products as need shall require: And add them together into one sum: And here for thy furtherance appeareth a note of the order of their parts, as they are to be taken. For pence 5. take. 3 & 2 or 4 and 1 7. take. 4 & 3 or 6 and 1 8. take. 4 & 4 or 6 and 2 9 take. 6 & 3 or 4.4 & 1 10 take. 6 & 4 or 4.4 & 2 11 take. 6▪ 4 & 1 or 4.4 & 3 Here in the first note of this table at 5 d, you shall first take for 3 d the ¼ of the number that is to be multiplied: And likewise for 2 d: the ⅙ of the same number, adding together both the products. But if you will work by 4 and 1 you must for 4 d first take the ⅓ of the number that is to be multiplied: And for 1 d take the 1/12 of the whole sum or rather, which is more better for 1d. you may take the ¼ of the producte which did come of the 4 d: Because the 1 d is the ¼ of 4 d: The total sums of these two numbers shall be the solution to the question. And in like manner is to be done of all others: As by these examples following shall appear. I. At 5 pence the yard What will— 758 yards amount to Otherwise. At 5 d the yard what are 758 yards worth II. At 7 d the el what 562 elles III. At 8 d the lb what 112 pounds Otherwise. What comes— 112 pound at 8 d the pound FOUR At 9 d the Ell What comes— 356 else to V At 10 d the piece What comes— 795 pieces to VI At 11 d the pound What— 757● pound maketh Here in this first example where it is demammaunded (at 5 d the yard) what will 758 cost: First for 3 d I take the ¼ of 758: And thereof cometh 18● s— 6 d: Then for 2 d I take the ⅙ of the same product which amounteth to 126 s 4 d: these two sums added together do make 315 s 10 d: And so much are the 758 yard's worth at 5 d the yard. Item also for the same again: First for 4 d I take the ½ of 758: and thereof cometh 252 s— 8 d: then for 1 penny I take the ¼ of the same product, that is to say of 252 s— 8 d, and it yieldeth me 63 s 2 d: which both added together make 315 s— 10 d, as before. Item, for 7 d there is take then ½ and the 1/● of the whole sum: which is to be multiplied, and add them together, that is to say, first, for 4 d there is taken the ⅓ of 563: which comes to 187 s 8 d as appeareth by the work: and for 3 d there is taken the 1/●● of the whole sum which amounteth to 140 s 9d. Both which products added together maketh 328 s— 5 d: And so much comes 563 else to at 7 d the Ell. Item, for the first 8 d there is taken for 4 d the ⅛ of the whole sum: and an other ⅓ for the other 4 d, which added together as in the example doth evidently appear, amounteth to 74 s— 8 d. Again, for the second work of 112 lb, there is taken first the ½ of the whole sum for 6 d, which comes to 56 s: then for the 2 d you have to take ⅙ of the whole sum, or if you will the ½ part of the product that came of 6 d either which maketh 18 s 8d. These two sums being added together do make 74 s 8 d: as in the third example appeareth. Item, for 9 d there is taken for 6 d, the ½ of the whole sum: and the ¼ of the whole sum for 3 d, or otherwise for the 3 d you may take the ●/2 of the product that came of 6 d, because 3 d is the ½ of 6 d: which added together as plainly appeareth in the fourth example, amounteth to 267 s— 0 d. Item, for 10 d, first there is taken for 6 d the ½ of the whole sum, which amounteth to 397 s— 6d. Then for 4 d there is found 265 s: both which added together maketh 662 s— 6 d as appeareth in the fift example: it may also be wrought, as appeareth by the second note in the table by 4 d twice taken, and the ½ of the product of 4 d: or else by the ⅙ of the whole sum, etc. Item, for 11 d, there is first taken the ½ for 6 d: then the ⅓ of the whole sum for 4 d: lastly, the ¼ of the last producte for 1 d: All which 3 sums added together maketh in s 6947-5 d, & in pounds 347-7 s— 5 d. Item, likewise by the same reason, 3. Rule. when you will multiply (by shillings) any number that is under 20 s you shall have in the product pounds, if you know the even or aliquot parts of 20, which are here in this little table set down to sight. Item s 10 is the 1/● of one lb 5 is the 1/● of one lb 4 is the ●1/5 of one lb 2 is the 1/10 of one lb 1 is the ●/20 of one lb So that for 10 s which is the ½ of a pound you may take the ½ of the number which is to be multiplied: and you shall have in your product pounds: if a unity do remain, it shall be worth 10 s. Likewise for 5 s you must take the ¼ of the number which is to be multiplied: And if there do remain any Unities, they shall be fourth parts of a pound, every Unity being in value 5 s. For 4 s take the ⅕ of the number which is to be multiplied: And if there do remain any Unities, they shall be fift parts of a pound, each unity being worth 4 s. For 2 s you must take the 1/10 of the number to be multiplied: wherefore to take the 1/10 of any number: you must cut off the last figure of the same number (which is nearest your right hand) from all the other figures with a small right down line or dash with a pen, and so have you done: for all the other figures which do remain toward your left hand from the same figure that you do separate shall be the said 1/10 of a pound: And that figure so separated towards your right hand shall be so many pieces of 2 s the piece: the which figure you must double to make thereof the true number of s, as by the example shall appear. Finally, for 1 s, needeth small work, for it is so many shillings as be proponed in the sum, which to bring into pounds hath been already taught in the first Rule. Example. At 10 s the piece At 5 s the Ell At 4 s the yard At 2 s the pound weight At 1 s the piece 4. Rule. Nextly, now followeth in order to be understood, that if the number of shillings be not some even, or aliquot part of 20, you must then convert the same number of shillings into the aliquot parts of 20: And thereof make two or three products, as need shall require: which done, add them together, and bring them into pounds. And here for thy furtherance I have set down a note of the order of their parts, as they are to be taken. s s 3 of 2 & 1 or 13 of 10.2 & 1 6 4 & 2 5. & 1 14 10. & 4 7 5 & 2 15 10. & 5 8 4 & 4 5.2.1 16 10.5.1 9 5 & 4 4.4.1 17 10.5.2 11 10 & 1 18 10.4.4 12 10 & 2 19 10.5.4 For 3 s according to the tenor that you see is expressed in the Table, you must first take for 2 s the 1/1● of the number that is to be multiplied: Then for 1 s you must take the ½ of the product which did come of the same 1/20 part, and add those two sums added there, produceth the effect desired. Item, for 6 s according to the note set forth in the table, first for 4 s I take the ⅕ of the number that is to be multiplied: Then for 2 s the ½ of the product that came of 4 s, and add them together. Or else, as appeareth also in the table, for 5 s you may take the ¼ and the 2/● part of the product that came of 5 s, and add them together. Item, for 7 s, first take for 5 s the 1/● of the producte, that is to be multiplied, then for 2 s, take the 1/● of the number that is to be multiplied, and add them together, etc. Item, for 8 s, according to reason, and the intent of the Table, for the first 4 s take the ⅕ of the product, and the same number again for the other 4 s: and add them together. Item, for 9 s: first for 5 s take the 1/●: then for 4 s take the ⅕: and add them together. Otherwise as you see by the intent of the table, work twice for 4 s, as was taught even now for 8: and then take the ¼ of the last product for the 1 s: But 5 and 4 is the shorter. Iten, for 11 s: first dispatch 10 s: for which you must take the ½ of the product: then lastly for 1 s take the 1/10 part of the sum produced of the ½ of the product and add them together. Item, for 12 s where I will end with the first part of my Table: First take the ½ for 10 s: And then for 2 s take the ⅕ of the sum that came of 10 s, and add them together: or else, if you please for 2 s you may take the ½ of the whole given number. To write more of the manner of taking the true parts, I think superfluous. The desirous practitioner will (no doubt) conceive it. Also the Table is some aid to help the unperfect: whereupon by & by I will set down three or four of these notes in examples: and the rest I will leave to thine own industry & practice to labour upon. This is the order most commonly used in Practice when the number of the s is not an aliquot part of a pound. But loving Reader) after I have touched the even or aliquot parts of a lb that falleth out in d and s, I will deliver 2 new Rules that shall drown this common order quite and clean: wherein shall be comprehended in one line, or working both even and odd part of s under ●0: without regard whether it be an aliquot or not an aliquot part: which 2 Rules, when they come in place, I commit to thy friendly judgement in working: Now followeth the examples upon the notes before said. At 6 s the yard Otherwise by multiplication of 6: At 7 s the Ell Otherwise by multiplication of 7: At 8 s the piece what 7563 pieces Otherwise by Multiplication. At 13 s that piece what 401 pieces Otherwise by Multiplication. These & such like questions of Compound numbers, which I have here in this fourth rule for order's sake set down, I count but as superfluous. For, in the second part of my new promised Rules shall appear, that the given price of any odd number of Shillings, either under or above 20: shall be wrought at two wor-kings at the most how difficult so ever the question be. Item, there resteth yet a kind of practise, how to bring pence into pounds at the first working: whereupon you must understand, that 240 pence maketh one pound, or 20 s, I cut off the last figure or 0: and there remaineth but 24 (of which 24) 8 d is the ⅓ part thereof: 6 d is the ¼ part, 4 d the ⅙ part: and 2 d is the 1/12 part thereof. Whereupon if it were demanded what 1486 yards or pounds of any thing cometh to: at 8 d the yard, in pricking or cutting off the first figure towards your right hand: for the 0 that appertaineth to 240: There is remaining of the said sum 148: whereout I take the ⅓ part: and it cometh to 49 lb: and there resteth one: which 1 I put to the 6: that I prick or cut off, and it maketh 16 pieces of 8 pence, which I double to make into groats and they make 32. whereof the ● part maketh 10 s and there remaineth ● s: which is 8 pence, whereby it followeth, that the 1486 yards at 8 pence the yard, maketh 49 lb 10 s 8 d: as by the example shall appear. Item for ● pence, take the 1/● part of the number from the pricked figure: And if any unities do remain, they are so many sixepences, whereof taking the 1/●, they are shillings, if there do remain yet one, it is in value 6 pence. Item for 4 pence, take the ⅙ part of the number from the prickte figure: If any unities remain, they are so many groats, which to convert into shillings, take the ⅓ part: And if any thing yet remain, they are thirds of shillings, echcone in value being worth 4 pence. Item, for 3 pence, take the ● part from the pricked figure, if any unities remain, they are so many pieces of ● pence whereof in taking the 1/● part, maketh shillings: If any thing yet remain, they are fourth parts of shillings, eachone being in value 3 pence. Item, for 3 pence, as appeareth also by the table, take the 1/12 part of the number from the pricked figure: If any thing remain, they are so many pieces of 2 pence: which by taking the ⅛ part, you shall turn into shillings: and if any unities remain, they are so many sixth part of shillings, or pieces of 2 pence, whether you will. If one pound cost— 8 d If one cost 6 d If one yard cost 4 d At 3 d the yard At 2 d the el what 7894 6. Rule. But if your number of pence be not an aliquot or even part of 24: then must you bring them into the aliquot parts of 24, and make thereof divers products, which must be added together, as by the questions hereafter following shall appear. Item, for 5 d, first take for 3 d, then for 2 d: and add them together according to the instruction of the second Rule: Or else first take for 4 d, then for 1 d. Item for 7 d, first take for 4 d: then for 3 d and add them together. Item, for 9, first take for 6 d: then for 3 d, and add them together. Item, for 10 d, first take for 6 d: then for 4 d, and add them together. Item for 11 d first take for 8 d then for 3 d and add them together: as by these examples following doth appear. Examples. If one yard cost 5 d what 7596 Otherwise. If one cost 7 pence what 987 Otherwise. If one cost 9 pence what 987 Otherwise. If one yard cost 10 d what 987 If one cost 11 pence what 987 But if you have any shillings, & pence to be multiplied together: Then are you to take for the shillings according to the instruction of the third Rule: And for the pennies according to the first Rule before mentioned: unless you can spy the advantage thereof: and thereby help yourself: as appeareth in this second example, where first I work for 6 d: which is to be rebated out of the given number, and I have 719 lb 11 s my desire. At 10 s 6 d: the yard What 738 yards The like again is done by rebating as by these 2 examples appeareth: Item, 418 else at 18 s Item 517. at 16 s 7. Rules And now I will touch a little the even parts of a pound that falleth out in pence and shillings, whereof for those parts you shall take such like part of the given number that is to be multiplied, as the price of that given number beareth in proportion to a pound which also for thy better aid is here set down. 1 s. 8 d is the 1/12 of the lb. 2. 6 is the ⅛ of a lb. 3. 4 is the ⅙ of a lb. 6. 8 is the ⅓ of a lb. Item first for 1 s 8 d take the 1/12 part of the given number & if any thing do remain, they are twelve parts of a pound, each one being in value 1 s 8 d. Item for 2 s 6 d take the ⅛ part of the number that is to be multiplied. And if any thing do remain they are eight parts of a pound each one being in value 2 s 6 d. Item for 3 s 4 d as appeareth by the table, you must take the ⅙ part of the given number. And if any thing do remain they are 6 parts of a lb: each one being in value 3 s 4 d. Item for 6 s 8 pence take the ⅓ part of the number that is to be multiplied: And if any unities do remain, they are thirds of a pound every one being worth 6 s 8 pence. Other infinite numbers there are, that may be reduced by abbreviation into the proportionate parts of a pound: as 16 s 8 pence maketh ⅙: which 16 s 8 d is easily reduced into groats by multiplying 16 by 3: & thereto add 2: which maketh 50 groats: Then set 60 the groats of a pound under 50, cutting of the 2 Ciphers, as is here performed in the margin. And then have you brought 16 s 8 pence into the known parts of a lb which maketh ⅚. But yet gentle Reader, for thy further instruction, I have hereunto annexed in a table, how pence and shillings beareth proportion to a lb: which I commit to thy friendly benevolence, it will be some aid unto the ungrounded practitioner: but I count him the best workman that can presently reduce his given price unto the known and proportionate parts of a lb. s d lb 0 2 1/120 0 3 1/80 0 4 1/60 0 6 1/40 0 8 1/●0 1 0 1/20 1 3 1/16 1 8 1/12 2 0 1/10 2 6 ⅛ 3 0 3/20 3 4 ⅙ 3 9 2/●6 4 0 ⅕ 5 ¼ 6 1/10 6 3 20/16 6 8 ⅓ 7 7/20 7 6 ⅜ 8 ⅖ 8 4 5/12 8 9 7/16 9 9/20 10 ½ 11 11/2● 11 3 7/16 11 8 7/12 12 ⅗ 13 1●/2● 13 4 ⅔ 13 9 1/1 14 7/1 15 ¾ 16 4/5 16 8 ⅚ 17 ●7/2● 17 6 ⅞ 18 9/10 18 4 11/12 18 9 15/16 19 19 19/20 Here followeth 4 examples upon the 4 notes delivered. At 1 s 8 pence the yard What 3884 yards At 2 s 6 pence the yard What 4562 yards At 3 s 4 pence the yard What 583● yeardes' At 6 s 8 pence the yard What 7562 yards Now by custom you are able to work by all sorts of summmes, being delivered in shillings & pence, as 1 s 1 penny: ●s ● pence 3 s 3 pence, and so of all other: wishing you to have some considerations of your questions, when they are set down, for there are many subtle abbreviations, and great advantages to be gotten, and easily to be perceived As ● s. 8 d of 2 s & 1 lb 8 d. 4 s ● d: of ● s 4 d: and 10 d which 10: is ●/4 of 3 s 4 d 5 s 8d. of 4 s: and 1 s. 8 d. 5 s. 10 d, of 5 s and 10 d: which 10 d is ⅙ of 5s·s And by this mean when you have taken one product, you may oftentimes upon the same take an other more briefly then upon the sum which is to be multiplied etc. 8. Rule. Now gentle Reader that you have seen the virtue of the even or aliquot parts of a lb: in shillings alone, And also in the aliquot parts of shillings and pence: according to my promise hereafter followeth a briefer and easier method for any even number of shillings either under or above 20, then ever yet hath been published: Notwithstanding Master Humphrey Baker, whose travel is worthy commendation, And whom for knowledge sake I reverence hath in some part touched this first part: though not in this method: The work of the Rule is both pleasant, ready and brief. As by the variety of the examples delivered thereupon shall appear. And first I will set forth a question: Thereby the better to express or teach you the order thereof: which is this. If one yard cost 6 s what 8574 To the understanding of this example, after you have set down your given number in form of the rule of 3, with a line drawn under it: you shall presently set a prick under your first figure 4, towards your right hand, drawing from the prick as heretofore hath been practised, a little short line, thereto set down the shillings anon, which done, multiply the first figure 4 by 6. the value of your price, (which here you see standeth in sight above the line.) it maketh 24: which is 1 lb 4 s. The 1 lb keep to carry to the next place, & the 4 s set down at the end of the prescribed line towards your right hand: Thus have you done now with 6 above the line, and also with 4 in the first place (for the prick under the 4. doth represent that 4 hath done his office.) Then secondarily for a general rule take but the ½ of the given price which here is ●, which 3, Note. A general rule. is the number that shall now continue the rest of the multiplication and end the work, whereupon I multiply 3 into 7 standing in the second place it maketh 21, and with the 1 lb I kept in mind 22, set down 2 & keep 3 in mind working according to the rule of multiplication, delivering the tens in mind in their due place, which done, the product from the prick to your left hand representeth the pounds and the other at the end of the line the shillings: as appeareth by the examples. If one yard cost 2 s what 7536 If one yard cost 4 s what 8792 If one piece cost 6 s what 9537 If one cost 8 s what 7509 If one cost 12 what 5794 If one cost 14 s what 3705 If one cost 18 s what 5703 If one cost 22 s what 953 Let these suffice gentle reader for an entreance into even numbers: And now I will show the like rule for any odd or uneven parts of a pound. To help you to the understanding of these other questions that hereafter followeth: where in my first example the given number is 6487. 