De Numeris Geometricis. OF THE NATURE AND PROPRIETIES OF GEOMETRICAL NUMBERS. First written by Lazarus Schonerus, and now Englished, enlarged and illustrated with divers and sundry Tables and observations concerning the measuring of Plains and Solids: All teaching the fabric, demonstration and use of a singular Instrument, or ruler, long since invented and perfitted by THOMAS BEDWELL Esquire. LONDON, Printed by RICHARD FIELD. 1614 TO THE HONOURABLE KNIGHT, Sir Robert Coke. HONOURABLE SIR, HIstorians report, that the Scythians were so far out of love with learning, that it was a shame and disgrace for any man amongst them to be esteemed a scholar. The name of a Scythian, all Christians cannot choose but detest and abhor. And yet is this their opinion, as we hear, maintained by the Nobility of some Nations. An argument, as I think, sufficient alone, if historians were silent, to show from what race such men are descended. Barbarous, I am sure it is,& no way consonant to the nature of man. For as the Philosopher saith, Omnes homines naturâscire desiderant. And, Prudentia( saith the Orator) quae in vericognitione consistit, maxim naturam attingit humanam. The Nobility therefore of England, are in this most highly to be commended, who do generally so much esteem of learning, that they think nothing more honourable then either to be accounted learned, or great patrons of learning: Knowing indeed that, as the Poet saith, In prudentia Nobilitas sita est. This generality first: and then next, your honourable affection and favour to me in particular, hath emboldened me to offer this small Pamphlet unto your patronage, ut aliquod saltem extaret meae inte obseruantiae& gratitudinis monumentum. Vale, V. O. ac nos tibi iampridem devinctissimos, devinctiores, si fieri potest, reddito. Londini Anglorum viij Kalend. Martias, an. MDCXIIII. Your H. in all true affection, Wilham Bedwell. TO THE courteous READER. IT is now, as far as I remember, well near forty years since a ruler, for the measuring of Plains and Solids, was first devised by mine uncle M. Thomas Bedwell. The use of this, himself hath taught in a little Treatise, which I have seen long ago in his study amongst other his writings of like nature and argument. The fabric or manner of making it, he did not, to my knowledge, impart to any. That I suppose be reserved for a larger volume of diverse other such inventions of his, which he bade purposed to have published, if God had spared him longer life. But these all being suppressed by some, who have, as it seemeth, more respected their own private gain, then either his honour or the common good; I did, about a dozen years ago, as some learned lovers of these studies can testify, labour to shadow out that work of his. And at last having finished a Treatise, both of the construction and use of the same, I offered it unto some to have it imprinted: But they seeing in the copy divers figures and diagrams, refused to meddle with it, except myself would bear a great part of the charge of that first impression. That I being unwilling to do, I was forced to defer the publishing of it until I might meet with some one that would of himself undertake the business. In this interim, I found yet another scruple, which I thought necessary first to be removed. For as the construction of the first Author was merely Geometrical, so that of ours was wholly Arithmetical, and could not be done without the help of numbers: and therefore was such as might not easily be taught or conceived, except first the nature and proprieties of such like numbers were well understood. Whereupon, before that Treatise of the ruler, this of GEOMETRICAL NUMBERS was first to be published. Geometrical numbers I call them; not Figurate or Cossicke numbers, as commonly they are called: For of those infinite sorts of this kind of numbers, we have to do with none but such as do represent and express some true Geometrical figure, such as the Quadrate, Parallelogramoblong, Prisma, and the Cubic are. Now whereas haply some man may object and say, that these things have been often handled and sufficiently taught by others; let him know, that although in general we do confess it to be true, yet no man, to our knowledge, hath applied those rules to that use that we have done: And therefore those Treatises of theirs, do in no ways fit our purpose. For first out of these rules, as afterward shall appear, we have framed certain Tables for sundry uses in measuring of Plains and Solids. One of these, called TRIGONUM ARCHITECTONICUM, The Carpenter's Squire, we caused for the public good, more than a twelvemonth since, to be imprinted at our own cost and charges. Moreover, sundry practices and works of these rules more plain and easy then commonly are used, are here showed and discovered. Lastly, we have here given thee a taste or sample of a form and manner of resolving any intricate demonstration mathematical, whereby it shall be made so plain and easy, that it may be conceived of the simplest. This we have used these seven or eight and twenty years; and because we see it to be approved of the most learned Mathematicians of this kingdom, we shall ere long, God willing, more fully manifest it by many examples. OF A RATIONAL FIGURATE NUMBER. CHAP. I. Of the Extraction of a Quadrate root, or side. 1. A rational figure ate number is, a number that is made by the multiplication of numbers between themselves. Such a combination and affinity there is between Arithmetic and Geometry, that the whole nature& property of the one cannot well be taught and conceived without the other. For so Ramus saith, Geometria sui& generis& juris magnam quidem partem est, neque aliter quam geometricè tractabilis. Attamen part quadam numeris associatur,& eye explicatur, numerique, geometricarum affectionum interprete, geometricis vocabulis appellantur, ut Planus, Quadratus, Solidus,& Cubus, à geometrico