❧ Robert Norton to the courteous Readers. ALthough I have too often been an unwilling witness of the overrash disposition of divers unadvised censurers, that would have themselves esteemed skilful, and yet either will not, or rather cannot do any thing of worth themselves, not sparing to cavil, detract, and injuriously to burden other men's well pretended endeavours, with unworthy and undeserved scoffs and scandals: but in stead of reading to understand, and then to examine their true validity, that so with judgement they might censure them, have crittically played the right Momes: And though I hope not as jacke alone, to escape that which few or none have done before me: yet the respect I have to the public good, that you my Countrymen, such as either want leisure or language, may become partakers of these excellent inventions of that famous foreign Author, more prevailing with me, than the careless regard I have of such injuries could hinder, I have, as you see, adventured to provide for this worthy stranger, this English welcome, and have preferred some few of mine own friends (though unworthy) to accompany him: And so commending him to your courteous entertainments, do bid you heartily farewell. Yours in all courtesy, R. N. DEFINITIONS appertaining to Arithmetical whole Numbers. The first Definition. Arithmetic is the Science of Numbers. The second Definition. NVmber is that which expresseth the quantity of each thing. The third Definition. THe Characters by which Numbers are denoted, are ten; namely, 0 signifying the beginning of Number? and 1, and 2, and 3, and 4, and 5, and 6, and 7, and 8, and 9 The fourth Definition. EVery three Characters of a Number is called a Member, whereof the first are the three first towards the right hand, the second, the three Characters next following towards the left hand: And so by order, for the third Member and others following, as many as there shall be found thrées in the Number propounded. The Explication. AS in the number 357876297, the 297 is called the first Member: and 876 the second: and 357 the third. The fift Definition. THe first Character of the first Member, beginning from the right hand to the left, doth simply signify his own value: the second, so many times ten, as that containeth unities: the third, so many times a hundred, as that containeth unities: and the first Character of the second Member so many times a thousand, as that containeth unities: and so by the tenth progression of all the rest of the Characters contained in the number proposed. Explication. LEt the Number propounded be 756871387130789276. Then according to this definition, the first Character 6, maketh six: and the 7 following 〈◊〉 and the 2 following, two hundred: and the 9, nine thousand and so of the rest. To express thi●●umber, place over every first Character of each Member (except the first Member) a prick or point, as you see above: then say, seven hundred fifty six thousand thousand thousand thousand thousand, (namely, so many times thousand, as there are pricks or points from 7 to the end) eight hundred seventy one thousand thousand thousand thousand, three hundred eighty seven thousand thousand thousand, one hundred thirty thousand thousand, seven hundred eighty nine thousand, two hundred seventy six. The sixth Definition. A Whole number is either a unity, or a compounded multitude of unities. The seventh Definition. THe Golden Rule, or Rule of three, is that by which to three terms given, the fourth proportional term is found. ❧ The operation of Arithmetical whole Numbers. Of the Addition of whole Numbers. The first Problem. Arithmetical whole numbers being given to find their Sum, Explication propounded, let the Numbers given to be added, be 379, and 7692, & 4545, Explication required, to find their sum. Construction: the Numbers given, shall be disposed as followeth: so as their first Characters towards the right hand, stand directly one under another: and likewise their second Characters, and so also the rest following, drawing under them a line: then shall all the Characters of the first rank towards the right hand be added, saying, 9 and 2 make 11, and 5 make 16, whereof the 6 shall be placed under the first rank, and the 1 of the same 16, shall be added to the second rank, saying, 1 and 7 make 8, and 9 make 17, and 4 make 21. of which the 1 shall be placed directly under the second rank, and the 2 shall be added to the third rank, saying, 2 and 3 make 5, and 6 make 11, and 5 make 16, whereof the 6 shall be placed under the third rank, and the 1 shall be added to the fourth, saying, 1 and 7 make 8, and 4 make 12, which shall be wholly placed in their rank thus. I say, 12616 is the sum required. Demonstration: if from the three Numbers given, the two first be taken away, and there remaineth 4545. And if from the Sum 12616, the two first given be substracted also, there remaineth likewise 4545: But by the common Axiom, if from things equal, equal things be substracted, their rests shall be equal: And things substracted equal to things substracted, all shall be equal. Therefore, 12616 is equal to three Numbers given, which is the thing required. Conclusion: Arithmetical whole Numbers being given to be added, we have found their sum as was required. Substraction of whole Numbers. The second Problem. AN Arithmetical whole Number being given, out of which to subtract, and another Arithmetical whole Number to be substracted: to find their Rest. Explication propounded, be the Number out of which to subtract, 238754207: And the number to be substracted 71572604 given Explication required to find their Rest. Construction: the Number to be substracted, shall be to placed