9 Rule. At 3 s the yard: I multiply 3 above the line into 7. it maketh 21: The 1 shilling I set down & the 1 lb I keep: Now am I to take the ½ of 3: which because it is an odd number I cannot. Therefore I shall keep and continue my multiplication by 3 still: And work by the ½ of the rest of the given figures or numbers: To wit 648: And first the ½ of 8 which is 4 multiplied into ● maketh 12 thereto join the 1 lb in mind, it maketh 1●: set down ● keep one. Then again multiply by 2 the ½ of 4 it maketh 6, and with 1 in mind it maketh 7. Then lastly take the ½ of 6 which is 3, saying 3 times 3, is 9: which 9 set down and so is the question answered as appeareth by the practice, and the examples following. At 3 s the yard what 6487 If one yard cost 5 s what 4769 At 7 s the elle what 6489 If one elle cost 9 s what 2807 At 11 s the pistolet what 8263 If one piece cost 13 s what 4629 But now note gentle Reader, when the given price falleth upon any odd number. As 3.5.7.9.11.13. &c: Then it is to be presupposed, that the given sum to be multiplied must be a sum made of even numbers, as 2.4.6.8.0 etc. else can not that question be wrought at one line or working. Providing always that it may bear an odd figure in the first place towards your right hand: as appeareth in these 6 Examples which last were wrought, and such like etc. which may bear an odd number for the price, and be done at one line or working very well. But if the given price be an odd number, and the sum to be multiplied odd numbers also: Then can it not be done at one working, but requireth the aid of 2 workings: for odd with odd will not agree, which notwithstanding to bring to pass. Take this for a general rule: First work for the even number, contained in that question, or given price, according as you have learned, And then afterwards for the one odd shilling, A general rule. take the ½ of the sum given to be multiplied, omitting the first pricked place, As was taught for the working of one shilling in my first rule of practice, And add those two together. And you shall have your desire. Example. At 3 s the yard what 7539 At 7 s the el what 7539 At 13 s the yard what 7534 And thus have I abbridged into these two Rules how to bring any number of s: whatsoever they be into pounds, with a brieffer method, than ever yet hath been published, which I commend unto thy friendly censure and judgements in the use of practice thereof. If one cost 6 s 5 d what 1231 At 14 s 2, d what 2825 At 16 s 4 d what 2531 At 3 s the Pistolet what 8325 At 7 s the crown what 6529 At 9 s the piece what 6567 These three last questions may seem some thing hard, yet are they easy enough, if you mark them well, if I should explain them, then are they too easy: therefore I leave them to whet the minds of the desirous. ●0. Rule. Item, when any one of the sums which is to be multiplied, is composed of many Denominations: and the given number but of one figure alone: Then shall you multiply all the denominations of the other sum, by the same one figure, beginning first with that sum which is least in value towards your right hand, and bring the producte of those d into s, and the producte of the s into lb▪ as by this example doth appear. 11. Rule. But if in any of the sums that are to be multiplied there be a broken number: First work for the whole according to the instructions that you have learned: and then take such part of the given price: as that broken number beareth in proportion to the price, as in the example: after you have wrought, for 3 s and for 6 d: then are you to take the ●/● of 36 d for the ½ yard: and add that to the sum: So adding all 3 productes together which maketh 43 lb— 2 s— 11 d the just price of 246 ½ Else: and thus must you do of all other. At 3 s 6 d the Ell what 246 ½ At 16 s 4 d the piece What If one piece cost What The Proof. If 12 pieces cost 50 lb· 2s· 6 d what one piece 12. Rule. Item, touching the manner how to understand the order of this proof, and others the like: first seek how many times 12 is contained in 50: maketh 4: resteth 2 lb which converted into shillings, and joined with the other 2 s, maketh 42 s: wherein is found 12 three times: resteth 6 s which turned into pence, putting thereto the 6 d in the first place, it maketh 76: wherein 12 is found 6 times, resteth 6 d, which containeth 12 but ½ a time, put that ½ to the 6 d: And then the solution is 4 lb— 3 s— 6 ½ as appeareth by the practice thereof. 13. Rule. Item, the like is to be done of any thing that is bought or sold after 5 score to the hundredth, or the quintal: As for example. If 100 lb cost 27 lb— 13 s— 4 d What one pound. I have wrought this at length for the aid of the young learner, because he should understand how all the Multiplication is set down. But to works it more neatly, it is by a little understanding ended thus. Item to the understanding of this and such like questions, the right down line is all the guide, which is pulled down close by 20, as you see in the example, where 27 lb— 13 s is reduced all into s: maketh 552. The 5 towards your left hand being sepated with the hanging or right down line, is the just number of shillings: that aunsweareth to the question: Nextly, 53 s is multiplied by 12 to reduce them to pence, putting to the 4 d: it yieldeth for the multiplication of the first figure two: 1 10: the one beyond the line towards the left hand: is 1 penny towards the rest of the price: then 53 also multiplied by 1 yieldeth 53: but the 5 behind the line towards the left hand, is also 5 d more, towards the price, which 1 and 5: I add together under the line: it maketh 6 d: So is there found now as appeareth by the Titles of s and d: 5 s 6 d. Finally, I come now on this side the line, towards the right hand: and under 12: I find first 10: and then 3: which added together maketh 40: under which 40, you must put the 100: and it maketh— 40/1●0 which abbreviated cometh to ⅖: So the just price of one pound after 5 score to the hundredth, maketh— 5 s— 6 ⅖ d. One example more, and so will I leave this rule. If 100 cost 10 ¾ d What Also the like may be done of our usual weights here in England (which is 112 lb for every hundredth weight) in case you know the Aliquot parts of a hundredth weight, which are these, 56 lb, 28 lb, 14 lb, and 7 lb: For 56 lb is the ½ of 112 lb, 28 lb is the ¼ of 112 lb, 14 lb is the ⅛, and 7 lb is the 1/16. Therefore for 56 lb, take the ½ of the sum of money that 112 lb weight is worth. For 28 lb take the ¼ of the sum of money that 112 lb weight is worth. For 14 lb, take the ⅛ of the sum that 112 lb is worth. And for 7 lb, the 1/16 of the sum of money that C. is worth. As for example: at 17 lb— 19 s the hundredth pounds weight, that is to say, the 112 lb, what shall three quarterus and 7 lb cost? The second Chapter entreateth of the Reduction of divers measures to others value by Rules of Practice. 13. Rule. NOW will I show a few examples of Practice in reducing of measures: as yards, Braces, pawns of jeanes etc. Much more I would have touched but that I fear the book will rise to too great a volume. In 864 else of Antwerp, how many yards of London? Item, in these and such like questions of Flemish measure to be brought into yards English: first take the ½ of the given number, as appeareth in the first example towards your left hand: Then take the ½ of that product: or the ¼ of the given number: and add those 2 products together, they shall be yards English, as by the example you may perceive. The second example towards your right hand is yet briefer than the first, whose work is this: take the ¼ of the delivered number, and that product, subtray out of the given number: and the rest showeth your desire: of these two ways use which you think best. The Proof. In 648 yards of London, how many of Antwerp Item for the understanding of this work: first take the ⅓ part of the yards of London, which found, add that ½ part, and the yards together, as appeareth by the practice: and the product showeth the of Antwerp. In 320 yards of London, how many of Antwerp maketh 426 2/● . Proof. Other Reductions. Item you shall understand, that for as much as 6 Braces of Milan make 5 of Antwerp, whereupon, according to the Rules of Practice, you may reduce the one into the other by the like Reasons aforesaid in taking the ⅙ part, and then subtraye the same to make of Antwerp: And again by the contrary in taking the ⅕ part, with adding the given number, to turn the to Braces, as for example. In 876 Braces how many else of Antwerp Thus appeareth, that 876 braces by practice, make 730 Flemmishe: which Flemish reduced into English yards by the Rules aforesaid, make 547 d yards. ●. Rule. So again upon the same first Question of Braces: I would know how many yeardes' English they make. After the rate that 100 Braces are worth 62 ½ yards. Item, to the understanding of this work and such like, first take the ½ of the given Braces: And after take the ¼ of that half: or the ⅛ of the given number, and add them together: And the product are also yards English. 18. Rule. Item 3 of Rochel make 5 at Lisbon: So likewise 3 at Lions make 5 at Antwerp. To work these and such like, double the of Lions, and the of Rochel: and from their products Subtray the ⅙: And the rest shall be of Antwerp, or of Lisbon. Example. In 63. of Lions how many Else of Antw. In 100 of Rochel how many of Lisb. Touching the proof or return of these & such like questions for a general Rule, you shall first take the ⅕ of the given number: And add that ⅕ and the given number together. And the ½ of that producte shall be your desire. Example. In 105. of Antw. how many of Lions In 166. ⅔ else of Lisbon how many else of Roch. The third Chapter treateth of the order and work of the Rule of three in broken Numbers, after the trade of Merchants, digressing something from Master Records, which is comprehended in 3 Rules. NOw that I have somewhat entreated of the Rules of Practice, I will give a few instructions after my simple order, for the working of the Rule of three in broken numbers, wherein I shall need to say the less, because I hope the studious Learner, that hath traveled any thing in the Ground of Arts, is not unfurnished of knowledge capable to understand me. But before I deliver any instructions for broken numbers, I will propone a question, which shall be wrought 3 sundry ways, thereby to show as it were 3 degrees of comparison: how far the Rule of three, in broken, for more speed of work, differeth from whole, which I rather set down for a view, that the studious herein may be more desirous to attain broken: leaving any more so discourse in Dialogue form: but only to give instructions, where need is: and in the rest to put forth the questions with their answers. My first question is this. If one yard cost 6 s— 8 d What are— 789 worth at that rate Here the product of the sum are pence according to the nature of the middle number. I answer 263 lb Here the producte of the sum is s, according to the nature of the middle number. Here the producte is pounds according to the title of the second number. Now that you have seen the 3 former virtues of the rule of three, whose products hath first brought forth d, next s, and lastly, pounds: I will deliver 3 notes in order following: And with them a dozen questions: that shall show the work of the Rule of three in broken numbers or Fractions. 1 The first four shall be sundry Questions of a Fraction coming in the second place. 2 The second four shall be of 2 Fractions coming in the second or third place. 3 The third four of Fractions in all three places. My first question is this. If one yard cost● me 3 s— 4 d what are 756 worth at that price. 1. Rule. In setting down the Question to perform the work I turn 4 d into the part of ● s. TO the ready working of this question, & all such other like, my first note is this: which take for a general rule, that when any one Fraction shall come either in the second or third place: that the Denominator of that Fraction or Fractions, must always be brought unto the number or Numerator of the first place: and thereby multiplied the one into the other. And this benefit is always gotten by the virtue of bringing the Denominator of the second numbers fraction unto the first place. For the fraction in the middle number is now released: and the producte that cometh of the Multiplication, is of the nature and like Denomination of the whole number in the second place which here are shillings. Whereupon now to work the question I bring 3 the Denominator of the Fraction in the second place, unto my first number 1 with a line set under 1 thus: and the 3 under it ⅓ thus▪ saying, once 3 is my Divisor: That done, reduce 3 ⅓ saying, 3 times 3 is 9, and the 1 over 3 make 10: my second number in the Rule of 3: by which 10 I do multiply my last number 756 as appeareth by the work thereof: And it yieldeth 7560 s my dividend. Then dividing 7560 by 3 my divisor, it yieldeth in Quotient 2520 s: which, as appeareth by the work maketh 126 pounds. At 3 s 4 d the yard what 756 yards If one yard of cotton cost 8 ¼ d what 859. This question was also wrought like the first and bringeth forth 29 lb— 10 s— 6 ¾ d the price of 859 yards. If 7 pound of any thing cost 3 lb 10 s What comes 987 lb to. Item I will work upon my first question now again: but altered into the proportion it beareth to a lb: for that 3 s 4 d is ⅙ of a pound. As soon as I have carried 6 the denominator of my middle number unto my first place, as afore hath been taught: I pull down one, the numerator of 6, with a line under 6/1 thus: And that one of custom I pull down in sight being the figure that I shall multiply my last number by▪ According to the tenor of the Rule of 3: And because one can neither multiply nor yet divide (though here it is set down in form of multiplication,) the producte of the multiplication, according unto the declaration of my first note, is converted into the title of my second number which here are pounds: now followeth the division performed by my divisor 6, to make an end of the question. Notes upon my second Rule for two Fractions coming in the second and third place. 