Plano, Quadrato, Solido, Cubo; quorum umbr ae quoedam tales numers sunt. Item, Aristoteles affirmat gener aliser arithmeticam demonstrationem magnitudinum accidentibus conucnire, cum magnitadines numerifiunt: Et Proclus ait, Quicquid in Geometria 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 explicabile& cognobile sit, numeris explicari& cognosci. And indeed a great part of Euclides Elements are wholly arithmetical, that is, that book of Euclid, whose purpose is to teach the general grounds of Geometry, hath notwithstanding many arithmetical conclusions here and there intermeddled among; and to speak the truth, the seven, viij, ix, and x books of that his work, are in a manner wholly arithmetical. All this, I say, doth teeth us that Geometry cannot be understood without the knowledge and use of Arithmetic. Of this kind most properly are those figurate numbers, which do participate of both natures. And therefore one defineth a figurate number thus; Figuratus numerus est numerus qui per siguras& appellationes geometricas exprimitur, A figurate number is a number which is expressed by geometrical names and terms. Whereupon some have called it a Geometrical number. Now a Figure, as Euclid saith, is that which is contained on every side with one or more bounds, 14 dj. Such an one is a circle, contained with one line, which they call a periphery. Item, a Triangle, which is bounded with three lines: or a Quadrate, with four lines. Such in bodies is a Cube, bounded with six equal surfaces,& a Prisma, bounded with six unequal surfaces, etc. Now a Rational figure is a figure that is comprehended of the Base and Height rational between themselves, 9 e iiij R: And the Base and Height are said to be rational one to another, when as the rate or reason of both may be expressed by a number of the same measure given; by the 8 e j R. As for example, if the length of a quadrangle given be 14 inches, and the breadth 12, it is said that that quadrangle is a rational figure; because the length, that is, the Base, and again the breadth, or Height are both expressed by a number of the measure given, that is, by a certain number of inches, to wit, that by 14 inches, this by 12. Now to be comprehended of the Base and Height, is when the length is multiplied by the breadth. For this geometrical comprehension, which here is understood, is as it were a multiplication by numbers. So in the former example the Quadrangle before named is comprehended of the Base 14, and of the Height 12. Therefore if thou shalt grant that the Base and Height are rational between themselves, or that their rate may be expressed by a number of the measure assigned: then I affirm, that the numbers of those sides being multiplied the one by the other, shall show the content of that figure. Item that the product of them, that is, 168, is the figurate number expressing that content, as here thou seest. 2 The numbers thus multiplied, or the numbers which make the figurate number, are called Sides or Roots: and the art whereby the Sides of a figurate number given are found, is called The extraction of a root. That number which expresseth the area, or content of a rational figure, is called, as we have showed before, a Figurate number: and the numbers representing or expressing the Height and Base, that is, the numbers thus multiplied, or making this figurate, are by the geometrical term called Latera, Sides: but vulgarly of the Arithmeticians they are called Roots. For as plants and tres do spring from their roots: so these figurate numbers, whether plains or solids, do arise and have their beginning from their roots. If then the sides be given, the figurate is easily found by multiplication. But to find the sides, roots, or numbers whereof any figurate number was made, is a matter not so easy: That art or rule that teacheth to perform this, is called, The extraction of a root; of the Latines it is called, Analysis Lateris: as the multiplication aforesaid or manner of making a figurate number, is called of them Genesis figurati. 3 A figurate number is made by one multiplication, or by many: And either of them is equilater, or unequilater. 4 An equilater figurate is made of equal numbers, or of one number multiplied by itself; which multiplier is also specially called the Side or Root: An unequilater is that which is made of numbers which are unequal between themselves. As for example, the figurate 4, is made by one multiplication of one number by itself, to wit, of the side 2 by itself; Therefore 4 is a figurate equilater, and the side or root of it is 2: So 9 is an equilater, whose side is 3. This side of the equilater by the Arabians is called Radix, that is, The Root, as Schoner testifieth. Item 6 is a figurate of unequal sides, made, I say, of the multiplication of 2 by 3, and therefore 6 is an unequilater figurate. Here observe that an unity doth imitate every kind of equilater: For 1, by multiplication increaseth not, neither yet doth it diminish any whit at all. It remaineth therefore that an unity multiplying an unity, maketh but an unity, that is, it taketh upon it the nature of an equilater. 5 Moreover, an equilater figurate is twofold: either it is that whose true side is to be expressed by a number; or such whose true side may not be expressed by any number. 6 The equilater whose side is to be expressed by a number, is that whose rate or reason unto an 1, may be showed. This division although it be not altogether proper to this place, yet because it is commonly used by the vulgar sort of Artists in this case, we would not omit it. The first sort they call 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, effabile, that is, such as may truly be pronounced or spoken by some arithmetical number: As for example, 16 is an equilater, or a figurate of equal sides, and the true side is 4, that is, a number which may be spoken, uttered and set down by arithmetical figures; I mean whose quantity may be conceived by the reason of it unto 1. For in the quadrate or plain equilater, as the reason of the number given is unto the root thereof: so is the same root unto an unity. For the root or side is the mean proportional between the figurate given,& 1. As in our example; as 16 is unto 4, so is 4 unto 1. And indeed all absolute numbers are conceived& understood by the reason they bear unto an unity; and whatsoever they are, that they are said to be in respect and comparison of an unity. 