under the Number out of which it is to be substracted, as that the 4 stand directly under the 7, and the 0 under the 0, and so of the rest, drawing a line between the numbers given, and another under the number which is to be substracted, as hereunder appeareth. Then beginning at the right hand, subtract 4 from 7, and there resteth 3, which shall be set directly under the 4, and then say, 0 out of 0 resteth 0, placing 0 under the 0: then 6 from 2, which being impossible, say, 6 from 10, and 2 (which is 12) resteth 6, placing that under the 6: then 2 from 3, (true it is that you should have said 2 from 4, had it not been that you borrowed 1 from the 4 to make the other 2 to value 12) resteth 1, placing that under the 2: and so of all the other. The disposition of their Characters are as here appeareth. I say that 167181603 is the Rest required. Demonstration: adding the Rest 167181603 to the number to be substracted 71572604, the sum shall be equal to the number from which the substraction was made: wherefore seeing that 167181603 is the difference between the number from which the substraction was made, and the number to be substracted; therefore that is their Rest which was to be demonstrated. Conclusion. An Arithmetical whole Number from which to be substracted, and another to subtract, being given, we have found their Rest which was required. Multiplication of whole Numbers. The third Problem. AN Arithmetical whole Number given to be multiplied, and another to multiply, to find their product. Explication propounded: Be the Multiplicand or Number to be multiplied 546, and the Multiplicator or number to multiply 37. Explication required: To find their product, Note, that for the more easy solution of this proposition, it were necessary to have in memory the multiplication of the 9 simple Characters among themselves, learning them by rote out of the Table here placed, seeking the Multiplicand in the superior line of squares, and the Multiplicator in the diagonal or slope line of squares: and in the common Angle answering them both, you shall find their product. Pythagoras' Table. 1 2 3 4 5 6 7 8 9 2 4 6 8 10 12 14 16 18 3 9 12 15 18 21 24 27 4 16 20 24 29 32 36 5 25 30 35 40 45 6 36 42 48 54 7 49 56 63 8 64 72 9 81 As we would know the product of 3 and 8, seek 8 in the upper line, and 3 in the slope or diagonal: and in the common Angle you shall find 24 their product, and so of all the rest, as by the Table will plainly appear. Construction: place the first numbers on the right hand (of the given) one directly under another, and then draw a line, as heereunder is done. Then say, 7 times 6 make 42, place 2 under the 7, and retain the 4 (because of the 4 tenths) in memory: then say, 7 times 4 make 28, and the 4 which you had in mind, make 32, whereof place the 2 under the 3, and retain 3, and say, 7 times 5 make 35, and 3 which was borne in mind, make 38, which shall be placed in order under the line, as you see. In the like sort shall the 546 be multiplied by the 3 of the multiplicator, saying, 3 times 6 make 18, placing the 8 under the 3: and so of the rest. Then shall be drawn a line, adding all that is between the two lines in this sort. I say, that 20202 is the Product required. Demonstration. The 20202 containeth the 37 so many times as there is unities in the 546: therefore 20202 is the product which was to be found. Conclusion. An Arithmetical whole number being given to be multiplied, and another to multiply, we have found their required product. ❧ Division of Arithmetical whole Numbers. The fourth Problem. AN Arithmetical whole Number being given to be divided, and another to divide, to find their Quotient. Explication propounded: Be the number to be divided, 995, and the number to divide, 28 given. Explication required: to find their Quotient. Construction: The number to be divided (or divident) and the number to divide (or divisor) shall be placed in order, drawing a crooked line, as hereunder followeth, saying, how many times 2 in 9? three times, (true it is that there are 4 times 2 in 9, and 1 remaining) but we will show the reason hereafter why we must say but three times) set down 3 for the first Character of the Quotient, behind the crooked line, and the 3 remaining of the 9 canceling the 2 & 9: then multiply 8 by the divisor, by 3, the Quotient it maketh 24, which subtract from 39 (here appeareth the occasion why we said that 2 is but only 3 times in 9: for if we had said 4 times, resting of the 9, and had multiplied 8 by 4 it would have been 32 which should be substracted from 19 which then remained of the divident, which is impossible; therefore there must be such a number taken, & placed behind the crooked line, as that the product thereof may be substracted from the remainder) resteth 15, which place over 39, canceling the 39, and the 8, so shall the disposition of the Characters be in this manner. Now to find the second Character of the Quotient, the divisor must again be set under the divident, placing the 8 of the divisor under the 5 of the divident, and the 2 under the 8, saying how many times 2 in 15? five times, which 5 shallbe placed near the 3 at the obliqne line, for the second Character of the Quotient resteth 5 which shallbe placed over the 5 of the 15 canceling the said 15 and 2: then multiplying the divisor 8 by the Quotient 5 maketh 40, which subtract from 55 remaineth 15, canceling the 55 and the 8 and distinguishing the 15 with crooked lines from the other Characters: then draw a line near the Quotient 35, placing over the same the said remainder, and under the same the divisor 28, and the disposition of the Characters willbe as appeareth above, I say that 35 15/28 is