2. Rule. My first question is this. If one ell cost 13 s 4 d: what half a quarter or ⅛ of an ell? Answer. first bring 13 s 4 d into the parts of a lb: which is ⅔ and then will the question stand thus. 1— ⅔— 1/●. Item for the performance of this work do as before was taught in the first Rule, first bring 3 the denominator of the second fraction unto your first number 1: setting a line under 1 thus: Saying once 3 is 3: that done bring 8 the denominator of the third fraction setting it under 3: and multiply them together, saying 3 times 8 makes 24: which 24 is your divisor: (Now have you done with the denominator 3. and also with the denominator 8.) Therefore you shall put a line under 3 thus. And the like line also under 8: setting or pulling down under them their own numerators, that is 2 under 3, and also 1 under 8, as appeareth in the example, which Numetors for a general rule are evermore to be pulled down, of custom in sight, to multiply the one by the other according to the tenor of the Rule of three. Then I multiply the one by the other saying once 2 is 2: which signifieth 2 lb: being of the Nature and like denomination of the middle number, which 2 lb is to be reduced into shillings, otherwise it can not be divided by my first number 24: Then dividing 40 by 24: the quotient bringeth forth 1 ⅔ s: So much is ⅛ of an ell worth after that rate. Otherwise although 2 pound could not be divided by 24: yet it might have been abbreved to 1/12 of a pound: which is worth 1 s 8 pence, as before. Second question. If one pound of any weight cost 13 shillings 4 pence: what are ⅞ of the pound worth after that rate: Answer. Reduce the 13 shillings 4 pence into the parts of a pound: which is ⅔: and then will the question stand thus. Item for the understanding of this, if you mark well the last example, this and the rest lieth open, and needeth small instruction. For as you did last. So now again bring the Denominator of the second and third fraction unto the first figure, 1, multiplying the one into the other which maketh also 24, as before your Divisor. Then making a line under 3 thus, and a line under 8 thus, And pulling down their Numerators under each figure, that is, 2 under 3, and 7 under 8, which as I said before for a general Rule I pull down of custom in sight: to be the two numbers that of duty ought to be multiplied together, which done I bring 2 being the lesser figure under 7: multiplying them together, it maketh 14: which are of the nature of the middle number. That is to wit pounds: which 14 cannot aptly be divided among 24: Therefore are reduced into s, as is plainly to be seen in the example. Then 280 s parted among 24: yieldeth for his quotient 11 s 8 d: your desire: and the just price of ⅞ of an ell. Otherwise, 14, though it could not be divided by 24: might by mediation or division in broken numbers have been divided or abbreviated to 7/12 which in effect being reduced to his known parts maketh 11 s 8 d as before. But my good will & meaning is to aid young beginners. Therefore have I reduced the 14 lb into s: which is the easier way. Now followeth the Example. The third example. If one yard cost me 2 s 6 d what 345 ●/4 yards. Answer. First put 6 d into the parts of a s: and then the question standeth thus. Iten to the ready understanding of this & all such like, according as before hath been declared: Bring the denominators of the second & third fractions unto the first place, multiplying them the one into the other, all which maketh 8. your divisor common. Then next reduce your second number, saying 2 times 2 is 4: and 1, is 5: as appeareth in the example. Lastly, reduce your third number 345 ¼ all into fourth's, and they make 1381: which 1381 is to be multiplied by 5, according to the tenor of the Rule of three: which done, maketh 6905 s: And divided by 8 your Divisor: yieldeth in Quotient 863 ⅛ s which maketh in pounds 43 lb— 3 s— 1 d ½: And so much are the 345 and ¼ yeardes' worth at that price. The same question wrought again but 2 s 6 d: is now converted into the parts of a lb, and standeth thus: Item, after I have brought here my second and third fraction unto my first place, & found 32 to be my devisor: having thus furnished my first place with all things unto him belonging (which is meant of bringing and multiplying the Denominators of the second and third Fractions into him) I then go in hand to see what is to do in my second place, where presently of custom I pull down my Numerator 1 under 8: being the figure in sight that shall multiply my third number. Then lastly I reduce 245 ¼ all into fourthes, as afore was practised, which maketh 1381, the which 1281 I am to multiply by one my second number: they are nothing increased, but by the Methamorph of my work, they are now 1381 pound: being of the nature of the middle number, as I have often showed you, which divided by 32 my divisor, yieldeth 42 pound and 5/32: which 5/32 of a lb reduced into known numbers, make 3 shillings 1 d ½ as before. Example. Now followeth 4 other questions which are in all: ● Rule. places, broken numbers or whole and broken together. Item first for the finding out of your divisor: you shall take this for a most certain and general rule. That you must multiply the numerator of the first number in the question by the denominator of the second: And also all that again by the denominator of the third: and the total thereof shall be your divisor. Secondarily for a general rule to find out your dividend, multiply the denominator of the first number by the numerator of the second, and the whole thereof by the denominator of the third. And the total thereof shall evermore be your dividend. Now for an example I propone this question thereby to make my meaning the more plain, and to show you as I have done in the rest the manner and order of the work. If 3/2 of any weight or measure cost ●/8 of a lb or 20 s what are 2/8 of the like weight or measure worth after that rate. Example. Item for the more plainer understanding hereof, and all other the like, in broken numbers: First you shall pull down 2 the Numerator of the first number or fraction, with a line under 3/2 thus: that done according as you have learned before, bring 6 the denominator of the second fraction, and set it under 2 multiplying the one into the other which maketh 12. Then lastly bring 8 the denominator of the third fraction: and set it under 12, multiplying that 12 by 8: which amounteth to 96: (or else for more brief multiply 6 by 8: saying 6 times 8 makes 48: which 48 set under 2, and multiply the one into the other, maketh 96 as before) And this 96 is the first number in the rule of three. That shall always for a most general rule be your divisor. Secondly to work for your dividend you shall as hath been sufficienttie declared afore, pull down 5 the numerator of your second fraction. And set it under 6 with a line under 6 thus. That done as you know, you are to pull down 3 the numerator of the third fraction and set it under 8. with a line under 8 thus multiplying the one into the other according to the tenor of the rule of 3: which maketh 15: them according to my note, forget not to bring the Denominator of the first fraction which is 3 under 15: And multiply them together, which maketh 45: which 45, is your diuidend. Which 45 are of the nature or denomination of the middle number, as I have oft taught you before And therefore are 45 lb, which aptly cannot be divided by 96: Therefore you shall reduce that 45 lb into s as you see is performed in the example, which amounteth to 900 s which divided by 96, your divisor it yieldeth 9 s & ●6/96 of a s which in lesser terms is 3/8 which 3/8 in money maketh 4 ¼ d: And so much will the aforesaid 3/8 cost, as by the work following shall appear. Example. Otherwise though 45 could not be divided by 96: yet by division in broken numbers it might have been abbreviated to 15/32 of a lb, which redused into known parts will make 9 s 4 ¼ d as before. Now my second example shall be the proof of this question. If ⅜ yards cost 15/32 of a lb: or 20 s what shall ⅔ cost? Answer. Work as was taught you before, and you shall have your desire. Here as appeareth by the work, the multiplications being ended 240 is to be divided by 288: which to some perchance may seem hard, yet notwithstanding is the work good: Therefore abbreviate 240 by 288: as you see here is practised: and the end of your abbreviation shall come to ⅚ your desire. Otherwise. Otherwise. The third question. If 2/4 elles cost 13 s 4 d what 156 ½ else? Answer. To work this question the shortest way: reduce 13 s 4 d into the parts of a lb which is ⅔. Then as you did afore after, you have set down the question, the numerator of the first Fraction 3, is pulled down under 4. and the denominators of the other 2 fractions multiplied into him which maketh 18 your divisor. Then the Numerator of the second fraction 2: is pulled down under 3 of custom now in sight ready to multiply my third number by: which is performed as soon as the last number 156 ½ is reduced into halves. Then lastly I multiply that product by 4: the denominator of my first fraction: if yieldeth 2504: which I divide by 18: And my quotient is 129 lb & 1/● of a lb remaining which is worth 2 s 2 ⅔ d: And so much will 156 ½ else cost as by the work following doth appear. The fourth Example. If 2 ½ cost 1 ⅔ lb what cometh 29 ¼ else to? Item to the workmanship of this question: first reduce your first number to one direct Numerator: in saying, ● times 2 is 4, and 1 is 5: Then bring the Multiplication of the Denominators of the second and third Fractions, which maketh 12: and multiply that 12 by 5 your first Numerator, it maketh 60, which is your divisor. Then the reduction of the second number, which is 5 multiplied by 117 the product of the last numbers reduction, make 585, which 585 yet resteth to be multiplied by 2, the denominator of the fraction in the first place yieldeth 1170: which divided by your divisor 60: yieldeth 19 lb— 10 s as appeareth by the work thereof. Thus having now touched the 12 questions whereof I first pretended, which with diligence and oft practise I trust are sufficient to aid the desirous, unto the working of any broken numbers, therefore I will propone a question or two more, and so end this treatise of the Rule of three in broken numbers. Example. If ⅔ of an ell cost 3 s 4 d what one ell. Proof. If one Ell cost 5 s what ⅔ If one piece of Kersey cost 2 lb— 5 s, and one yard thereof sold for 2 s— 4 d, how many yards long was the whole piece. A Grocer hath bought a bag of Almonds weighing 385 lb: tore 4 ¼ lb at 27 s— 6 d the C: The question is what they amount to in money. The fourth Chapter treateth of Lost and gain, in the trade of Merchandise. IF one yard cost 6 s— 8 d: and the same is sold again for 8 s— 6 d: the Question is, what is gained in 100 lb laying out on such commodity. Answer, the Rule of three direct, apply two manner of ways to do the same: the one is to say: If 6 ⅔ give 8 ½, what giveth 100? Multiply and divide, and look what your quotient bringeth forth above your laying out, is the neat gains, and the solution to your Question: If you follow the work, your Quotient will bring forth 127 lb— 10 s. Item, to work it the other way, which I take the nearest, seek the difference betwixt the just price, and the overprice, which is 1 s— 10 d: Then say by the Rule of three: If 6 ⅔ s gain 1 ⅚ s what shall 100 lb gain: Multiply and divide, and you shall find— 27 lb 10 s: and so much is gained in 100 lb laying out. Use which of these two ways you think good. The Proof. If a yard of cloth be delivered for 8 s— 6 d, whereupon was gained after the rate of 27 lb— 10 s in 100 lb laying out: The Question is what the yard cost at the first hand? Answer. Put your gains to 100 lb, all maketh 127 lb— 10 s: then say, if 127 lb— 10 s give but 100 lb, what giveth 8 ½ s: work, and you shall find 6 s— 8 d, the true solution to your question. Yet an other Branch or proof upon the same first Question. If one yard cost 6 s— 8 d: the question is, what price the same is to be sold again for, to gain 27 lb— 10 s in 100 lb laying out. Answer, say by the Rule of three, if 100 lb give 127 lb— 10 s: what giveth 6 ½ s: multiply and divide, and you shall find 8 s— 6 d, your true solution. If one Ell cost 7 s— 8 d, and sold again for 8 s— 6 d Question: what is gained in 20 lb laying out, in such commodities. Answer, Seek the difference betwixt the just price & the overprice, which is 10 d, and then apply the Rule of three, as before is taught: saying, if 8 ½ give ⅚ s, what giveth 20 lb: multiply and divide, and you shall find 2 lb— 3 11/22 s: And so much is gained in 20 lb laying out. The Proof. A Merchant hath bought Holland cloth at 8 s— 6 d the Ell, which proveth not to his expectation: whereupon he is content to lose 2 lb— 3 s— 11/23 s in 20 lb laying out: The question is what price ought to be made of the Ell abating this loss. Answer, do as before in gains hath been taught: putting 2 lb— 3 11/2● s to your 20 lb, all together maketh 22— 3 11/2● s: Then say by the Rule of three: If 22— 3 11/23 s give but 20 lb, what shall come of 8 ½ s: work and you shall find 7 s— 6 d, the just price that the Ell ought to be sold for after the rate of this loss. Thus you see as in company the Rule is appliable as well to gains and loss. If 20 ¼ yards cost 36 lb— 10 s how shall I sell the same again to gain ⅓ of the principal: or to make of 3.4: which is all one. Answer, by the Rule of three: if 3 do do give 4: what will 36 1/● give? Multiply & divide, and you shall find 48 lb— ⅔: Then say again, if 20 1/● yards do give 48 ⅔ pounds, as well principal as gain: what will one yard be worth at that price? Multiply and divide, and you shall find 2 lb— 8 9/4 ●/3 ●/3 s. If one Ell of Cloth cost me 8 s— 8 d: And afterwards I sell, 10 ½ thereof for 5 lb— 12 s— 4 d: I would know whether I win or lose: and how much upon the 100 lb of money. Answer. See first, at 8 s— 8 d the Ell, what 10 ½ else comes to, and you shall find 4 lb— 11 s: and I sold the same for 5 lb— 13 s— 4 d: So that I did gain upon the 10 ½ yeardes' 1— 2-4 d: Then if you would know how much is gained in the 100 lb, say by the rule of three: If 4 lb— 11 s did gain 22 ½ s what will 100 lb gain: Multiply and divide and you shall find— 24 lb— 10 s— 10 1●/10 d: And so much is gained in the 100 lb of money. If 12 ½ yards cost me 11 lb— 5 s: And I sell the yard again for 16 s: the question is, whether I do win or lose, and how much in or upon the pound of money. Answer. Look what the 12 ½ yards come to at 16 s the yard, and you shall find 10 lb: But they cost 11 lb— 5 s: So there is lost upon the whole 1 lb— 5 s: Then to know how much is lost in the pound: Say by the rule of 3: if 11 ¼ lb do lose 1 ¼ lb, what will 1 lb lose: Multiply and divide, and you shall find 6 ⅖ d: and so much is lost in the lb of money. If I sell the C. weight of any commodity, for 4 lb: whereupon I do lose after 10 lb in the 10● lb: I demand how much I shall lose or gain in the 100 lb, if in case I had sold the same for 4 lb— 10 s. Answer. Say if 90 lb yield 100LS, how much will 4 give? Multiply and divide, & you shall find 4 ●/9: Then say again, if 4 4/9 give me 4 ½ what will 10 come to? Multiply & divide, and you shall find 101 lb ¼ which is more than 10● lb by 1 lb— 5 s: And so much is gained in the 100 lb. A Merchant hath sold Currans for the sum of 430 lb: & he hath gained therein after 10 lb in the 100 lb: The question is, to know how much he gained in all? Answer, Say by the Rule of three: If 110 lb do gain 10 lb, what will 430 lb gain: Multiply & divide, and you shall find 39 lb- 1— 9 ●/11 d, And so much hath he gained in al. If one yard be worth 28 ½ s, for how much shall 10 yards be sold to gain after 8 lb— 6 s— 8 d in the 100 lb. Answer. First add 8 lb— 6 s— 8 d to 100: them say, if 100 lb do give 108 1/● for principal and gain, what will 28 ½ s principal yield? Multiply and divide, and you shall find 30 ⅞ s: Then say again by the Rule of three: If 1 yard do give 30 ⅞ s (which is as well the principal as the gain) what shall 10 yards give? Multiply and divide, and you shall find 15 lb— 8 s— 9 d: And for the same price shall the 10 yards be sold, for to gain after the rate of 8 lb— 6 s— 8 d upon the 100 lb. A Branch or Proof out of this Question. A Merchant hath sold Clothes for 15 lb— 8 s— 9 d, and he hath gained in the whole, the sum of 1 lb— 3 s— 9 d: The question is to know how much he hath gained in the 100 lb. Answer, to know this, first rebate the gains from the price, and there will remain 14 lb— 5 s— 0 d: Then say by the Rule of three direct, if 14 lb ¼ give me— 1 lb— 3 s— ¼ d: what will 100 lb give? Multiply and divide, and you shall find— 8 lb— 6 s— 8 d the effect desired: the proof is apparent in the question before. Yet an other Branch or Proof of the first Question. If— 10 yards be delivered for— 15 lb 8— 9 whereupon was gained after the rate of 8 lb— 6 s— 8 d upon the 100 lb: the question is, what the yard did cost at the first hand. Answer, first say by the Rule of three, if 10 with principal and gain yield 15 lb— 8 ¾, what shall 1 yield? Multiply and divide, and you shall find 30 ⅞ s: Then say again by the Rule of three: If 108 ⅓ principal and gain give but 100: what shall 30 ⅞ s of principal and gain yield: work, and you shall find 28 ½ s: And so much did the yard cost at the first penny. If one yard cost 36 s how much shall 12 yards be sold for to gain after the rate of 10 lb in the 100 lb. Answer, first say, if 100 give 110 lb principal and gain, what will 26 s give? Multiply and divide, and you shall find 39 ⅗ s Then say again by the rule of three: If 1 yard of principal and gain yield 39 ⅗ s, what shall 12 yards gain? Multiply, and divide, and you shall find 23 lb— 15 ⅕ s, which ⅕ s is 2 ⅖ d: And for the same price shall the 12 yards be sold, to gain after the rate of 10 in the 100 The Proof. If 12 yards be sold for 23 lb— 15 s— 2 ⅖ d whereupon is gained after 10 lb in the 100 lb: The question is, what the yard cost at the first penny? Answer. First say, if 12 give 23 lb— 15 ⅕ s, what 1 yard? Multiply and divide, and you shall find 39 ⅗ s: Then say again by the Rule of three: if 110 lb give but 100, what shall 39 ⅗ give: work and you shall find 36 s, the just price of the yard at the first hand. Iten, when one Merchant selleth wares to another, and he giveth to the buyer 1 lb— 6 s— 8 d upon the score, or 20 lb: The question is, how much shall the buyer gain upon the 100 lb after that rate. Answer. first add 1 lb— 6 s— 8 d unto 20 lb, and they are 21 ⅓: Then say, if 20 lb give 21 ⅓, what shall 100 give: multiply and divide, and you shall find 106 ●/3: So the buyer getteth after the rate of 6 ⅔ lb upon the 100 lb. Gentle Reader other necessary questions appertaining to Loss and Gain, you shall have in the seventh Chapter of this Treatise. The fifth Chapter entreateth of Loss and Gain upon Time, wrought by the double Rule of three: or by the Rule of three composed, which is contained in four special selected branches or questions of divers forms, each one of them springing from the first question, and each one of them also being a proof to other, etc. IF one yard cost me 2 s 8 d ready money: and after I sell the same again for 2 s 10 d to be paid for it at the end of 3 months: the question is, what I gain upon the 100 lb in 12 months? Answer. First say, if 2 ⅔ gain 1/● what shall 100 lb gain: Multiply, and divide, and you shall find— 6 ¼ lb: Then say again by the Rule of three: if 3 months gain 6 ¼ lb what shall 12 months gain: Work, and you shall find 25 lb: and so much shall I gain in 12 months after that rate. Iten you may also work it at one working by the first part of the Rule of 3 composed: saying, if 2 ⅔ s in 3 months do gain ⅙ of a s, which is 2 d, what will 100 lb gain in 12 months: which for thy further encouragement, the work of this one Example I will here put down, to verify that I affirmed when I delivered in the first part of this Ground of Arts, that this Rule, and so all others more rejoiceth in Broken than in Whole. If a yard be delivered for 2 s— 10 d: whereupon was gained after the rate of 25 lb in the 100 for 12 months: The question is now, what the yard cost at the first hand. Answer. first say, if 12 months gain 25 lb, what shall 3 months gain: and you shall find 6 ¼ lb: Then say again the second time, if 106 ¼ lb give but 100, what shall 2 ⅚ s give? work, and you shall