the Quotient required. Demonstration: the 35 15/28 containeth the unity so often as the 995 containeth the divisor 28: therefore 35 15/28 is the Quotient required which was to be demonstrated, Conclusion: an Arithmetical whole number for divident, and one for divisor given, we have found their Quotient required. ❧ The Rule of Three, or Golden Rule of Arithmetical whole Numbers. The fift Problem. Three Terms of Arithmetical Numbers, being given to find their proportional Term. Explication propounded: Be the three terms given 234. Explication required: To find their fourth proporcionall Term: that is to say, in such Reason to the third term 4, as the second term 3, is to the first term 2. Construction: Multiply the second term 3, by the third term 4, that giveth the product 12: which dividing by the first term 2, giveth the Quotient 6: I say that 6 is the fourth proportional term required. Demonstration: there is from 6 to 4, Reason sesquialter, and the same Reason is there from 3 to 2: therefore 6 is the fourth proportional term to be demonstrated. Conclusion: three Arithmetical numbers being given, we have found their fourth proportional term required. The Preface of Simon Stevin. To Astronomers, Land-meaters, Measurers of Tapestry, Gaudgers, Stereometers in general, Monty-Masters, and to all Merchants, Simon Stevin wisheth health. MAny seeing the smallness of this Book, and considering your worthiness to whom it is dedicated, may perchance esteem this our conceit absurd: But if the proportion be considered, the small quantity hereof compared to human imbecility, and the great utility unto high and ingenious intendiments, it will be found to have made comparison of the extreme terms, which permit not any conversion of proportion. But what of that? is this an admirable invention? No certainly: for it is so mean, as that it scant deserveth the name of an invention: for as the countryman by chance sometime findeth a great treasure, without any use of skill or cunning, so hath it happened herein. Therefore if any will think, that I vaunt myself of my knowledge, because of the explication of these utilities, out of doubt, he showeth himself to have neither judgement, understanding, nor knowledge to discern simple things from ingenious inventions, but he (rather) seemeth envious of the common benefit: yet howsoever, it were not fit to omit the benefit hereof, for the inconvenience of such calumny. But as the Mariner having by hap found a certain unknown Island, spareth not to declare to his Prince the riches and profits thereof; as the fair fruits, precious minerals, pleasant champions, etc. and that without imputation of Philautry: even so shall we speak freely of the great use of this invention; I call it great, being greater than any of you expect to come from me. Seeing then that the matter of this Disme (the cause of the name whereof shallbe declared by the first definition following) is number, the use and effects of which, yourselves shall sufficiently witness by your continual experiences, therefore it were not necessary to use many words thereof: for the ginger knoweth, that the world is become by computation Astronomical (seeing it teacheth the Pilot the elevation of the Equator and of the Pole, by means of the declination of the Sun, to describe the true Longitudes, Lantitudes, situations & distances of places, etc.) a Paradise, abounding in some places with such things as the Earth cannot bring forth in other. But as the sweet is never without the sour: so the travail in such computations cannot be unto him hidden, namely, in the busy multiplications and divisions which proceed of the 60 progression of degrees, minutes, seconds, thirds, etc. And the Surveyor or Land-meater knoweth, what great benefit the world receiveth from his science, by which many dissensions and difficulties are avoided, which otherwise would arise by reason of the unknown capacity of Land: beside, he is not ignorant (especially whose business and employment is great) of the troublesome multiplications of Roods, Feet, and oftentimes of inches, the one by the other, which not only molesteth, but also often (though he be very well experienced) causeth error, tending to the damage of both parties, as also to the discredit of Land-meater or surveyor, and so for the Money-masters, Merchants and each one in his business: therefore how much they are more worthy, and the means to attain them the more laborious, so much the greater and better is this Disme, taking away those difficulties: But how? it teacheth (to speak in a word) the easy performance of all reckonings, computations, & accounts, without broken numbers, which can happen in man's business, in such sort, as that the four Principles of Arithmetic namely, Addition, Substraction, Multiplication, & Division, by whole numbers, may satisfy these effects, affording the like facility unto those that use Counters. Now if by those means we gain the time which is precious, if hereby that be saved which otherwise should be lost, if so, the pains, controversy, error, damage, and other inconveniences commonly happening therein, be eased, or taken away, than I leave it willingly unto your judgements to be censured: and for that, that some may say that certain inventions at the first seem good, which when they come to be practised, effect nothing of worth, as it often happeneth to the searchers of strong moving, which seem good in small proofs and models, when in great, or coming to the effect, they are not worth a Button: whereto we answer, that herein is no such doubt: for experience daily showeth the same: namely, by the practise of divers expert Land-meaters of Holland, unto whom we have showed it, who (laying aside that which each of them had, according to his own manner, invented to lessen their pains in their computations) do use the same to their great contentment, and by such fruit as the nature of it witnesseth, the due effect necessarily followeth: The like shall also happen to each of yourselves using the same as they do: mean while live in all felicity. The Argument. THe Disme hath two parts, that is, Definitions & Operations: by the first definition is declared what Disme is, by the second, third, and fourth, what Comencement, Prime, Second etc. and Disme numbers are: the Operation is declared by four propositions, The Addition, Substraction, Multiplication and Division of Disme numbers. The order whereof may be successively represented by this Table. The Disme hath two parts. Definitions, as what is Disme, Comencement, Prime, Second etc. Disme number. Operations or practise of the Addition, Substraction, Multiplication, Division. And to the end the premises may the better be explained, there shallbe hereunto an Appendix adjoined, declaring the use of the Disme in many things by certain examples, and also definitions and operations, to teach such as do not already know the use and practise of Numeration, and the four principles of common Arithmetic, in whole numbers, namely, Addition, Substraction, Multiplication, & Division, together with the Golden Rule, sufficient to instruct the most ignorant in the usual practise of this Art of Disme or decimal Arithmetic. The first Part. Of the Definitions of the Dimes. The first Definition. DIsme is a kind of Arithmetic, invented by the tenth progression, consisting in Characters of Ciphers; whereby a certain number is described, and by which also all accounts which happen in human affairs, are dispatched by whole numbers, without fractions or broken numbers. Explication. LEt the certain number be one thousand, one hundred and eleven, described by the Characters of Ciphers thus 1111, in which it appeareth that each 1 is the 10th part of his precedent character 1: likewise in 2378, each unity of 8 is the tenth of each unity of 7, and so of all the others: But because it is convenient that the things whereof we would speak, have names, and that this manner of computation is found by the consideration of such tenth or disme progression; that is, that it consisteth therein entirely, as shall hereafter appear: We call this Treatise fitly by the name of Disme, whereby all accounts happening in the affairs of man, may be wrought and effected without fractions or broken numbers, as hereafter appeareth. The second Definition. EVery number propounded, is called Comencement, whose sign is thus (0). Explication. BY example, a certain number is propounded of three hundred sixty four: we call the 364 Comencements, described thus 364 (0) and so of all other like. The third Definition. ANd each tenth part of the unity of the Comencement, we call the Prime, whose sign is thus (1), and each tenth part of the unity of the Prime, we call the Second, whose sign is (2), and so of the other: each tenth part of the unity of the precedent sign, always in order, one further. Explication. AS 3 (1) 7 (2) 5 (3) 9 (4) that is to say, 3 Primes, 7 Seconds, 5 Thirds, 9 Fourths, and so proceeding infinitely: but to speak of their value, you may note, that according to this definition, the said numbers are 3/10 7/100 5/1000 9/10000, together 3759/10000 and likewise 8 (0) 9 (1) 3 (2) 7 (3) are worth 8 9/10 3/100 7/1000 together 8 937/1000 and so of other like. Also you may understand, that in this Disme we use no fractions, and that the multitude of signs, except (0) never exceed 9: as for example, not 7 (1) 12 (2) but in their place 8 (1) 2 (2), for they value as much. The fourth Definition. THe numbers of the second and third Definitions before going, are generally called Disme numbers. The end of the Definitions. The second part of the Disme. Of the Operation or practise. The first proposition of Addition. DIsme numbers being given how to add them to find their sum. The explication propounded; there are 3 orders of Disme numbers given, of which the first 27 (0), 8 (1), 4 (2), 7 (3), the second 37 (0), 8 (1), 7 (2), 5 (3), the third 875 (0), 7 (1) 8 (2), 2 (3). The explication required, we must find their total sum. Construction. The numbers given, must be placed in order as here adjoining, adding them in the vulgar manner of adding of whole numbers in this manner: The sum (by the first Problem of Arithmetic following) is 941504, which are (that which the signs above the numbers do show) 941 (0) 5 (1) 0 (2) 4 (3). I say, they are the sum required. Demonstration: the 27 (0) 8 (1) 4 (2) 7 (3) given, make by the 3 Definition before 27 8/10 4/100 7/1000, together 27 847/1000, and by the same reason, the 37 (0) 8 (1) 7 (2) 5 (3) shall make 37 875/1000, and the 875 (0) 7 (1) 8 (2) 4 (3) will make 875 782/1000, which three numbers make by common addition of vulgar Arithmetic 941 304/1000. But so much is the sum 941 (0) 5 (1) 0 (2) 4 (3): therefore it is the true sum to be demonstrated. Conclusion: Then Disme numbers being given to be added, we have found their sum, which is the thing required. Note, that if in the number given, there want some signs of their natural order, the place of the defectant shall be filled. As for example, let the numbers given be 8 (0) 5 (1) 6 (2) and 5 (0) 7 (2): in which, the latter wanted the sign of (1), in the place thereof shall 0 (1) be put, take then for that latter number given 5 (0) 0 (1) 7 (2) adding them in this sort. This advertisement shall also serve in the three following propositions, wherein the order of the defayling figures