find 2 ⅔ s, which is the just price that the yard cost at the first hand. If one yard of cloth cost me 2 s— 8 d ready money, for what term shall I sell the same again for 2 s 10 d: So that I might gain after the rate of 25 lb upon the 100 lb in 12 months. Answer. First say, if 2 ⅔ gain ⅙ what shall 100 lb gain: Multiply and divide, and you shall find 6 1/● lb. Then say again for the second work, if 25 lb become of 12 months, what shall come of 6 1/● lb: Work, and you shall find 3 months: the just term of time that the cloth ought to be delivered at 2 s 10 d, to gain 25 lb, upon the 100 lb in 12 months. If one yard cost me 2 s 8 d ready money for what price shall I sell the same again to be paid at the end of 3 months: So that I may gain after the rate of 25 lb in the 100 lb for 12 months? Answer. first say, if 12 gain 25 lb what shall 3 months gain? Multiply, and divide, and you shall find 6 ¼ lb: then say for the second work: if 100 lb gain 6 ¼ lb, what shall 2 ⅔ s gain? work, and you shall find 2 d, which must be added to 2 s— 8 d: and then it maketh 2 s— 10 d: and for that price must the yard be sold to gain after the rate of 25 in the 100 for 12 months, you may also work it by the first part of the rule of three composed, saying if 100 lb in 12 months do gain me ●5 lb? what shall 2 ⅔ s gain in ● months: work and you shall find ●/2● of a lb, which in lesser terms is 1/1●, and is in known money worth 2 d, as before. Many other of these questions I might here have delivered, but for fear the book would rise to too thick a volume, and so to make the price so much the dearer, whereby it might not be so partable to my Countrymen as I wish it. But these four, I have of purpose framed in this order, having relation one to an other: Assuring you, that what questions soever may be proponed within the compass of this rule, you shall find by one of these four to make a solution: And moreover, divers other are yet to be delivered: where the Creditor giveth divers days of payment, which can never be well wrought, nor yet understood, unless you can first find by Art, the just time that all those payments, how different soever they be, aught to be paid at once: whereupon first I think good here to give some instructions into such a Rule, for it is the only aid for the finishing of such questions as hereafter shall follow. The sixth Chapter entreateth of Rules of Payment, which is a right necessary Rule, & one of the chiefest handmaids that attendeth upon buying and selling, etc. Example. A Merchant doth owe a sum of money, whereof the ½ is to be paid at 6 months, the ½ at 8 months, and the rest at a year, If he would pay at one payment, the question is, what time ought to be given him. Answer I have omitted the quantity of the sum, for you shall understand, the Rule is appliable, and yieldeth a true solution to what sum soever shall be proponed. But now for order's sake in teaching, I do imagine the sum to be 60 lb: whereupon the manner of this work is to multiply the proportionate part of the money by the time, as in company: Then 20 being the first payment, and the ⅓ of 60, which ⅓ multiplied in broken numbers by 6: his time of payment, maketh 6/● which in whole numbers, as appeareth by the example in the margin, maketh 2 months: next 30, which is the ½ multiplied by his term 8, yieldeth 4 months, then the rest which is 10 lb must needs be abbreviated into the proportion it beareth to 60. which is ⅙: which ⅙ multiplied by his time 12 months produceth 12/6: maketh 2 months. All which added together as appeareth in the margente. maketh 8 months: which is the just time that all those payments ought to be paid at once. A Merchant hath 800 lb to pay, the ⅙ thereof ready money the ¼ at two months, the ½ at four months, and the rest at a year. The question is, if he would pay all at one payment, what time ought to be given him. Answer▪ The ready money is never multiplied: then ¼ multiplied by 2 months as you did before, maketh ½ then ½ by 4 produceth 2 months, as appeareth here in the margin: But now for the rest of the money you can not multiply it until you have sought what proportion it beareth to 800 lb. Therefore you must subtract the ready money: the ¼ and the 1/● out of the principal: The rest will be 66 ⅔ lb: which you must look what part it beareth to the principal, which you shall find to be 1/12: The same you must also multiply by his time 12 months. And it yieldeth 1 month: So all make 3 ½ months as appeareth in the margin. A Merchant is to pay 1200 lb: in ● terms, That is to wit, 400 lb. at 2 weeks And 100LS lb at 4 months, lastly 200 lb. at 5 months. The question is in what time, they ought to be paid at once. Answer. Proportionate the parts, and you ●hal find for 400 ⅓ and for 600 ½. And 200 ●s the ⅙ part: which multiplied by their times ●s before, and you shall have ⅔ weeks: more 〈◊〉 weeks, and lastly 3 ½ weeks, which together maketh 12 weeks, or 3 months your ●esire. A Merchant is to pay 600 lb in 6 terms, whereof 100 lb is paid present, more 300 lb at 20 days: And the rest at 5 months, accounting 30 days to a month. The question is, what time ought these payments to be paid at once. I answer two months. The seventh Chapter entreateth of buying and selling in the trade of Merchandise, wherein is taken part ready money, and diverse days of payments given for the rest: & what is won or lost in the 100 lb▪ forbearance for 12 months more or less according to the quantity of money, or proportion of time etc. A Merchant hath bought Satins which cost 8 s the yard ready money: And he selleth the same again to another man for 10 s the yard: But he giveth two days for the payment: That is to say three months for the one half, and five months for the other half: The question is to know how much the seller doth gain upon 100 lb in 12 months after that rate. Answer. Seek first by the Rules of payment, at what time those two payments ought to be paid at once, and you shall find 4 months, at which time the second Merchant ought to have paid the whole entire payment: And therefore say by the first part of the rule of three composed: If 8 s in 4 months do gain two shillings, what will 100 pound gain in twelve months multiply and divide and you shall find 75 lb as appeareth in the example, and so much doth the first merchant gain upon 100 lb in 12 months. A Merchant hath sold 50 clothes at 9 ½ lb the piece, to be paid the one ½ at 4 months: the ⅓ at 5 months, & the ⅙ at 7 months: And the sellers mind is to take no more but after 8 lb in the 100 for 12 months. The question is now what the first Merchant gaineth in the sale of these clothes after that rate. Answer. First look what the 50 clothes come to at that price: and you shall find 475 lb. Then secondly, according to your direction in the Rules of payment, seek at what time all the payments are to be performed at once. And you shall find 4 ⅚ months. Then thirdly say by the first part of the Rule of 3 composed: If 100 lb in 12 months gain 8 lb what will 475 lb gain in 4 ⅚ months: work and you shall find 13 lb and 7/36 of a lb: which is the neat gains that the first Merchant hath after the rate aforesaid: A Merchant hath bought Holland at 7 s 3 d the el ready money: And he selleth the same again, for 8 s 4 d the ell, to be paid ¼ part in ready money, more ⅓ parts at three months, and the rest at four months. The question is now to know how much the first Merchant doth gain● upon the 100 lb in 12 months after th● same rate. Answer. According to the direction deli●uered you in the rule of payment, the ready money is not to be multiplied: Then wor●king for the other 2 payments, to find ou● the true proportion at what time they ought to be paid at once, you shall find for ⅓ at months, ⅔ of a month: And the rest of the mo●ney which is 5/12 multiplied by his term 〈◊〉 months, yieldeth 1 ⅔ months: both which added together make 2 and ⅓ months: The● just time, that both the payments ought to be performed at once. And therefore say by the first part of the Rule of three composed, if 7 ¼ in 2 ⅓ months do gain 13/240 of a lb: what shall 100 lb gain in 12 months after that rate, work and you shall find 76 172/20● pounds. And so much doth he gain upon 10 pounde● in 12 months. A Merchant hath bought 30 clothes at 6 lb the piece for ready money: afterward he selleth 10 of them for 7 lb the piece, for three months term. And the other 20 he selleth for 8 lb the piece for four months term: The question is now, what he gaineth upon 100 lb in 12 months? Answer. first find the value of the 30 , which amount to 180 lb: Secondarily, seek what the ten pieces come to, at 7 lb: and what the 20 pieces come to at 8 lb the one comes to 70, and the other to 160: both which together make 230, which is 50 lb more than they cost: Thirdly, as I have taught you in the Rule of payment, proportionate the first & two prices, unto the proportion they bear unto 230: the producte of their two prices ● and you shall find 7/2● for the first, and ½ 6/● for the latter. Then four, multiply those parts, by their times: and you shall have 21/23 and 6●/2●: both which together maketh whole months, and 1●/23 of a month, which is the just time that both those payments are to be paid at once. Then say by the first part of the Rule of 3 composed: If 180 lb in 3 ●6/2● months do gain 50 lb, what shall 100 gain in 1 ● months'? multiply and divide and you shall find 90 10/51 lb. And so much doth he gain upon a 100 lb in 12 months. A Merchant hath bought Cinnamon which cost him 9 s the lb ready money: The question is now at what price he ought to sell the C weight: To wit 112 lb: to be paid the ¼ at 2 months, and the residue at the end of 3 months, so that he may gain after the rate of 9 lb upon a 100 lb for 12 months. Answer. Seek first by the Rules of payment what term both the payments ought to be paid at once, where the ¼ multiplied by his term 2 months maketh ½ months: Likewise the next payment which is ¾ multiplied by his term 3 months maketh 2 ¼ months: both which added together maketh 2 ●/4 months: which is the time, that both the payments ought to be paid at once. Then say by the Rule of 3: if 12 months do give me 10 lb what will 2 ●/4 months give? Multiply and divide and you shall find 2 ⅝ lb. Then say again by the rule of 3: If one pound cost me 9 s what will 112 pound cost: multiply and divide and you shall find 50 lb 8 s. Then say once again if 100 lb do give 102 ⅝ what will 50 ⅖ lb give? multiply and divide & you shall find 20 lb— 18— 8 13/25 d: And for that price ought I to sell 112 lb of Cinnamon to be paid at two several payments aforesaid: To gain thereby after the rate of 10 lb upon the 100 lb in 12 months. Brief Rules for our hundredth weight here at London which is after 11● lb for the C. Item who that multiplieth the pence that one pound weight is worth by 7: & divideth the product by 15: shall find how many pounds in money the 112 pound weight is worth. And contrariwise, he that multiplieth the pounds that 112 lb weight is worth by 1●. And divideth the product by 7: shall find how many pence the pound weight is worth. Example. At 10 d the pound weight, what is 112 lb weight worth? Answer. Multiply 10 by 7 and thereof cometh 70: the which divide by 15: and you shall find 4 ⅔ lb. And thus the 112 lb is worth 4 lb— 13— 4 d after the rate of 10 d the lb aforesaid. At 6 lb the 112 lb weight what is one lb worth? Answer. Multiply 6 lb by 15 and thereof cometh 90: the which divide by 7: And you shall find 12 d 6/7: So much is one pound worth when the 112 lb did cost 6 pounds. The eight Chapter entreateth of tars and allowances of merchandise sold by weight, and of losses and gains therein etc. AT 16 lb the 100 subtle what shall 895 lb subtle be worth in giving 4 lb weight upon every 100 for treat? Answer. Add 4 unto 100, and you shall have 104. Then say by the Rule of three if 104 be worth 10 lb what are 895 lb worth? Multiply and divide, and you shall find 237 lb ●13-10 2/13d. And so much shall the 895 lb weight be worth. Item at 3 s 4 d the pound weight what shall 754 ½ be worth in giving 4 lb weight upon every 100 for treat? Answer. See first by the Rule of three, what the 100 pound is worth, saying, if 1 cost 3 ⅓ s what 100? multiply and divide and you shall find 16 lb 2/●: Then add 4 unto 100 and they are 104: Then say again by the rule of three, if 104 be sold for 16 ⅔: for how much shall 754 ½ be sold for? Multiply and divide and you shall find 120 lb— 18-3 1●/53 d: And for so much shall the 754 ½ be sold for, at 3 s 4 d the pound in giving 4 upon the 100 Item if 100 lb be worth 36 s 8 d what shall 860 lb be worth in rebating 4 pound upon every 100: for tore and cloff? Answer Multiply 860 by 4: and thereof cometh 3440: the which divide by 100 and you shall have 34 2/● lb: abate 34 ⅖ from 800: and there will remain 825 ⅗: Then say by the rule of three. If 100 lb cost 36 ⅔ s what will 825 ⅖ cost after that rate? Multiply and divide: and you shall find 13-15 ⅕ s And so much shall the 800 cost in rebating 4 pound upon every 100: for tore and cloff. Item whether doth he lose more that giveth 4 lb upon the 100: or he that rebateth 4 lb upon the 100? Answer. First note that he that giveth 4 lb upon 100, giveth 104 for 100: And he which rebateth 4 lb upon the 100 giveth the 100 for 96: Therefore say by the Rule of three, if 104 be delivered for 100: for how much shall the 100 be delivered? multiply and divide and you shall find 96 2/12: and he which rebateth 4 in the 100, maketh but 96 of 100, so that he looseth 4 in the 100, And the other which giveth 4 upon the 100, looseth but 3 ●/1 1/● upon the 100: Thus you may see that he which rebateth 4 in the 100: looseth more by 11/12 in the 100 lb. than the other which gave 4 upon the 100: for tore and cloff. If 100 lb of any thing cost me 22 s 4 d: The question is how I shall sell the lb to gain after the rate of 10 lb upon the 100 pound. Answer Say by the rule of 3, if 100 lb give 110 lb. what shall 23 ½ s give? multiply & divide, and you shall find 1 lb 17/60: Then say again if 100 lb be worth 1 17/60 lb: what is one pound worth? multiply and divide, and you shall find 3 d 6/75: And so much is the pound worth in gaining 10 pound upon the 100 pound. Item a Grocer hath bought C weight of a commodity for 6 lb 10 s: The question is now to know how many pounds there of he shall sell for 33 s 4 d: to gain 20 shillings in the C. weight. Answer. Add 20 s unto 6 lb 10 s: and they make 7 lb 10 s Then say if 7 ½ yield me 112 lb, what shall 1 ⅔ lb yield? multiply and divide and you shall find 24 lb 8/9. And so many pound ought he to sell to gain 20 shillings in his C. weight. If one pound weight cost 3 s 4 d and I sell the same again for 4 s what is gained in 100 pound? Answer. You may say if 3 ⅓ s give 4 s what will 100 lb give? But then when you have sound you must subtract the product out of 100 lb, the rest is your neat gain: Or else to produce the neat gain in your work at the first: Then subtract the just price out of the over price, as I taught before in the first beginning of loss and gain: And your conclusion shall be all one, multiply & divide by which of the two ways you think good, and you shall find that he gaineth 16 lb 13 s 4 d in the 100 pound. Item if the pound weight which cost 4 s be sold again for 3 s 4 d: I demand what is lost in the 100 lb of money? Answer. Say if 4 s lose: ⅔ what shall 100 lb lose? Multiply and divide and you shall find 25 lb: and so much is lost upon the 100 lb of money. Item if C weight of any commodity cost 45 lb: And the buyer repenting would lose 5 lb in the 100 lb of money: I demand how the pound may be sold: his loss to be neither more nor less than after the rate aforesaid of 5 by the hundredth? Answer. By the Rule of three, if 100 lb lose 5 lb what shall 45 lb lose? Work and you shall find 2 ½ lb: which rebated from the principal 45: resteth 42 lb 15 s: Lastly say if 112 lb yield but 42-15 s: what 1 pound? multiply & divide, and you shall find 7 s 7 d 17/28. And so much is the pound worth after that loss. A Grocer hath bought 2 pieces of reasons: weighing 175 ½ lb 182 ¼ lb 191 lb: tore for each frail 2 ¼ lb: at 25 ½ s the C weight. The question is what they amount too in money? I answer 6 lb 8s ●/40. A Grocer hath bought 3 sacks of Almonds weighing 267 ½ lb tore 2 lb: 257 ½ lb tore 2 ½ lb 252 lb tore 3 lb at 2 s 10 ½ the pound what amount they too in money? I answer 110 lb— 11-7 ⅛ d. The ninth Chapter treateth of lengths and breadthes of Arras, and other clothes with other questions incident unto length and breadth. IF a piece of Arras be 7 ells and ¼ long: & 5 ells and ⅖ broad: how many else square doth the same piece contain? Answer. Multiply the length by the breadth that is to say 7 ¼ by 5 ⅔: And thereof will come 4● 11/12 elles: so many else square doth the same piece contain. Item more a piece of Arras doth contain 22 else square, And the same being in length 3 1/● elles I demand how many else in breadth the same piece doth contain? Answer Divide 22 else by 2 1/● and thereof cometh 6 10/13: So many else doth the same contain in breadth Item more a Merchant hath 3 ¼ else of Arras, at 1 ⅔ else broad which he will change with another man for a piece of Arras, that is 7/8 el square. The question is how many else of that squareness ought the first Merchant is to have. Answer. Multiply the first Merchants piece his length by his breadth, and you shall find it containeth 5 5/12 elles, which 5 5/12 else you shall divide by ⅞ and you shall find 6 4/21 else: and so many elles of that squareness, ought the latter Merchant to give the first. Item, a student hath bought 3 ●/2 yards of broad Cloth, at 7 quarters broad, to make a Gown: and should line the same throughout with Lamb, at a foot square each skin: the question is now how many skins he ought to have? Answer. Seek first the number of yards square that his cloth containeth, which to do, multiply 3 ½ his length, by 1 ¾ his breadth, and you shall find 6 ⅛ yards square: then say by the rule of three: if 1 yard square give 9 foot, what shall 6 ⅛? work, and you shall find 55 ⅛ skins. Item more, a Lawyer hath a rich piece of seeling come home which is 34 foot, and three inches long: and 7 foot and 2 ½ inches high: the joiner is to be paid by the yard square: the question is, how many yards this piece containeth? Answer. Multiply his length by his breadth, that is to wit, 24 ¼ foot by 7 5/24 foot and you shall find 174 77/96 foot square, which 174 you shall divide by 9 (for so many foot make a yard square) and you shall find 19 yards, 3 foot, and 77/96 of a foot: and so many yards doth this piece hold. Item, bought a piece of holland Cloth containing 36 else ⅓ Flemmishe: The question is, how many English it makes? Answer. You must note, that 5 Flemish do make but 3 English: Therefore say by the Rule of three: if 5 else Flemish make but 3 else english, how many English will 36 ⅓ Flemish make? Multiply and divide, and you shall find 21 ⅘: and so many English doth 36 ⅓ Flemish contain. The like is to be done of all others. Item more, I have bought 342 Flemish, of Arras work at two broad, Flemish, and I would line the same with Ell broad Canvas of English measure: The question is, how many English will serve my turn? Answer. For as much, as 3 elles English are worth 5 else Flemish: therefore put 3 else English into his square: in multiplying 3 by himself, which maketh 9: Likewise multiply the English ell which is 5 quarters every way into himself squarely, and you shall find 25: Then multiply 342 which is the length of the piece, by 2 which is the breadth, and thereof cometh 684: then say by the rule of 3, as before: if 25 else square of Flemmishe measure be worth 9 else square of English measure, what are 684 of Flemish measure? multiply and divide, and you shall find 246 6/25 elles English. The same is also wrought by the backer Rule of 3, in seeking the squares contained in the Flemish ell of 2 else broad (which are 18): and also in seeking the squares contained in the English ell (which are 25:) then say by the Rule of 3 backward: if 18 quarters require 34 else, what shall 25 quarters give? Multiply, & divide by the Rule of 3: Reverse, & you shall find as before 246 6/25 elles English. Item more, at 3 s 4 d the Flemmishe Ell what is the English Ell worth after the rate? Answer. Say, if 5 Flemish be worth 3 English, what is one Ell Flemish worth? Multiply, and divide, and you shall find ⅗ of an English Ell: Then say by the Rule of three, if ⅗ of an Ell be worth 3 ½ s, what is one ell English worth? multiply and divide, and you shall find 5 s 6 4/9 d? Item more, at 8 s 4 d the Flemish El square, what is the English El worth after that rate. Answer. Say by the aforesaid reason: if 25 Flemish square be worth 9 else square English, what is one Ell square Flemish worth? work, & you shall find 9/25 of a an Ell square English: Then say, if 9/25 of an English Ell be worth 8 ½ s, what is 1 square Ell worth? Multiply and divide, & you shall find 2● s— 1 7/9 d: and so much is the English Ell worth. Iten, more at 6 s 8 d the el square, what shall a piece of cloth cost that is 7 ½ long, and 3 ¼ broad? Answer. Multiply the breadth by the length, and you shall find 24 ⅜ else square: Then say by the Rule of three: if 1 Ell square costa 6 ⅔ s what 24 ●/8? Multiply and divide and you shall find: 8 lb— 2 s s6d: and so much the same piece of cloth shall cost. Item more, a Mercer sold 2 pieces of silk. To wit, 24 ¼, 24 ⅓, and 25 yards at 9 ¾ s the yard: and was glad to receive in part of payment again, a Cloth containing 34 ½ yards at 7 ⅔ s the yard: The question is now, what the Debtor is in the Creditors debt: Work, and you shall find, he oweth the Mercer 22 lb— 3 s— 2 ¼ d. The tenth Chapter entreateth of the reducing of the Pawns of Geanes into English yards. NOte, that 100 Pawns do make 26 yards: whereupon ● Pawns 11/13 do make 1 yard, and 1 Pawn after that rate and proportion is 13/5● of a yard. In 4563 Pawns of Geanes, how many yards English? Answer. Say by the Rule of three, if 100 Pawns do make 26 yards, what will 4563 Pawns make? Multiply, and divide, and you shall find 1186 yards 19/5●: so many yards do 4563 Pawns make. Otherwise, take some other number at your pleasure, as 10 Pawns, which is the 1/1● part of 100: then to find his proportion take the 1/10 part of 26, which is 2 ⅗: and then say also by the Rule of three: if 10 Pawns give 2 ●/5 yards, what will 456 Pawns give? work, and you shall find 1186 19/50 yards as before. More, at 2 s 6 d the Pawn of Geanes what will the English yard be worth after that rate? Answer. Say by the Rule of three, if 1●/5● of a yard cost 2 ½ s what one yard? Multiply, and divide, and you shall find 9 s— 7 5/13 d. More, if 346 ½ Pawns cost 30 lb— 13 s— 4 d sterling, what is that the English yard after the rate? Answer. say by the Rule of three: if 346 ½ Pawns cost 30 ⅔ lb, what are 3 11/13 Pawns worth (for so many Pawns make a yard:) Multiply, and divide, and you shall find 3804/27017 parts of a pound, which in known numbers is worth 2 s— 9 d— 20439/●0●77. The eleventh Chapter entreateth of Rules of Loane and Interest, with certain necessary Questions and Proofs incident thereunto, etc. Item, lent my friend 326 lb for 5 ½ months simply without any Interest, upon condition, to have the like courtesy again when I need: But when I came to borrow, he could spare me but 149 lb— 8 s— 4 d: the question is now how long time, I ought to have the use thereof, to countervail my friendship before time showed him? Answer. say by the Backer Rule of three: if 326 give 5 ½ months, what time will 149 lb 5/12 give? Multiply, and divide, and you shall stade 12 months: and so long time ought I to use his money. The Proof. Item, lent my friend 149 lb— 8 s— 4 d, for 12 months: The question is now, how much money he ought to lend me again for 5 ½ months to recompense my friendship showed him? Answer. Say by the backer or reverse Rule of 3: if 12 months give 149 5/12, what shall 5 ½ months give? work: and you shall find 326 lb: and so much ought he to lend me to requite my gentleness or good turn. Two other Branches yet more for Proof out of the same Questions. Item, lent my friend 149 lb— 8 s— 4 d— for 12 months, to have the like friendship again when I need: And coming to borrow of him he very courteously took me 326 lb (for that he could well then spare the same:) The question is now, how long I ought to occupy it, not usurping frienshippe, but in his due time to restore it again. Answer. Say by the Rule of three Reverse, if 149 5/12 give 12 months, what shall 326 lb give? Multiply, and divide, and you shall find, that at 5 ½ months term, I ought to restore it again. Proof. Item lent my friend 326 lb for 5 ½ months: The question is now, how many pounds he ought to lend me for 12 months to recompense the pleasure again. Answer. Work by the rule of three Reverse, as you have done before, and you shall find 149 lb— 8 s— 4 d. Again, four other selected questions of Loane and Interest, all out of one branch, and each one also a necessary Question, and a particular proof to other. ITem, lent my friend 430 lb at Interest for 3 months to receive after the rate of 8 lb in the 100 for 12 months: The question is, what the Interest cometh it? Answer. Say by the first part of the Rule of 5 numbers forward: if 100 lb in 12 months gain 8 lb, what shall 430 gain in ● months? Multiply the first by the second for your Divisor: and the other three the one into the other for your diuidend: and you shall find 8 lb— 12 s. Proof. Item a friend of mine received of me 8 lb— 12 s for the Interest and Use of 430 lb for 3 months term: the question is now, what he took in the 100 lb for 12 months after that rate. Answer. Say by the first part, or rule of 5 numbers forward: if 430 lb in 3 months did pay 8 lb— 12 s, what doth 100 lb in 12 months, take after that rate? work, and you shall find 8 lb: and so much he took upon the 100 lb for 12 months. A third question and Proof also wrought by the backer rule of 5 Numbers. Item, lente my friend 430 lb to receive for the Interest thereof: after the rate of 8 lb in the 100 lb for 12 months: The question is now, how long time my friend ought to have the use thereof, that it may be returned with 8 lb 12 s gains. Answer. Say by the Backer Rule of 5 numbers: if 100 lb in 12 months do gain 8 lb: how long time shall 430 lb be a gaining of 8 lb— 12 s? Multiply the first and the second into the last for your dividend: and the third and fourth multiply together for for your divisor: And then divide and you shall find 3 months: the just time that my friend ought to use it, to return it with 8 lb— 12 s gain. A fourth derived question out of this branch which is a Proof of this last, and also of the other two going before. ITem, how much money ought a Merchant to deliver after 8 lb in the 100 for 12 months that in 3 months I may gain 8 lb— 12 s. Answer. You may also if you please work it by the Golden Rule of three at twice, first saying if 3. months gain 8. lb. what 12 months gain? you shall find 34 lb: Then say again, if 8 lb— become of 100 lb what shall come of 34 lb— 8 s? work, and you shall find: the answer to the question, which is 430 lb: and so much aught the Merchant to deliver. But most briefly it is answered by the Backer Rule of 5 numbers, where I argue thus, saying: if 100 lb be 12 months a gaining of 8 lb: then but for 3 months term, only to take 8 lb— 12 s must needs be a good round sum: to work it, set your number thus, 100— 12— 8— 3— 8 ⅗: multiplying the first into the second: and also by 43 the product of the fift for your dividend: and the third and fourth together with 5 the Denominator of your Fraction, for your Divisor: then divide, and you shall find as before 430 lb: The true solution to your question. The twelfth Chapter treateth of the making of Factors which is taken in two sorts. THe first is when the estimation of the Factor, is taken upon the sending of the Merchant: as if the estimation of his person be 1/●, it is understood, that he shall have ¼ of the gain, and the merchant the other ¼. The other sort is when the estimation of his making is out of the sending of the Merchant: as if the order and agreement between them were such, that the Merchant shall put in 800 lb: and the Factor for his making shall have ¼: nevertheless he shall have but ⅕ of gain or profit: for the ¼ of 800 is 200 (for the estimation of his making) which with the 800 lb make 1000 lb; whereof the 200 lb is ⅕. A Merchant doth put in 800 lb into the hands of his Factor: under such condition, that the said Factor shall have ¼: And after certain time, they find in profit 124 lb— 6 s— 8 d I demand how much the Merchant shall have hereof, and how much ought the Factor to have? Answer. When the estimation of the Factor is out of the sending of the Merchant, it maketh for the Merchant for the Factor But if that his estimation be at the sending of the Merchant, than it maketh but for the Merchant for the Factor. For the Merchant is then to have ¾ and the Factor ¼. A Merchant doth put into the hands of his Factor 800 lb, & the Factor 400 lb to have the ⅕ of the profit: I demand now, for how much his person is esteemed: when the same is counted upon the sending of the Merchant. Answer. According to the tenor and order before prescribed in the first Rule: That is, if his estimate be ¼, he shall have the ●/4 of the gain. Therefore say by the Rule of 3 direct: If ¼ taken put in 400 lb what is the estimate, or putting in of ⅕ taking? Multiply, and divide, and you shall find 320 lb: and so much is the person of the Factor estimated. Otherwise. To find the estimation of the person of the Factor, you shall consider, that seeing it was agreed between them, that the Factor should take the ⅕: then the Merchant shall have the residue, which are ⅘: wherefore the gain of the Merchant, unto that of the Factor is in such proportion as 5 unto 4: Then if you will know the estimation of the person of the Factor: Say, if 5 give 4, what will 400 give? multiply and divide, and you shall find 320 lb: And so much is the person of the Factor esteemed to be worth. Other conditions than these aforesaid, may also be between merchants and Factors without respect, either of sending or not sending of the Merchant: where most commonly the estimation of the body of the Factor is in such proportion to the stock which the Merchant layeth in, as the gain of the said Factor is unto the gain of the merchant. As thus: if a Merchant do deliver into the hands of his Factor 400 lb & he to have half the profit: The person of the said Factor shall be esteemed to be worth 400 lb And if the Factor do take but ⅓ of the gain: he should have but ½ so much of the gain as the Merchant taketh: which must have ⅔: wherefore the person of the Factor is esteemed but the ½ of that which the merchant layeth in. That is to say 200 lb. And if the Factor did take the ⅖ of the gain, than the Merchant shall take the residue which are ⅖: wherefore the gain of the Merchant unto the Factor is then in such proportion as 3 unto 2: whereupon if you will then know the estimation of the person of the Factor. Say if 3 give 2: what shall 400 give? work and you shall find 266 ⅔ pounds: And so much is the person of the Factor esteemed to be worth. And if the merchant should deliver unto his Factor 400 pound and the Factor would lay in 80: and his person to the end he might have the ½ of the gain I demand how much shall his person be esteemed? Answer. Abate 80 from 400: and there will remain 320. And at so much shall his person be esteemed. A Merchant hath delivered unto his Factor 900 lb to govern in the trade of merchandise, upon condition that he shall have the 1/● of the gain, if any thing be gained: and also to bear the ⅓ of the loss if any thing be lost: Now I demand how much his person was esteemed at? Answer. Seeing that the Factor taketh the ●/3 of the gain, his person ought to be esteemed as much as ½ of the stock, which the Merchant layeth in. That is to say the ½ of 900 pound which is 450: The reason is, because ⅓ of the gain that the Factor taketh: is the ½ of the 2/● of the gain that the Merchant taketh, and so the Factor his person is esteemed to be worth 450 pound. A merchant hath delivered unto his Factor 600 pound. And the Factor layeth in 250 pound and his person. Now because he layeth in 250 pound, and his person: it is agreed between them, that he shall take the ⅖ of the gain. I demand for how much his person was esteemed? Answer. For as much as the Factor taketh ⅖ of the gain, he taketh ⅔ of that which the Merchant taketh: for ⅖ are the ⅔ of ⅗. And therefore the Factors laying in, aught to be 400 pound, which is ⅔ of 600 pound that the Merchant laid in: Then Subtract 250, which the Factor did lay in from 400 pound which should have been his whole stock: And there remaineth 150 pound: For the estimation of his person. More, a Merchant hath delivered unto his Factor 840 lb upon condition that the Factor shall have the gains of 160 lb as though he laid in so much ready money: I demand what portion of the gain the said Factor shall take? Answer See what part the 160 lb (which the Factor laid in) is of 950 which is the whole stock of their company: And you shall find 3/19: And such part of the gain shall the Factor take. But in case, that in making their covenants, it were so agreed between them: that the Factor should have the gain of 160 lb of the whole stock which the Merchant layeth in: That is to say of the 800 lb: then should the Factor take ⅕ of the gain: for 160 is ⅕ of 800 pound. The thirteenth Chapter entreateth of Rules of Barter, and exchanging of Merchandise, which is distinct into 7 Rules, with diverse other necessary Questions incident thereunto. 1. Rule. TWo merchants willing to change their merchandise, the one with the other: The one hath 24 broad clothes at 10 lb 10 s the piece: The other hath Mace, at 12 s the pound. The question is how many pound of Mace, he ought to give him for his clothes, to save himself harmless and be no loser. Answer. Seek first by the Rule of three, what the 24 clothes cost at 10 lb 10 s the piece. And you shall find 252 lb, Then to find the quantity of Mace: Say again by the rule of three, if 12 s buy one pound what shall 252 lb buy me? Work and you shall find: 420 lb of Mace. And so many pound ought he to give for his clothes. The proof. Two barter, the one hath 420 lb of Mace at 12 s the pound, to barter or change for broad clothes at 10 lb 10 s the piece, The question is how many broad clothes he ought to give for all his mace. Answer. First say if 1 cost 12 s what 400? you shall find 5040 s: then say again if 10 ½ lb give 1 cloth, what shall 5040 s give? work and you shall find 24 clothes: your desire. 2. Rule. Two change merchandise for merchandise the one hath Pepper at 2 s 4 d the pound: to sell for ready money: But in barter he will have no less than 3 s the pound. And the other hath Holland at 5 s 6. d the ell ready money. The question is now at what price he ought to deliliver the ell in barter to save himself harmless. Answer. say by the Rule of three direct: if 2 ⅓ s ready money give 3 s in barter what shall 5 ½ s give in barter? you shall find 7 1/14 s: and at that price ought the second merchant to sell his holland in barter. The proof. Two barter the one hath Holland at 5 s 6 d in the ell to sell for ready money: And in barter he will have 7 1/14 s: the other hath Pepper at 2 s 4 d the lb to sell for ready money. The question is now how he ought to sell it in barter. Answer. Say by the Rule of three direct if 5 ½ ready money give 7 1/14 s in barter, what ought 2 ⅓ s to take in barter? multiply and divide and you shall find 3 shillings your desire. 