must be supplied, as was done in the former example. The second Proposition. Of Substraction. A Disme number being given to subtract: another less Disme number given out of the same to find their rest. Explication propounded: be the numbers given 237 (0) 5 (1) 7 (2) 8 (3) & 59 (0) 7 (1) 3 (2) 9 (3) The Explication required; to find their rest. Construction: the numbers given shallbe placed in this sort, substracting according to vulgar manner of substraction of whole numbers, thus The rest is 177839 which valueth as the signs over them do denote 177 (0) 8 (1) 3 (2) 9 (3), I affirm the same to be the rest required. Demonstration: the 237 (0) 5 (1) 7 (2) 8 (3) make by the third Definition of this Disme, 237 5/10 7/100 8/1000 together 237, 578/1000 and by the same reason, the 59 (0) 7 (1) 4 (2) 9 (3) value 59 749/1000 which substracted from 237 578/1000 there resteth 177 839/1000 but so much doth 177 (0) 8 (1) 3 (2) 9 (3) value: that is then the true rest which should be made manifest. Conclusion: a Disme being given, to subtract it out of another Disme number, and to know the rest, which we have found. The third Proposition: of Multiplication. A Disme number being given to be multiplied, and a multiplicator given to find their product: The Explication propounded: be the number to be multiplied 32 (0) 5 (1) 7 (2), and the multiplicator 89 (0) 4 (1) 6 (2) The Explication required: to find the product. Construction: the given numbers are to be placed as here is showed, multiplying according to the vulgar manner of multiplication by whole numbers, in this manner, giving the product, 29137122: Now to know how much they value, join the two last signs together as the one (2) and the other (2) also, which together make (4), and say that the last sign of the product shall be (4) which being known, all the rest are also known by their continued order. So that the product required, is 2913 (0) 7 (1) 1 (2) 2 (3) 2 (4). Demonstration: The number given to be multiplied, 32 (0) 5 (1) 7 (2) (as appeareth by the third Definition of this Disme) 32 5/10 7/100 together 32 57/100: and by the same reason the multiplicator 89 (0) 4 (1) 6 (2) value 89 46/100 by the same, the said 32 57/100 multiplied, giveth the product 2913, 7122/10000 But it valueth 2913 (0) 7 (1) 1 (2) 2 (3) 2 (4). It is then the true product which we were to demonstrate. But to show why (2) multiplied by (2) giveth the product (4) which is the sum of their numbers, also why (4) by (5) produceth (9), and why (0) by (3) produceth (3) etc. Let us take 2/10 and 3/100 which (by the third Definition of this Disme) are 2 (1) 3 (2) their product is 6/10000 which value by the said third Definition 6 (3), multiplying then (1) by (2) the product is (3) namely a sign compounded of the sum of the numbers of the signs given. Conclusion. A Disme number to multiply, and to be multiplied, being given, we have found the product, as we ought. Note. IF the latter sign of the number to be multiplied, be unequal to the latter sign of the multiplicator, as for example, the one 3 (4) 7 (5) 8 (6), the other 5 (1) 4 (2), they shall he handled as aforesaid, and the disposition thereof shallbe thus. The fourth Proposition: of Division. A Disme number for the divident, and divisor, being given to find the Quotient. Explication proposed: let the number for the divident be 3 (0) 4 (1) 4 (2) 3 (3) 5 (4) 2 (5) and the divisor 9 (1) 6 (2). Explication required: to find their Quotient. COnstruction: the numbers given divided (omitting the signs) according to the vulgar manner of dividing of whole numbers, giveth the Quotient, 3587; now to know what they value; the latter sign of the divisor (2) must be substracted from the latter sign of the divident which is (5), resteth (3) for the latter sign of the latter Character of the Quotient, which being so known, all the rest are also manifest by their continued order, thus 3 (0) 5 (1) 8 (2) 7 (3) are the Quotient required. DEmonstration: the number divident given 3 (0) 4 (1) 4 (2) 3 (3) 5 (4) 2 (5) maketh (by the third Definition of this Disme) 3 4/10 4/100 3/1000 5/10000 2/100000 together 3 44352/100000 and by the same reason, the divisor 9 (1) 6 (2) valueth 96/100, by which 3 44352/100000 being divided, giveth the Quotient 3 587/1000; but the said Quotient valueth 3 (0) 5 (1) 8 (2) 7 (3): therefore it is the true Quotient to be demonstrated. Conclusion: a Disme number being given for the divident and divisor, we have found the Quotient required. Note, if the divisors signs be higher than the signs of the divident, there may be as many such Ciphers 0 joined to the divident as you will, or many as shallbe necessary: as for example, 7 (2) are to be divided by 4 (5), I place after the 7 certain 0 thus 7000, dividing them as aforesaid, & in this sort it giveth for the Quotient 1750 (7). It happeneth also sometimes, that the Quotient cannot be expressed by whole numbers, as 4 (1) divided by 3 (2) in this sort, whereby appeareth, that there will infinitely come from the 3 the rest of ⅓ and in such an accident you may come so near as the thing requireth, omitting the remainder, it is true, that 13 (0) 3 (1) 3⅓ (2) etc. shallbe the perfect Quotient required: but our intention in this Disme is to work all by whole numbers: for seeing that in any affairs, men reckon not of the thousandth part of a mile, grain, etc. as the like is also used of the principal Geometricians, and Astronomers, in computations of great consequence, as Ptolemy & johannes Monta-regio have not described their Tables of Arches, Chords, or Sines, in extreme perfection (as possibly they might have done by Multinomall numbers,) because that imperfection (considering the scope and end of those Tables) is more convenient than such perfection. Note 2. the extraction of all kinds of Roots may also be made by these Disme numbers: as for example, To extract the square root of 5 (2) 2 (3) 9 (4), which is performed in the vulgar manner of extraction in this sort, and the root shallbe 2 (1) 3 (2), for the moitye or half of the latter sign of the numbers given, is always the latter sign of the root: wherefore if the latter sign given were of a number imper: the sign of the next following shallbe added, and then it shallbe a number per; and then extract the Root as afore. Likewise in the extraction of the Cubique Root, the third part of the latter sign given shallbe always the sign of the Root: and so of all other kind of Roots. The end of the Disme. The Appendix. The Preface. seeing that we have already described the Disme, we will now come to the use thereof, showing by vi. Articles, how all computations which can happen in any man's business, may be easily performed thereby: beginning first to show how they are to be put in practise, in the casting up of the content or quantity of Land measured as followeth. The first Article, of the Computations of Landmeating. CAll the Perch or Rood also Comencement, which is 1 (0), dividing that into 10 equal parts, whereof each one shallbe 1 (1); them divide each prime again into 10 equal parts, each of which shallbe 1 (2); and again each of them into 10 equal parts, and each of them shallbe 1 (3); proceeding further so, if need be; but in Landmeating, divisions of second willbe small enough: yet for such things as require more exactness, as Fathom of the Lead, Bodies etc. there may be thirds used: and for as much as the greater number of Land-meaters use not the Pole, but a chain line of three, four or five Perch long marking upon the yard of their cross staff certain feet 5 or 6 with fingers, palms etc. the like may be done here: for in the place of their five or six feet with their fingers, they may put 5 or 6 primes with their seconds. THis being so prepared, these shallbe used in measuring, without regarding the feet and fingers of the Pole, according to the Custom of the place: & that which must be added, substracted, multiplied or divided according to this measure, shallbe performed according to the doctrine of the precedent examples. AS for example, we are to add 4. triangles or surfaces of Land, whereof the first 345 (0) 7 (1) 2 (2), the second 872 (0) 5 (1) 3 (2), the third 615 (0) 4 (1) 8 (2) the fourth 956 (0) 8 (1) 6 (2); THese being added according to the manner declared in the first Proposition of this Disme in this sort, their sum will be 2790 (0) or Perches 5 (1) 9 (2), the said Roods or Perches, divided according to the custom of the place; (for every Acre containeth certain Perches) by the number of perches you shall have the Acres sought. But if one would know how many feet and fingers are in the 5 (1) 9 (2) (that which Land-meater shall need to do but once, and that at the end of the casting up of the proprietaries, although most men esteem it unnecessary to make any mention of feet and fingers) it will appear upon the Pole how many feet and fingers (which are marked, joining the tenth part upon another side of the Rood) accord with themselves. In the second, out of 57 (0) 3 (1) 2 (2) substracted 32 (0) 5 (1) 7 (2) it may be effected according to the second proposition of this Disme, in this manner: In the third (for multiplication of the sides of certain Triangles and Quadrangles) multiply 8 (0) 7 (1) 3 (2), by 7 (0) 5 (1) 4 (2) 2 this may be performed according to the third proposition of this Disme, in this manner: And giveth for the product or superfices 65 (0) 8 (1) etc. Divide 367 (0) 6 (1) by 26 (0) 3 (1) according to the fourth proposition of this Disme: so the Quotient giveth from A, towards B, 13 (0) 9 (1) 7 (2), which is A E. And if we will, we may come nearer (although it be needles) by the second note of the fourth Proposition, the demonstrations of all these examples are already made in their propositions. The II. Article: of the Computations of the measures of Tapestry, or Cloth. THe Ell of the Measurer of tapistry or cloth, shall be to him 1 (0), the which he shall divide (upon the side whereon the partitions, which are according to the ordinance of the Town, is not set out) as is done above on the Pole of the Land meater, namely into 10 equal parts, whereof each shall be 1 (0), than each 1 (1) into 10 equal parts, of which each shall be 1 (2) &c. And for the practice seeing that these examples do altogether accord with those of the first Article of Landmeating, it is thereby sufficiently manifest, so as we need not here make any mention again of them. The III. Article: of the Computations, serving to Gaudging, and the measures of all Liquor vessels. ONe Am (which maketh 100 pots Antwerp) shallbe 1 (0), the same shall be divided in length and deepness, into 10 equal parts (namely, equal to respect of the wine, not of the Rod; of which the parts of the pro●…ditie shallbe unequal) & each part shallbe 1 (1) containing 10 pots, than again each 1 (1) into 10, parts equal as afore, and each will make 1 (2) worth 1 pot, than each 1 (2) into 10. equal parts making each 1 (3). Now the Ro● being so divided, to know the content of the Tun, multiply and work as in the precedent first Article, of which (being sufficiently manifest) we will not speak here any farther. It remaineth yet to divide each length as B O & O C, etc. into five, thus: Seek the mean proportional between B M: & his 10 part which shallbe B R: cutting B S: equal to B R: Then the length S R. noted from B towards A: as B T: and likewise the length T R: from B