3. Rule. Two barter, the one hath cloth of Arras at 30 shillings the ell ready money: but in barter he will have 35 ½ shillings. And the other hath white wines, which he delivered in barter for 16 lb for a Tun. The question is now what his wines cost the Tun in ready money. Answer. Say by the rule of three direct, if 35 ½ s in barter give but 30 ready money: what did 16 lb in barter cost? Work and you shall find 13 lb— 10 ●●/●● s: and so much cost his wines for a Tun ready money. The proof. Two barter merchandise for merchandise. The one hath wines white: at 13 lb 10 10/●1 s the Tun to sell for ready money: But in barter he delivered it for 16 lb. The other to make his match good and save himself harmless: delivereth Arras at 35 ½ s the ell: The question is now what an ell of his Arras cost in ready money. Answer. say by the Rule of three direct if 16 lb in barter give but 13— 10 30/71 s in ready money: what shall 35 ½ s yield in barter? work and you shall find: 30 shillings your desire. 4. Rule. Two barter, the one hath Carsies at 14 lb the piece, ready money: But in barter he will have 18 lb. And yet he will have the ⅓ part of his overprice in ready money: And the other hath Ginger at 8 groats the puund to sell for ready money. The question is how he ought to deliver the Ginger by the lb in barter to save himself harmless and make the barter equal. Answer Item, for the working of this question and such other the like, you must understand, if the party overselling his wares require to have also some portion in ready money: as etc. Then shall you rebate the same demanded part whatsoever it be from the over price And also from the just price. And those two numbers that shall remain after the subtraction is made, shall be the two first numbers in the Rule of three. And the just price of the same merchandise shall be the third number, which by the operation of the Rule of three direct shall yield you a true solution: how and at what price, you shall oversel that your merchandise to save yourself harmless, and make the barter equal. Example. Take the ⅓ (of eighteen) which is the overprice of his cloth: which ⅓ of eighteen: is six, which as appeareth here in the margin you must subtract from 18: there resteth 12. And also abate it from 14 which is the just price of the cloth: and there remaineth 8: which 8 and 12, are the two first numbers in the Rule of three. Then take 8 groats or 2 ●/1 shillings for the third number. Then say by the Rule of three direct: if 8 s give 12 s what shall 2 ⅔ s give? Multiply and divide, and you shall find 4 s: And for so much shall the second Merchant sell his Ginger, or his commodity in barter, to balance the same equal. The proof. Two barter, the one hath Fine Carsies at 14 lb the piece ready money: But in barter he will have 18 lb. And yet he will have the ⅓ part of his overprice in ready money: And the other hath Ginger, which he having cunning enough to make the barter equal, delivered for 4 s the pound. The question is now what his Ginger cost him in ready money. Answer. After you have made the subtraction, abating 6, the ⅓ part of 18: both from 18 and 14: as before was taught you, then will there remain 8: and 12 for your two first numbers in the Rule of three. Then say, if twelve give but eight: what shall come of 4 the overprice of the pound of Ginger? Multiply and divide, and you shall find 2 s 8 d your desire. Two merchants barter Merchandise for merchandise, the one hath Denshire whits at 7 lb— 13 s 4 d the piece ready money: but in barter he doth them away for 8 lb— 3 s— 4d. And yet he will have the ⅓ part of his overprice in ready money. And the other hath Cottons at 3 lb the piece ready money. The question is now at what price he ought to sell or exchange his Cottons in barter to save himself harmless, and make the barter equal. Answer. First seek ⅓ part of 8 lb— 3 s— 4 d: which is 2 lb— 14— 5 ⅓ d which rebated from 8— 3— 4 d, there resteth as appeareth by the example above said 5— 8— 10 2/2 d, which ⅓ of 8— 3— 4 d also rebated from 7— 13— 4 d there resteth 4 18— 10 ⅔ the two first numbers in the Rule of three. And the 3 lb which is the neat price of the piece of Cotten is the third number. Then say by the Rule of three direct: as was taught before: if 4— 18— 10 ⅔ d give 5-8— 10 ⅔ d what shall 3 lb give? multiple and divide and you shall find 3 lb— 6 s— 1 91/292 d the just price that he ought to deliver his Cottons in barter. 5. Rule. Two merchants will change merchandise for Merchandise, the one hath Carsies at 40 s the piece to sell for ready money: And in barter he will sell them for 56 s 8 d and he will gain after 10 lb upon the 100 lb: And yet he will have the ½ of his overprice in ready money: The other hath Flax, at 2 d the pound ready money: The question is now how he shall sell the pound of his Flax in barter. Answer. See first at 10 lb upon the 100 lb what the 56 ⅔ s cometh to, in saying by the Rule of three direct, if 100 lb give 110 lb what 55 ⅔? multiply and divide and you shall find 3 lb— 2 s— 4 d of which the ●/2 that he demandeth in ready money, is, 1 lb— 11 s— 2 d: the same 31 s 2 d abated from 40 s, and also from 56 s 8 d: there will remain 8 shillings 10 pence: and 25 shillings 6 pence, for the two first numbers in the Rule of three: And 3 pence the price of the lb of flax for the third number. Then Multiply and divide, and you shall find, 9 ●8/5●d. And for so much shall he sell the pound of flax in barter. 6. Rule. Two are willing to exchange merchandise: the one hath Norwiche Grograines at 35 shillings the piece ready money: And in barter he will have 30 shillings, and he will have the ¼ part of his over price in ready money: The other hath Norwich Stockings at 40 shillings the dozen to sell for ready money: But in as much as the first merchants Grograins are no better, he would deliver them so to balance the barter, that he may gain after 10 lb in the 100 lb, The question is now how he shall sell his hose the dozen in barter. Answer. Say, if 100 give 110, what shall 40 s give, which is the just price of the dozen of stockings? multiply, and divide, and you shall find 44 s: Then take the ¼ of 30 s which is 7 s— 6 d: And subtract it from 25 s, and also from 30 s: And there will remain 17 s— 6— and 22 s— 6 d, for the two first numbers in the Rule of three: and 44 s which is the just price (with his gain in the dozen of stockings) for the third number: Then multiply and divide, and you shall find 56 s— 6 6/7: and for so much he is to sell his dozen of stockings in barter. 7. Rule. Two Merchants will change their merchandise one with the other: the one hath 720 of Cambric at 5 s the Ell to sell for ready money: but in barter he requireth 6 s— 8 d: And yet notwithstanding he loseth by it after 10 lb upon the 100 lb: whereupon he requireth the ½ of his over price in ready money: And the other Merchant having skill enough to make the barter equal delivereth English Saffron at 30 s the lb: The question is now what his Saffron cost the pound in ready money. Answer. You must first seek what is lost upon the 100 lb, which to do you may say if you please, if 100 lb lose 10? what shall 6 ⅔ s lose? work and you shall find ⅔ shillings (or eight pence) which must be rebated from 6 s 8 d: So resteth 6 shillings still: or you may say if 100 lb give me but 90 lb what shall 6 s 8 d give? Work this way either and you shall find also as before, directly in your quotient 6 s: your desire. Then are you next to cast up what the 720 else of Cambric cometh to at 6 s 8 d the el and you shall find 240 pound: the ½ whereof the Cambric Merchant will have in ready money (which is 120 pound) nextly you must cast what the Cambric cometh to after his loss in the 100 lb which as you found is but 6 s an ell: and you shall find 216 lb: now must you subtract (his ready money which is 120 lb) out of 240 lb: & also of 216 lb. And there will remain 120 lb & 96 lb for your two first numbers in the Rule of three, and 30 s the overprice of your Saffron for the third number. Then multiply and divide, and you shall find 24 s: and so much did his Saffron cost in ready money. Two Merchants barter, the one hath 50 Clothes, to put away for ready money at 11 lb the Cloth: and in barter putteth them away for 12 lb: taking holland Cloth at 20 d the Flemish Ell, which was worth no more but 18 d: The question is now, what Holland payeth for the Cloth: and what he winneth or looseth by the bargain? Answer. 50 Clothes at 11 lb the Cloth, cometh to 550 lb: and put away at 12 the piece, maketh 600 lb: Then to find what Holland payeth for the Cloth: Say by the Rule of three direct, if 20 d buy 1 Ell: what 600 lb? work, and you shall find 7200 : Now to find the estate of his gain or loss, you must seek what his 7200 cometh to at 18 d the Ell: work by the Rule of Proportion direct, and you shall find 540 lb, which is not so much as his Clothes were worth in ready money, by 10 lb: and so much lost the first Merchant by his exchange. A Venetian hath in London 100 pieces of Silk, to put away for ready money at 3 lb the piece. But in Barter he delivereth them for 4 lb the piece, taking wools of a Fell-monger at 7 lb 10 s the C. weight which was worth no more but 6 lb the C. in ready money: The question is now, what wools payeth for the Silks, and which of them winneth or looseth by the barter. Answer. 100 pieces of Silks at 3 lb, is 300 lb: and at 4 lb is 400 lb: Then to find what wools payeth for the Silk: Say by the rule of three direct: If 7 ½ d buy me 1 C. weight, what 400 lb? work, and find 53 ⅔ C. weight of wool: Now to find the estate of their gain or loss, cast up his wool at 6 lb the C. (for so much they were worth ready money) and you shall find 320 lb, which is 20 lb more than the Silks were to be sold for in ready money: whereupon the Venetian gained 20 lb by the Barter. A Merchant hath 53 ⅓ C. weight of wool at 6 lb the C. to sell for ready mony● but in barter he will have 7 lb— 10 s: and an other doth barter with him for Silks which are worth 3 pounds a piece ready money: The question is now, how he ought to deliver his Silks the piece in barter? and how many payeth for the wools. Answer. Say by the Rule of Proportion, or the rule of three direct: If 6 lb for a C. weight ready money yield me 7 lb— 10 s what will 2 lb yield, which is the just price of a piece of Silk in barter? To make the truck equal: work, and find 3 lb— 15 s the price of a piece of Silk in barter: then say, if 3 lb— 15 s require 1 piece of Silk, how many pieces of Silk are bought with 400 lb, which is the value of the 53 ½ C. weight of wool at 7 lb— 10 s: work by the Rule of three direct, and you shall find 106 pieces of Silk, and ⅔ of a piece: and so many pieces of Silk payeth for the wools, and neither party hath advantage of other. Two Merchants will change merchandise the one with the other: The one of them hath beer, at 6 s 8 d the Barrel to sell for ready money: but in barter he will sell the barrel for 8 s: and yet he will gain moreover after 10 lb upon the 100 lb: and the other hath white Spanish wool at 20 s the Rove to sell for ready money: the question is now, how he shall sell the Rove of wool in barter? Answer. say if 6 ⅔ s which is the just price of the barrel of Beer be sold in barter for 8 s, for how much shall 20 s (which is the just price of the Rove of wool) be sold in barter? work by the Rule of three direct, and you shall find 24 s: Then for because the first Merchant will gain after 10 lb upon the 100 lb, he maketh of his 100 lb— 110 lb: And therefore say by the Rule of three, if the second Merchant of 110 lb do make but 100 lb, how much shall he make of 24 s? Multiply, and divide, and you shall find 21 s— 9 d 9/11 of a penny: And for so much shall he sell the Rove of wool in barter. Two Merchants will change their commodities the one with the other: The one of them hath white Paper at 4 s the resme to sell for ready money: And in barter he will do it away for 5 s: and yet he will gain moreover, after the rate of 10 lb upon the 100 lb: and the other hath Mace at 14 s— 6 d the pound weight, to sell in barter: Now I demand what the pound did cost in ready money? Answer. Say, if 5 s which is the over price of the paper in barter be come of 4 s the just price, of how much shall come 14 s ½, which is the surprice of the pound of Mace in barter? Multiply, and divide, and you shall find 11 s— 10 d: Then for because the first Merchant of Paper will gain after the rate of 10 upon the 100: Say, if 100 do give 110, what shall 11 ⅚ s give? work, and you shall find 13 s— 0 d ⅕: and so much did the pound of Mace cost in ready money. The fourteenth Chapter entreateth of Exchanging of money from one place to an other. Exchange is no other thing then to take or receive money in one City, to render or pay the value thereof in an other City, or else to give money in one place, and receive the value thereof in an other, at term of certain days, months, or fairs, according to the diversity of the place. But this practice chief consisteth in the knowledge of the Money or Coins in divers places, of which for thy benefit, (after a few examples given to the Introduction to this work) I will set down by certain notes of diversity of the common and usual Coins in most places of Christendom for traffic. And first I will begin at Antwerp, where they use to make their Accounts by Deniers de gros: that is to say by pence Flemish, whereof 12 do make 1 s Flemish, and 20 s do make 1 lb de Gros. Item, a merchant delivered at Antwerp 400 pounds Flemmishe, to receive in London, 20 s sterling for every 23 s— 4 d Flemmishe: The question is now, how much sterling money is to be received at London for the said 400 pounds Flemish? Answer. say by the Rule of three, if 23 ⅓ Flemmishe give 20 s sterling, what 400 pounds Flemmishe? work, and you shall find 342 lb— 17 s— 1 5/7 d: and so much sterling shall I receive in London for the 400 lb Flemmishe. Otherwise also wrought by Rules of Practice in taking the 1/7 of the Flemish money delivered: and abating the same from the principal, the rest is English money sterling, as before. A Merchant at London delivereth 200 pound sterling for Antwerp at 23 s 5 d Flemmishe the pound sterling: The question is, how much he must receive at Antwerp. Answer. Say by the Rule of three, if 1 lb sterling give 23 s— 5 d Flemish, what 200 lb sterling? work, and you shall find 234 lb— 3 s 4 d: So many pounds Fleshmish shall he receive at Antwerp for the said 200 lb sterling. Otherwise also by Practice. In London 200 lb sterling is delivered by exchange, for Antwerp at 23 s— 9 d Flemmishe the lb sterling: The question is, what rate the Flemmishe money ought to be returned to gain 4 lb upon the 100 lb sterling at London? Answer. First say by the rule of 3 direct: if 1 lb sterling give 23 2/4 Flemmishe, what 200 lb sterling? Multiply, and divide, and you shall find 237 lb— 10 s: The which to return to gain 8 lb sterling in London. Say by the Backer Rule, if 200 lb sterling require the exchange 23 s— 9 d Flemish, what the exchange to make 208 lb styrling? work by the Rule, and find 22 s— 10 d 1/26 d Flemish, the effect in the question required. If I take up money at Antwerp after 19 s— 4 d Flemish, to pay for the same at London 20 s sterling: and when the day of payment is come, I am forced to return the same money again in London to pay my Bill of Exchange: So that for 20 s which I take up here at London, I must pay 19 s— 6 d at Antwerp, I demand whether I do win or lose: and how much in or upon the 100 lb of money? Answer. Say by the rule of three: If 19 ½ give 19 ⅓ what will 100 lb give? Multiply, and divide, and you shall find 99 lb— 2 s 106/117 which being abated from 100 lb, there will remain 9 s 11/117: and so much do I lose upon the 100 lb of money. If I take up at London 20 s sterling to pay at Antwerp 22 s 4 d: and when the day of payment is come my Factor is constrained to take up money again at Antwerp, wherewith to pay the foresaid sum: and there he doth receive 23 s 4 d Flemish, forthwith I must pay 20 s at London: the question is now, whether I do win or lose, and how much upon the 100 lb of money after that rate. Answer. Say by the rule of Proportion: if 22 ⅓ s give 23 ⅓ s, what will 100 lb give? Multiply, and divide, and you shall find 104 lb— 9 111/201 s: from the which abate 100 lb, and there will remain 4 lb 9 5●1/201 s, and so much is there gained upon the 100 lb of money. In Antwerp is delivered 200 lb Flemish by exchange for London, at 20 s sterling for every 23 s 4 d Flemmishe: The question is, at what rate the same is to be returned to gain 5 lb upon the 100 lb Flemish in Antwerp. Answer. First say by the rule of three: if 23 ⅓ Flemish give 20 s, what shall 200 lb give? work, and you shall find 171 lb 8 s 6 6/7 d: Then say again by the Rule of three direct, if 171 lb— 8 s— 6 6/7 sterling give me 210 lb flemish, what shall 20 s sterling give? work and you shall find 24 s— 6 d Flemish. And at the same rate ought the same to be returned at Antwerp to gain 10 lb upon the 100 Flemish. A Merchant of Antwerp delivereth 234 lb— 3 s— 4 d Flemish, to receive at London 200 lb sterling: the question is now, how the Exchange goeth after this rate. Answer. Say by the Rule of 3 direct, if 200 give 20, what giveth 234 ⅙? Multiply, and divide, and you shall find 23 s 5 d and for so much goeth the Exchange. Item, the Exchange from London into France, is not like as it is in Flanders, but is delivered by the French Crown, which is worth 50 soulx Turnois the piece. Whereupon also you must note, that in France they make their accounts by Franckes, Soulx and Deniers Tournois, whereof 12 Deniers maketh 1 Soulx Tournois, and 20 soulx maketh 1 lb Tournois, which they call a Liure or Franc. But the merchants to make their accounts do use French crowns, which is currant among them for 51 soulx Tournois: But by exchange it is otherwise, for it is delivered but for 50 Soulx Tournois the Crown, or as the taker up of the money can agree with the deliverer. And note that this 🜄 Character representeth the Crown by exchange, and is ever 50 soulx Tournois or French money. A Merchant delivereth in London 240 lb sterling after 5 s— 6 d sterling the Crown, to receive at Paris 50 soulx Tournois for every Crown, I demand how much Tournois or French money payeth the bills for the said 240 lb sterling. Answer. say by the Rule of three, if 5 ½ s sterling give me 50 s Tournois, what shall 240 lb sterling give? Reduce the pounds into shillings: then multiply and divide, and you shall find 2181 Livers— 16 soulx,— 4 Deniers, and 4/11 Tournois, and so much payeth the bills at Paris for the said 240 lb sterling. A Merchant delivereth in Rouen, or elsewhere in France, 1430 lb, or Francs: the which Francke or lb is 20 soulx or pound Tournois, to receive in London 6 s 4 d sterling for every 🜄 of 50 soulxe Tournois: The question is how much sterling money I ought to receive at London for my 1430 pound Turnois? Answer. Say, if 2 ½ lb give me 6 ⅓ what will 1430 give me? work, and you shall find 3622 s ⅔ sterling, which maketh 181 lb 2 s— 8 d: and so much money is to be received at London for the said 1430 Livers Tournois, after 6 s— 4 d for every crown of 50, soulxe. In London is delivered 200 lb sterling by exchange for Paris, at 5 s— 9 d the 🜄 of 50 soulx Tou●nois: the question is at what price, the said Crown is to be returned to gain 6 lb upon the 100 lb sterling at London. Answer. first say by the Rule of three direct, if 5 ¾ sterling give 50 soulx Turnois, what shall 200 lb sterling give? work, and you shall find 1739 Franckes, or Livers, 2 soulx 14/2●. Then the which to return and gain ● lb upon the 100 lb in London: Say by the Rule of three direct: if 17●9 Frank's 2 soulx 14/23 yield 1 lb. what the 🜄 of 50 soulx? work, and find 6 s— 1 d 7/50 the effect required in the question. A Merchant delivered in London 160 lb sterling, to receive in biscay for every 5 s— 6 d 1 Duckate of 374 Marueides, the question is, how many Marueides I ought to receive at Biskate. Answer. Say, if 5 ½ s sterling give 374 Marueides, what shall 100 lb sterling give? Multiply, and divide, and you shall find 217000 Marueides, and so many I ought to receive at Biskie for my 160 lb sterling. A Merchant delivereth in Bayon, 20000 Marueides. to receive in London 5 s— 8 d sterling, for every Duckate of 374 Marueides: the question is now how much sterling money payeth the Bills of Exchange for the said 20000 Marueides. Answer. Say, if 374 Marueides make 1 Duckate, what 20000 Marueides? Multiply, and divide, and find 106 Ducats 178/187. Then say again, if 1 Duckate give 5 2/● s: what giveth 106 178/187 Ducats? work, and find 30 lb— 6 s and 34/56● s which is worth ●●●/561 parts of a penny. Otherwise it is wrought more briefer at one working, as in the last question before, in considering, that 5 s— 8 d containeth 1 Duckate, or 374 Marueides. Therefore say by the rule of 3: if 374 Marueides give 5 ⅔ s what 40000 Marueides: work, and you shall also find in your quotients: 30 lb— 6 s 34/561: and so many pound sterling is to be received for the 40000. ducats. In London 200 lb delivered by Exchange for Vigo 374 Marueids the Duckate, of 5 s— 10 d sterling, maketh 256457 1/7 Marueides: the which io return and gain 10 lb upon the 100 lb in London: Say by the rule of three direct, if 220 lb require 256457 1/● Marueides, what 5 s— 10 d? work, and find 340 Merueides, prizes of every Duckate in return, which is the effect in the question required. These may seem sufficient for instructions. NOtwithstanding for thy further aid and benefit hereafter followeth 6 special and most brief Rules of Practice for English, French, and Flemish money. 1 teacheth how to turn Flemish to English sterling. 2 teacheth how to turn English sterling to Flemish. 3 teacheth how to turn Flemish to French. 4 teacheth how to turn French into Flemish. 5 teacheth how to turn sterling into French. 6 teacheth & lastly, how to turn French into sterling. The fifteenth Chapter entreateth of the said 6 Rules of brevity, and of valuation of English, Flemish, and French money, and how each of them may easily be brought to others value. How briefly to reduce lb, s, and d Flemish, into lb, s, and d English, Sterling. IT is to be noted, that 7 pound Flemish maketh but 6 lb. sterling, 7 s flemish, maketh 6 s sterl. and 7 d flem. 6 d sterl. So that 7 yieldeth but 6. Wherein is evident, that there is lost 1/7, (if it may be so called) when it is reduced into English money. Wherefore to know how much 233 lb. 13 s, 4 d, flemish maketh English, you must subtract from it 1/7, beginning with the pounds, etc. and that which resteth after this subtraction is the sum required: so that 233 lb, 13 s, 3 d flem. maketh 200 lb, 5 s, 8 4/7d. sterling. Example. An other Example. To reduce lb s. andd. star. into lb s d flem. Note that a lb sterling, maketh 1 lb, 3 s, 4 d flem. that is 1 lb ⅙: 1 s star. maketh 1 s ⅙ flem. and 1 star. maketh 1 ⅙ Flem. So that there is gained (if it may so be called) ⅙ of the sum being thus reduced to Flem. For of 6/6 is made 7/6▪ which is 1 whole, and ⅙. Then to know how much 237 lb 7 s 6 d star. maketh Flem. subtract from your star. the ⅙ of the whole sum. and add it to the same sum, and it maketh 276 lb 18 s 9 d, which is the sum required. Example. An other example. Ye shall note, that the equality of Flemish and French money is this, that is to say, the lb Flemish, maketh 7 lb ⅕ French or Turnois 1 s Flemish maketh 7 s ⅕ French, & a great Flemish, maketh 7 d ⅕ French. Wherefore to know how much 143 lb 4 s 9 d Flemish maketh French. Ye must multiply the whole number twice by 6, beginning at d, and so forward: and the product of your second multiplication, divide by 5, so that work is finished. Or multiply the said sum by 7, and take out of it ⅕ adding it to the producte of your multiplication by 7, and that is your number required. So that as well by the one, as by the other, 143 lb 4 s 9 d Flemish, maketh 1031 lb 6 s 2 d ⅖ French or Tournois. Example. An other Example. Another example. or thus. A brief Reduction of lb s and d French, into lb s and d Flemish. Multiply 233 lb, 8 s, 4 d, fr. by 5, and divide the product twice by 6, that is the said number by 6, and the product again by 6: and the quotient of this second division is the thing required. So that 233 lb, 8 s, 4 d, fren. maketh 32 lb, 8 s, 4 d, 5/9 flemish. Example. Another. To reduce lb, s and d Sterling, into lb s and d French or Tournois. The lb star. maketh 8 lb, 8 s french, that is to say, 8 lb ⅖: the s maketh 8 s ⅖, and the penny 8 d ⅖ french. Wherefore to know what 231 lb, 13 s, 4 d star. maketh french, ye must multiply your whole sum by 42, that is by 7, and the product of it by 6, and divide this second product by 5, and that is the sum required. Otherwise multiply the sum star. by 8, and add twice to the product ⅖, and it shall produce the sum required. So that both ways, 231 lb, 13 s, 4 d star. maketh 1946 lb french. As here under followeth. The same otherwise. An other Example. The same. To reduce lb s and d fren. into lb s d star. To know how much 1256 lb 12 s 6 d fren. maketh in sterling money, multiply the sum by 5, and divide the producte by 7 and 6 at twice, and the last quotient shall be the thing required that is to say, 1256 lb 12 s 6 d maketh 149 lb 11 s 11 d 4/7 sterling. Example. An other Example. Note, that when any money is given by exchange at London for Rouen at 71 d ½: or rather 71 1/7, for the crown of 50 s french, there is neither gain nor loss, for it is one money for an other, accounting 8 lb 8 s French for 1 lb sterling So the giver loseth the time of payment which is about 15 days, & he that taketh it, hath gain of the same. They of Rean that put forth, or take money by exchange for London, aught to have like consideration. Item, when any man giveth at London 64 d ⅓, or rather 64 d 2/7, to have at one of the Fairs of Lions a crown de Marc, he that so giveth his money, loseth the time, and he that taketh it, gaineth the same: for 62 d 2/7 is equal in value to 45 s French. He that putteth or taketh money at Lions for London, aught to consider the same. Item, when any deliver in Antwerp 75 d, to receive at Lions a crown of Mark, he that putteth it forth, looseth the time, and he that taketh it, gaineth the same: For 75 groats Flemish is equal in value to 45 s French. Thus for this time I make an end of the practice of exchange and the instructions thereunto belonging, and according to my promise gratify such as are desirous to know the common coins used for traffic among merchants in these Cities following: Here followeth a brief declaration of their moneys and the reckonings and accounts of them. The sixteenth Chapter containeth a declaration of the valuation and diversity of coins of most places of Christendom for traffic: And the manner of exchange in those places from one city or town to an other, which known is right necessary for merchants, by means whereof they do find the gain or loss upon the exchange. ITem for as much as the greatest diversity of money of exchange is at Lions Therefore I will begin duly of the money of that place. At Lions they use Franckes, Soulxes and Deniers Tournois: a Francke maketh 20 Soulx, and one Soulx 12 Deniers: But the Merchants to keep their books of accounts do use French Crowns of the mark at 45 Soulx the piece, and do divide it into 20 s, 1 s and 12 d. Item a Mark of gold 65 🜄 of the Mark, which serveth for exchange. And divide it into 8 ounces. The ounce into 24 pennies or Deniers, the Denier into 24 grains: And so the sum or whole by imagination or guess. Also at Lions there are 4 fairs in a year, at the which they do commonly exchange, which are from three months to three months. At Geanes they use the Soulx: on Ducat maketh lb. 3. At Naples they use Ducats, Tarry, and Grains: The Ducat maketh 5 Taris, & one tarry 20 grains, but they take 6 Ducats (which maketh thirty tarries) for the ounce. A Ducat maketh 10 Carlins', & a Carlin 10 grains, so that 2 Carlins' make a Tarry, and 100 grains make a Ducat. At Rome they use Ducats of the Chamber, one Ducat is worth 12 Guylis, and a Guili 10 Soulx. At Venice they use Ducats Curraunts at 124 Soulx a piece or 24 Deniers, & one Denier maketh 32 picolis. At Falerine and Messine they writ, after ounce, tarry, & grains, and 1 ounce is worth 6 Ducats, or 30 taris, and 1 tarry is twenty grains, and 1 grain 6 picolis, One Ducat is also worth 24 Carlins', At Milan they use lb. s.d. of Ducats imperials, and 🜄 of exchange is worth 4 lb. At Lucques, Florence and Aucone they use the 🜄 of gold: in gold the French Crown is worth lb. 7: but at Buloigne, lb. 3.10 s At Barselon they use the soulx, the Ducat of exchange is worth 22 soulx. At Valence and Saragosse they use the Liver, Soulx and Denier, the French crown of exchange is worth 20 soulx, and 1 soulx is 12 Deniers. At the Fairs of Castill they use the Meruaidies, the Ducat is worth 375 Meruaidies. At Lisbon they use the Rays, one Ducat of exchange is worth 400 rays. At Noremburge, Frankford, and August in Germany they use the Krentzers, whereof 60 make a Floryn. At Antwerp they use lb. s and d de Gros, and they exchange into the Denier de Gros. To wit our English penny. At London they use the 1 lb sterling and 1 d sterling and they exchange in 1 d sterling. The exchange of Lions at sundry places. Item at Lions there is exchange in three sorts, at the cities and towns following. first they deliver at Lions one Mark to have or receive at Naples almost 41 ½ Ducats: at Venice, 70 Ducats corrant, at Rome 63 Ducats of the Chamber, Luques and Florence 65 🜄 of Gold, at Milan 82 🜄. And contrariwise at the said Cities aforesaid they do give so much of money to have a mark at Lions. secondly they give at Lisbon one 🜄 of Mark of 45 soulx Turnois a piece to have at Gennes, almost 68 Soulx. At Palerme and Messine almost 24 Carlins': at Barselone 22 soulx at Valence or Saragosse 20 soulx. At the fair at Castili 350 Meruaides: at L●sbone 360 Rats: in Antwerp 57 Deniers de Gros: and at London 70 d sterling. And contrariwise they do give in the said Cities almost as much of their money to have a French Crown of the Mark at Lions. thirdly they do give at Lions a 🜄 of the sun to have almost 93 Krentzers at Franckforde, August, Norenberge, or other Cities in Almain. Also at Lions only they do pay the change the 2/● in gold, and 1/● in money, or else all in money in giving 1 ½ for the hundredth. Changes at Naples and other Towns. Item at Naples they give or deliver almost 112 Ducats to receive at Rome, 100 Ducats of the chamber at the old value. Through Luques and Florence they deliver 100 Ducats Carlins' to receive there almost 86 🜄 of gold. Through Palerme & Messine one ducat of 5 Tarry to receive there almost 164 grains. Through Milan one Ducat to receive there almost 90 Soulx. Through Geanes one Ducat to receive there almost 65 soulx. The whole sum to be paid within 10 days after the sight of the bill of exchange. Also at Naples they deliver one Ducat to receive in Antwerp almost 67 d or Deniers de Gros. within 2 months: At London almost 60 d sterling, in ● months': At Barselone almost 20 Soulx within 2 months: At Valence almost 18 soulx within 2 months: At Lisbon 333 Rays, within 3 months: and at the fair at Castil, almost 340 Merueydes at the same fair. Change of Venice to other places. At Venice they deliver 100 Ducats curraunte to receive in Almaigne almost 140. Florenes at 60 Krentzers the piece. At Lucques and Florence almost ●8 🜄 of gold in 10 days. Likewise at Venice they deliver a Ducat current to receive at Palerme and Messine, almost 21 Carlins', at Milan almost 93 Soulx. At Geanes almost 62 soulx, the whole at 10 days end. Of the Pair or Pari. As touching the exchange, it is necessary to understand or know the Pair which the Italians call Pari which is no other thing than to make the money of the change of one City or town to or with the money of an other, by means whereof they do find the gains or loss upon the exchange. Example. Item having received letters of credit of one of Antwerp that the 🜄 of the sun is there worth 7 soulx. The question is what the same is worth at London when the Pair or exchange goeth for 23 s? Answer. say if 23 give but 20 what giveth 7: work and find 6 s 1 1/23 d: and so much is the 🜄 of the sun worth at London. The seventeenth Chapter containeth also a declaration of the diversity of the weights and measures of most places of Christendom for traffic. At the end of which discourse are two tables, the one for weight, and the other for measure, proportionate and reduced to an equality of our English measure and weight, by the aid whereof the ingenious may easily by the Rule of three, convert the one into the other at their pleasure etc. AT London and so all England through are used two kinds of weights and measures, As the Troy weight and the Haberdepoize: from the Troy weight is derived the proportion and quantity of all kind of dry and liquid measures, as Pecks, Bushels quarters etc. wherewith is bought and sold all kind of grain and other commodities met by the Bushel. And in liquid, Ale, Beer, Wine, Oil, Butter, Honey, etc. upon these grounds and statutes, is Bread made and sold by the Troy weight. And so is gold, silver, pearl, precious stones, and jewels. The least quantity of this Troy weight is a grain, 24 of these grains make a penny weight, 20 penny weights an ounce: and 12 ounces a pound: 2 lb or 2 pints of this weight maketh a quart. And so ascending into bigger quantities, is produced the Measures, whereby are sold our other natural sustenance, uz. Ale, or Beer, with also other necessary commodities, as Butter, Honey, Herrings, Eels, Soap, etc. All which last before rehearsed, though their Measures (wherein they are contained) be framed and derived from the Troy weight: yet are they in traffic, with divers commodities, as Led, Tin, Flare, Wax, with all other commodities both of this Realm, & of other Foreign Countries whatsoever, bought & sold by the Haberdepoise weight, after 16 ounces to the pound, and 112 lb to the C. weight. And unto every C. is allowed 12 lb weight at the common beam. From hence is also derived the weigh of Suffolk Cheese which containeth 32 Cloves, 8 lb to a Clove, and weigheth in all 256 lb: And also the barrel of Suffolk Butter is or should be of like weight with the weigh of cheese: uz. 256 lb. More 14 of these lb make a stone: And 26 stone containeth a sack of english Wool. Foreign wools, to wit, French, Spanish, & Ostrich, is also sold by the pound or C. weight, but most commonly by the Rove, 25 lb to a Rove: other commodities of tale are bought and sold by the C. five score to the C. Except headed ware, to wit, cattle, nails, & Fish, which are sold after six score to the C. There is also two other sorts of Measures, to wit, the Ell, and the Yard. By the Ell is usually met linen Cloth, as Canvas, etc. And by the Yard, Silks, , etc. Antwerp. At Antwerp are also 2 sorts of weights, their gold & silver weight, & their common weight gold & silver is weighed by the Mark, the Mark is 8 ounces, the ounce 20 Esterlings, and the Esterling 32, as our grains, The Goldsmiths divide that into smaller, but not the Merchants: the proof of Gold is made by Caractes, whereof 24 maketh a Mark of fine Gold: the Karact is 24 grains: the proof of the money is made by Deniers, 12 deniers is 1 s fine: that is a Mark of fine silver, the Denier also is divided into 24 grains, and the grain