to V: & so of the others: & in like sort proceeding to divide B S: and S T: etc. into (3), I say that B S: S T: and T V: etc. are the desired (2) which is thus to be demonstrated. For that B N: is the mean proportional line (by the Hipothesis between B M: and his moiety, the square of B N: (by the 17. proposition of the sixth book of Euclid) shallbe equal to the Rectangle of B M: & his moiety: But the same Rectangle is the moiety of the square of B M: the square then of B N: is equal to the moiety of the square of B M: But B O is (by Hipothesis) equal to B N: and B C: to B M: the square then of B O: is equal to the moiety of the square of B C. And in like sort it is to be demonstrated, that the square of B S is equal to the tenth part of the square of B M. Wherefore &c. we have made the demonstration brief, because we writ not this to learners, but unto masters in their science. The FOUR Article: of Computations of stereometry in general. TRue it is, that Gaudgerie which we have before declared, is stereometry (that is to say, the Art of measuring of bodies) but considering the divers divisions of the Rod, Yard, or Measure of the one and other, and that and this do so much differ, as the Genus and the Species: they ought by good reason to be distinguished. For all stereometry is not Gaudgerie. To come to the point, the Stereometrian shall use the measure of the town or place, as the Yard, el etc. with his ten partitions, as is described in the first and second Articles, the use and practise thereof, (as is before showed) is thus: Put case we have a Quadrangular, Rectangular Column to be measured, the length whereof is 3 (1) 2 (2), the breadth 2 (1) 4 (2), the height 2 (0) 3 (1) 5 (2), The question is, how much the substance or matter of that Pillar is: Multiply (according to the doctrine of the 4. proposition of this Disme) the length by the breadth, & the product again by the height in this manner, And the product appeareth to be 1 (1) 8 (2) 4 (4) 8 (5). NOte, some ignorant (and understanding not that we speak here) of the Principles of Stereometry, may marvel wherefore it is said, that the greatness of the abovesaid colunn is but 1 (1) &c. seeing that it containeth more than 180 cubes, of which the length of each side is 1 (1), he must know that the body of one yard is not a body of 10 (1) as a yard in length, but 1000 (1) in respect whereof 1 (1) maketh 100 Cubes, each of 1 (1) as the like is sufficiently manifest amongst Land-meats in surfaces: for when they say 2 roods, 3 Feet of Land, it is not barelymeant 2 square Roods, and three square feet, but two Roods (and counting but 12 feet to the Rood) 36 feet square: therefore if the said Question had been how many Cubes each being 1 (1) was in the greatness of the said Pillar, the solution should have been fitted accordingly, considering that each of these 1 (1) doth make 100 (1) of those; and each 1 (2) of these maketh 10 (1) of those etc. or otherwise, if the tenth part of the yard be the greatest measure that the Stereometrian proposeth, he may call it 1 (0), and so as abovesaid. The fift Article; of Astronomical Computations. THe ancient Astronomers having divided their Circles each into 360 degrees, they saw, that the Astronomical Computations of them with their parts was too laborious: and therefore they divided also each degree into certain parts, and those again into as many, etc. to the end thereby to work always by whole numbers, choosing the 60th progression, because that 60 is a number measurable by many whole measures, namely, 1, 2, 3, 4, 5, 6 10, 12, 15, 20, 30: but if experience may be credited (we say with reverence to the venerable antiquity, and moved with the common utility) the 60th progression was not the most convenient, (at least) amongst those that in nature consist potentially, but the tenth which is thus: we call the 360 degrees also Comencements, expressing them so 360 (0), and each of them a degree 12 1 (0) to be divided into 10 equal parts, of which each shall make 1 (1), and again each 1 (1) into 10 (2) and so of the rest, as the like hath already been often done. Now this division being understood, we may describe more easily that we promised in Addition, Substraction, Multiplication, and Division; but because there is no difference between the operation of these, and the four former propositions of this book, it would but be loss of time, and therefore they shall serve for examples of this Article: yet adding thus much, that we will use this manner of partition in all the Tables & computations which happen in Astronomy, such as we keep to divulge in our vulgar German Language, which to the most rich adorned and perfect Tongue of all other, & of the most singularity, of which we attend a more abundant demonstration, than Peter and john have made thereof in the Bewysconst and Dialectique, lately divulged, and have in the lease following placed a necessary Table, for the reducing of the minutes, seconds, etc. of the 60, progression, into primes, seconds, etc. of the tenth progression: the use whereof followeth. The use of this Table. WHen any number of minutes, seconds, thirds, fourth's &c. of the 60th progression, are given to be reduced into the primes, seconds, thirds, etc. of the tenth progression, seek the given number in this Table, or if the number be not there to be found, take the nearest: if none be there great enough, take half or one quarter of the given: if there be none small enough, double, triple, or quadruple, the given, and then as aforesaid seek the nearest number thereunto in the Table, and the two numbers in whose common Angle the given number is found, or nearest found, shall show you the quantity and