into 4 quarters. Item, 100 Marks in Antwerp, Troy weight, maketh at Lions 103 Marks, 2 ½ ounces, and 20 grains 23 lordship At Noremburg 103 Marks, 2 ½ ounces, 2 quints, 3 Deniers: at Frankford 105 Marks: at Ausburge, 104 Marks, 3 ounces, 1 quint: At Venice 103 Marks, 1 ounce, 7 deniers, 18 grains: At London 66 lb. The Mark of Gold or Silver at Antwerp Troy weight, which is 8 ounces maketh 7 ½ ounces common weight with which all other merchandise is weighed: So that the Troy weight is greater than the common weight by 6 ¼ in the C. By this weight of Troy they also weigh Musk, Amber, Pearl, etc. All Silks are bought at Antwerp, by the Bruges Ell, which is greater than the common measure, by which they retail by 2 in the C. Their common Ell is ●/4 of our yard, and ⅖ of our Ell. Lions. At Lions is used 3 sorts of weights, whereof the first is the common Town weight, with which they weigh all kind of Spicery, and divers other merchandise. The second is called Geneva weight, which is 8 in the 100 greater than the common weight, with which they weigh Silks, etc. The third is French weight, called commonly the Mark weight, and 100 lb thereof, maketh 106 ¼ lb Geneva and 114 ¾ of their common weight, with which French weight is weighed all things that payeth custom or toll. At Lions is also used two sorts of or aulns: The one wherewith they measure gross Clothes, as Canvas, and such like. The other is called the French El or Aulne, with which they measure all other kind of merchandise, whereof 7 common Town maketh 11 ordinary French . Rouen. At Rouen 6 ½ Muydes of Salt, being the measure of the place, make an hundredth at Armuiden in Zealand, and the C. of Bronage, measure of Armuiden maketh at Rouen, 11 Muids, 30 Mines maketh a Last of Corn, & 16 a last of Dates, 100 lb weight there maketh at London 114 ¼ and 109 2/8 at Antwerp: And 100 make at London 115 ●/8. Noremburge. A 100 lb weight of Noremburg, maketh at London 111 ¾ at Antwerp 107 ½: And 100 ells of Noremburg make at London 75 ⅖ at Antwerp 95 ⅖ etc. Lisbon. The C. weight at Lisbon maketh 4 Roves, every Rove 32 lb: So that their C. weight is 128 lb, and their pound containeth 14 ounces, and 100 lb of their weight maketh at London 113 ⅛. Their Silk, cloth of Gold and is measured with a measure which they call a cubit, containing about 3/● of a Varre of Castille: Howbeit, their common Measure is called a Varre which maketh 5 Palms & containeth 1 1/● of a Varre of Castille: our ell of London is equal with the Varre of Lisbon. All kind of Merchandise brought from Flaunders, Roan, or Britain, payeth at Lisbon, as a duty or custom to the King, 20 in the C. which they call the tenth in Merchandise, and the other tenth in money. Note also, that all kind of Merchandise coming to Lisbon by land, payeth less in custom than that that cometh by water. Civil. The Rove of Civil is 30 lb, 4 Roves make their C. weight which is 120 lb: The 100 lb of Civil maketh at London 102 lb: Their other common Measure is a Varre, whereof 100 maketh at London 74 ells: and at Rome 40 Canes, etc. Venice. At Venice be 2 sorts of Weights, the one called Lafoy Gross, the other La subtle, with the Gross is weighed all kind of great wares: and with the small, all kind of Spicery, and such like: 96 lb of gross weight there maketh at London 100 lb: And 100 lb of Spicery there without any tore, or allowance, make at London 64, and with tore 56. Their other common measure are Braces, whereof 100 make at London 55 ½ , at Antwerp 92 ½, etc. Florence. At Florence the 100 lb weight maketh at Aquila, for Saffron 110: and 145 lb of Florence make at Rouen but 100 lb: the weight of Florence, and that of Luke is all one. Their other measures are Braces, whereof 100 maketh at Antwerp Burges measure 81 ⅔ , 100 Braces there make at London 49 , etc. Lucque. The Lucque Satins are commonly sold at Lions by weight, and 133 ⅓ lb, maketh at Lions 100 lb: So that 1 lb ⅓ maketh at Lions but 1 pound. Their other measures are Braces, whereof 100 of them make at London 50 , at Antwerp 83 ⅓ , etc. Aquila. At Aquila their 100 lb maketh at London 71 ¼, their 136 ⅔ lb of Saffron maketh at Geneva but 100: And 11 pound of Geneva maketh 15 pound at Aquila. Valentia. At Valentia be 2 sorts of weights, a great and a small: The C. weight of great weight containeth 4 Roves: The Rove 36 lb: So the C. great weight is 144 lb: and the C. weight small containeth but 120 lb: and is also parted into 4 Roves which is 30 lb to a Rove: By the small is sold the Scarlet grain, with all other kind of Spicery, and by the great is sold Wool with all such like gross wares: The 1 ½ lb of Silk at Valentia maketh at Lions 1 lb Geneva weight: The Charge of great merchandise at Valentia, containeth 432 lb, and in small wares 360 lb: The weight here, and at Barcellone is all one: Their 100 lb weight maketh at London 78 lb: and at Antwerp 75. Dansicke. At Dansicke or Spruce land the Rule is, that whosoever buyeth any merchandise there, buyeth it by the Shippound, which is 320 lb: 20 Lispoundes make a Shippounde, and the Lispounde containeth 16 pound: which Shippound of Dansicke maketh at Antwerp 266 ⅔ lb: Their 100 lb weight maketh at London 86 ⅝, etc. Their other common measures are , whereof 100 make at London 72 ¼: And at Antwerp 120 ½ . Toulouse. At Toulouse 6 Cabes of Woad, maketh a Charge, 2 cisterns of Corn, and all kind of grain maketh a Charge: the Cester weigheth 160 lb weight of that place. Their 100 lb in weight maketh at London but 91 ¼ lb. Geanes. At Genua or Geanes, a 100 lb of their weight maketh at London, 71 ¼: And at Antwerp, 68 2/8: 100 lb weight at Genua maketh at Venice, to wit, subtle 106 lb. Their other common Measures are Palms, whereof 100 make at London 20 ⅘ : & at Antwerp 34 ⅗. The rest are supplied in 2 Tables which hereafter followeth: whereby the ingenious may gather his desire. The agreement of the Weights of diverse Countries, the one with the other, being reduced to an Equality, and drawn into a Table, as followeth. 112 pounds' weight at London, make at Antwerp 107 ⅝ Frankford 099. colen and Ausburge 102 ¼ Noremburge 100 ⅛ Rouen 098 Paris 102 ¼ Lions 118 ½ deep 100 ¼ Geneva 090 ⅛ Towlouse 122 ¾ Rochel 124 7/8 Marseilles 124 ¼ Civil, etc. 109 ¾ Venice gross weight 105 ⅜ Venice subtle weight 166 ⅞ Aquila 157 ¼ Vienna 089 ⅜ Preslaw 134 ⅝ Leibzig 101 ¼ Dansicke 129 ¼ Lubeck 097 ⅜ Barcellon 144 ½ Lisbon 099 Geanes 157 ¼ The other Table of the agreement of Measures of divers Countries reduced unto an equality, by the aid whereof you may with the use of the rule of three convert either more or less of any one measure unto other. or ●eards ●ondon ●e at Antwerp 100 Norenburg 104 ½ Frankford 125 Leibsig 125 Preslaw 125 Dansicke 83 Vienne in Austi 87 Lions in France 60 ●0/41 aulns Paris in France 57 aulns Rouen in Normandy 52 aulns Lisbon 60 aulns Civil in Spain 81 Varres Castille in Spain 81 Varres Methera Iles 62 Varres Venice 108 Braces Luques 120 Braces Florence 122 ½ Braces Milan 138 Braces Rome 90 Canes Geanes. 288 6/13 Palms The eighteenth Chapter treateth of Sports and pastimes, done by Number. IF you would know the number that any man doth think, or imagine in his mind, as though you could divine. Bid him triple it, or put twice so much more to it, as it is: which done, ask him whether it be even or odd: if he say odd, bid him take 1 to it, to make it even: and for that 1 keep 1 in your mind: now after he hath taken 1 to it, to make it even, bid him give away half, and keep the other half for himself, which when he hath done, bid him triple that half: and again after he hath tripled it, ask him whether it be even or odd: if he say odd: then bid him take 1 to make it even again: and for that last 1 keep 2 in your mind, now after he hath made his number even, bid him cast away the one half, and keep the other still: From which half that he keepeth, cause him sutlely to put away or give you 9 out of his number, & for each 9 that he giveth you keep 4 in your mind, and thereunto join the 3 which I bade you keep, and you shall have your desire. Example. Imagine he thought 7: the triple whereof to 21: and because it is odd, he is to take 1 to make it even: which first 1 given is 1 for you to keep in mind: Then the half of his 22 being cast away, he reserveth still 11: which after you have bid him triple, it maketh 33: then in giving of him 1 again to make it even: upon that last 1 reserve 2 in your mind: then his half of 34 maketh 17: From whence he can give you 9 but once: Therefore that yielding to you 4: and the 3 that you keep make 7, your desire. another kind of Divination to tell your friend, how many pence, or single pieces, reckoning them one with an other, he hath in his purse, or should think in his mind. Which to do first bid him double the pieces he hath in his purse, or the number he thinketh: if he participate his number or secreacie unto some one friend that setteth by him that can but multiply, & add never so little: if their number be great, then shall they work as you bid them so much the surer. Now after he hath doubled his number, bid him add thereunto 5 more, which done, bid him multiply that his number by 5 also, which done, bid him tell you the just sum of his last Multiplication, which sum the giver thinking it nothing available, because it is so great above his pretended imagination: yet thereby shall you presently with the help of Subtraction tell his proposed number. The Rule is this. Imagine he thought 17: double 17, and it maketh 34, whereunto if you add 5, it maketh 39: which multiplied by 5, as here is practised in the margin, it yieldeth 195: which 195 is the sum delivered you in the work: then for a general rule, you shall evermore cut off the last figure towards your right hand, with a dash of your pen, as here is performed, as a figure nothing available unto your work: and then rebate 2 from your first figure, after 5 is cut off: and the rest shall evermore be your desire, as by this example doth appear. If in any company, you are disposed to make them merry by manner of Divining, in delivering a Ring unto any one of them, which after you have delivered it unto them, that you will absent yourself from them: and they to devise after you are gone, which of them shall have the keeping thereof: And that you at your return will tell them what person hath it, upon what hand, upon what finger, & what joint. Which to do, cause the persons to sit down all on a row, & to keep likewise an order of their fingers: now, after you are gone out from them to some other place, say unto one of the lookers on, that he double the number of him that hath the Ring, and unto the double bid him add 5: And then cause him to multiply that Addition by 5: And unto the product bid him add the number of the finger, of the person that hath the Ring. And lastly, to end the work beyond that number towards his right hand, let him set down a figure, signifying upon which of the joints he hath the Ring, as if it be upon the second joint, let him put down 2: Then demand of him what number he keepeth. From the which you shall abate 250. And you shall have three figures remaining at the least: the first towards your left hand shall signify the number of the person which hath the Ring: the second or middle number shall declare the number of the finger, and the last figure towards your right hand, shall betoken the number of the joint. Example. Imagine the seventh person is determined to keep the Ring upon the fifth finger, and the third joint: first double 7, it maketh 14, thereto add 5, it maketh 19, which multiplied by 5 yieldeth 95: unto which 95, add the number of the finger, and it maketh 100: and beyond 100 toward the right hand, I set down 3 the number of the joint, all maketh 1003, which is the number that is to be delivered you, from which abating 250, there resteth 753, which prefigureth unto you the seventh person, the fifth finger, and the third joint. But note, that when you have made your subtraction, if there do remain a 0: in the place of tens, that is to say, in the second place, you must then abate 1, from that figure which is in the place of Hundreds, that is to wit from the figure which is next your left hand, and that shall be worth 10 tenths, signifying the tenth finger: as if there should remain 803, you must say, that the seventh person upon his tenth finger, and upon his third joint, hath the Ring. And after the same manner, if a man do cast 3 Dice, you may know the points of every one of them. For if you cause him to double the points of one die, and to the double to add 5: and the same sum to multiply by 5: and unto the product add the points of one of the other Dice. And behind the number towards the right hand to put the figure which signifieth the points of the last Die: and then to ask what number he keepeth, from which abate 250: and there will remain 3 figures, which do note unto you the points of every Die. Another. If three diverse things are to be hidden of three diverse persons, and you to divine which of the three persons hath the three diverse things do thus: imagine the three things to be represented by A. B.C Then secondly keep well in your mind which of the persons you mean to be the first, second and third: Then take 24 counters or stones, and your three things, And give A. unto the party whom you imagine to be your first man: and there withal give him one of your 24 counters in his hand: And B. unto your second man: and there withal 2 counters: And C unto your third man, and there withal 3 counters. And leave the rest which are 18 still among them, which done, separate yourself from them, & afterwards bid them change the things among them as they shall think good, which done after they are agreed, bid him that hath such a thing, as before you have represented by A for every counter that he hath in his hand to take up as many more, And for him that hath B for every one in his hand to take up two: And for him that hath C. for every one in his hand to take up 4: And the rest of them to leave still upon the board. These three things and the three persons being fully printed in your mind, come to the table, and you shall evermore find one of these six numbers 1.2.3.5.6 or 7 If therefore one remain still upon the board, then have they made no exchange, but keep them still as they were delivered unto them: So that the first man hath A. the second B and the third C. But if 2 remain, than the first man hath B. your second man A and your third man C: the rest of the work and the order thereof are here apparent by the table following. 1 1 A 2 B 3 C 2 1 B 2 A 3 C 3 1 A 2 C 3 B 5 1 B 2 C 3 A 6 1 C 2 A 3 B 7 1 C 2 B 3 A Another Divination of a number upon the casting of two Dice. First let the caster cast both the Dice, and mark well the number: then let him take up one of them it maketh no matter which: and look what number it hath in the bottom, and add altogether: then cast the die again, and keep in his mind what altogether maketh: then let the Dice stand: bring 7 with you and thereunto add the rest of the pits that you see upon the upper side of the Dice, and so many did the caster cast in all. FINIS. I. D. To the earnest Arithmetician. MY loving friend to Science bend, Something thou hast by this book won: But if thou wilt be excellent, Another race thou must yet run. Supplies thereto but (few do) need, And none but such as in our phrase, (By Records pen) thou mayst well read: Proceed therefore: Be not stunt dwase. The ground most sure, whereon this race With speedeful courage must be past, Of late hath turned his Greekish face, By English tilth, which aye will last. The famous Greek of Plato's lore, EUCLID I mean Geometer: So true, so plain, so fraught with store, (As in our speech) is yet no where. A treasure strange, that book will prove, With numbers skill, matched in due sort, This I thee warn of sincere love, And to proceed do thee exhort. Plus oultre. THus gentle Reader I end this my treatise, with earnest request to accept it in good part. And let my good will countervail the baseness of the style: praying thee also to thy better furtherance of perfection, to amend in the Margente of the book, these faults that have escaped in the Printing, which I have here in this table of errata following collected. Vale I.M. Norwicensis. Faults escaped in the Printing, which I desire thee good Reader to correct with thy pen, for thine own ease and benefit. E. 16. page. line. 7 there lacketh 4 lb 2 s. S. 12. page. line. 26. read as you work into s. Dd. 1. page line. 5. for 5/8 read ⅚. Gg. 2. page. line. 8. for ⅙ read 3/6. Gg. 6. page. line. 11. there lacketh. 666 ⅔. Ll. 10 page. line. 10. for ⅛ read ⅓. Mm. 1. page. line. 13. read for 2 s take the 1/10. Mm. 2. page. line. 12. for ½ read 1/10. Mm. 6. page. line. 2. read whereof the ⅓. Mm. 9 page. line. 23. Item for 3d. take ●/8. Mm. 15. page. line. 2. What 3884 yards. Mm. 15. page. line. 26. read which 10 d is ⅙ of 5 s. Nn. 7. page. line. 4 read 23. Nn. 9 page. line. 12. mend 48. to 4 s 4 d. Nn. 13 page. line 7. under 2/4 put ¼. Oo. 8 page, line 13. read saying once 3 is 3. Oo. 9 page. line. 6. in the example, say if 1 yard cost 8 ¼. d. Oo. 10. page. for 7/17 mend it to 7/14. Oo. 11. page. line. 24. for ⅓ make it ⅔. Pp. 8. page. line. 3 for 18 s make 10 s. Qq. 3. page. line. 5. for the seventh make it the eight. Qq. 4 page. line. 13. read when I delivered it. Qq. 5 page. line. 23. for 9 ¼ read 6 ¼. Qq. 9 page. line. 5. for ½ make it 8/2. Qq. 11. page. line. 1. put a prick at 400. to sever him & ⅓ asunder, the like do at the end of 600: for to sever ½: Qq. 16. page. line. 11. after the rate of 9 lb. read 10 lb. Rr. 1. page. line. 5. for 20.18.8 13/25: read 51 lb. 14.5 13/25. Tt. 14. page. line. 10 put out, by: read notes of the. Xx. 1. page. line. 7. read of gold maketh 65. 🜄