quality of the subdivisions of the ten progressions proper to that given number, namely, the number standing in the top or front of the table directly over it, shall show the quantity, and the number directly against it in the first Column toward the left hand, shall denote the quality; as for example, be the pronumber given , seek it in the Table, and you shall find to stand in the front directly over it the figure 7, and in the first Column directly against it toward the left hand (5): therefore according to the rule above mentioned, I conclude, that of the 60 progression valueth just 7 (5) of the tenth progression etc. This example I think sufficient to enlighten the ingenious practizer: only this, that if there be no number to be found in the Table, just or near the number given, you may take two, three or more of those that will come nearest, and so work as before: as for example also, be the number given of the 60 progression; you shall find them all by taking 4 of the numbers of the colunne under 3, to be of the tenth progression: and so with a small diligence may any other number of the one progression be reduced into the other, which I omit to speak any further of at this time. The sixth Article; of the Computations of Money-masters, Merchants, and of all estates in general. TO the end we speak in general and briefly of the sum and contents of this Article, it must be always understood, that all measures (be they of length, liquors, of money etc.) be parted by the tenth progression, and each notable species of them, shallbe called Comencement: as a Mark, comencement of weight, by the which Silver and Gold are weighed, Pound of other common weights, Livers-degros in Flanders, Pound sterling in England, Ducat in Spain etc. Comencement of Money: the highest sign of the Mark shallbe (4), for 1 (4) shall weigh about the half of one Es of Antwerp, the (3) shall serve for the highest sign of the Liure de gros, seeing that 1 (3) maketh less than the quarter of one DS. The subdivisions of weight to weigh all things, shallbe (in place of the half pound, quarter, half quarter, ounce, half ounce, esterlin, grain, Es, etc. of each sign, 5, 3, 2, 1, that is to say, that after the pound or 1 (0) shall follow the half pound or 5 (1), than the 3 (1) than the 2 (1) than the 1 (1), and the like subdivisions have also the 1 (1) and the other following. WE think it necessary, that each subdivision, what matter soever the subject be of, be called Prime, Second, Third, etc. and that because it is notable unto us, that the Second, being multiplied by the Third, giveth in the product the Fifth (because two and three make five, as is said before) also the Third divided by the Second, giveth the Quotient Prime etc. that which so properly cannot be done by any other names: but when it shallbe named for distinction of the matters (as to say, half an Ell, half a pound, half a pint etc.) we may call them Prime of Marc, Second of Marc, Second of Pound, Second of Ell, etc. But to the'nd we may give example, suppose 1 Mark of Gold value 36 lib. 5 (1) 3 (2) the Question what valueth 8 Marks 3 (1) 5 (2) 4 (3): multiply 3653 by 8354 giving the product by the fourth Proposition (which is also the solution required) 395 lib. 1 (1) 7 (2) 1 (3); as for the 6 (4) and 2 (5) they are here of no estimation. suppose again, that 2 els and 3 (1) cost 3 lib. 2 (1) 5 (2) the question is, what shall 7 els 5 (1) 3 (2) cost: multiply according to the custom the last term given by the second, and divide the product by the first, that is to say, 753 by 325 maketh 244725, which divided by 23, giveth the Quotient and Solution 10 lib. 6 (1) 4 (2). WE could also more amply demonstrate by easy examples of broken numbers, the comparison and great difference of the facility of this more than that, but we will pass them over for brevity sake. LAstly, it may be said, that there is some difference between this last sixth Article, and the 5 precedent Articles, which is, that each one may exercise for themselves the tenth partition of the said precedent 5 Articles, thoughst be not given by the Magistrate of the place as a general order, but it is not so in this latter: for the examples hereof, are vulgar computations, which do almost continually happen to every man, to whom it were necessary that the solution so found, were of each accepted for good and lawful: Therefore considering the so great use, it would be a commendable thing, if some of those who expect the greatest commodity, would solicit to put the same in execution to effect, namely, that joining the vulgar partitions that are now in weight, measures, and moneys (continuing still each Capital measure, weight and Coin in all places unaltred) that the same tenth progression might be lawfully ordained by the superiors, for every one that would use the same it might also do well, if the values of Moneys, principally the new Coins, might be valued and reckoned upon certain Primes, Seconds, Thirds etc. But if all this be not put in practise so soon as we could wish, yet it will first content us, that it will be beneficial to our successors, if future men shall hereafter be of such nature as our predecessors, who were never negligent of so great advantage. Secondly, that it is not unnecessary for each in particular, for so much as concerneth him, for that they may all deliver themselves when they will, from so much and so great labour. And lastly, although the effects of the sixth Article appear not immediately, yet it may be; and in the mean time may each one exercise himself in the five precedent, such as shallbe most convenient for them; as some of them have already practised. The end of the Appendix.