7 The equilater whose true side cannot be expressed, is a figurate, the reason of whose side unto 1, cannot be told or declared. As for example, 3 may be conceived to be an Eequilater, that is, to be a product made of two equal numbers, or of one and the same number multiplied by itself, which number thus multiplied is, by the former, the side of the said Figurate made. Now this side or number multiplied, is greater than 1. For 1, multiplied by itself, by the 4 e, doth make the Equilater 1; which is lesser than 3. Again the same side is lesser than 2. For 2 multiplied by 2, doth make 4, which is an Equilater greater than 3. Therefore the root or side of 3, doth fall to be some mean quantity between 1, and 2. And yet what that number or difference is, or how specially it is to be understood or conceived, no man may possibly tell. Therefore such a like side is called arrheton, inexplicabile, irrationale, Surde, irrational, or not to be uttered, as the Arithmeticians do call it. 8 A figurate made by one multiplication, is called a Plain: which is a figurate number made by the multiplication of two sides equal one to the other. A figurate made by multiplication, as before is declared, is a number representing a parallelogram, and yet not any indifferently, but that only which is right-angled. Now A Parallelogram is a Quadrangle whose opposite sides are parallel, 6 e x R. And A right-angled parallelogram is a parallelogram all whose angles are right-tangles, 2 e x j R. And because that all parallellograms are plains; and for that plains have but two dimensions, to wit Length& Breadth: therefore, by the 1 e, a plain figurate is made by one multiplication only; that is, by multiplying of the length by the breadth. As for example, 9 is a figurate made of two sides, to wit, of 3 multiplied by 3; and therefore 9 is a plain figurate. Item 12 is a figurate made of two sides; to wit, of 3 multiplied by 4. Therefore 12 is plain figurate, representing a right-angled prarallelogramme surface. Therefore 9 If thou shalt divide an inequilater parallelogram by one side given, the quotient shall be the other side desired. As for example, Suppose the figurate plains were 12, and the one side thereof given were 3: here I say that the other side shallbe 4, to wit, the quotient of 12, divided by the same 3. This is plain by the 1 e 6 j lib. Arithmet. Salignaci. The argument is thus concluded: If a number be made of two numbers given, the one of the numbers given shall divide the product by the other: and chose. But a figurate plain is made by the multiplication of two sides one by the other: 8 e. Ergo, The one side shall divide the figurate plain by the other. This rule only hath place in those examples where the one side of the unequilater plain is given: that which followeth is more general. 10 All the roots of the squares contained in a Figurate plain given, which shall measure the said figurate, with their quotients, shall be all the sides of the said plain given. As for example; Let the figurate plain given, all whose sides are to be found, be 20: the squares contained in the same, whose sides 1, 2, 4, do measure 20, let them be 1, 4, 16: And the quotients by the same sides, let them be 20, 10&, 5: I say that 1, 2, 4, 5, 10, 20, are all the sides measuring 20, the figurate plain given. For although 9, be a quadrate also contained in the said figurate; yet because that 3, the side of the same, doth not measure the same figurate given, it is neglected as nothing pertaining to this our purpose. By this rule, as you see, may be performed that which the 2 verse of the 13 Chapter of the first book of Salignacus his Arithmetic, doth teach; to wit, How to find out all the measures of many, of any compound number given. Let the compound number given, all whose measures of many are to be found, be 60: Here all the squares contained in 60, whose sides may measure the same, be 1, 4, 9, 16, 25, 36: And the quotients of 60, by 1, 2, 3, 4, 5, 6, the sides of the said squares, are 60, 30, 20, 15, 12, 10: Therefore these twelve numbers 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, are all the measures of 60, the compound number given. 11 Like-plaines have a doubled reason of their correspondent sides; and one mean proportional comprehended of the extreme, or mean proportional sides. 20 p. v j: Item 11& 18 p. viij E. Figurates have their denomination of Figures,( as we have showed at the 1 e:) And therefore their nature cannot otherwise be conceived& taught but from the same. Like figures therefore, as Ramus at the 14 e iiij, teacheth, are figures whose corners are equal and proportional to the shanks of the equal angles. Now plains have but two dimensions, and solids three. Wherefore they shall have a doubled reason, these a trebled reason of the correspondent sides. Again here, as Salignacus at the 2 e of the 14 Chap. of the ij book of his Arithmetic, teacheth, to double, treble, or quaduple a reason, is not to add the same reason unto itself twice, thrice, or fouretimes: But to multiply it by itself, twice, thrice, or four times. For example, let 8 and 18 be the two figurates given: And let their correspondent sides be 2 and 3: Item 4 and 6: those the Bases, these the Heights. I say, that the reason of 8 unto 18, is the reason of 2 unto 3, or of 4 to 6, doubled: The reason is thus doubled, : or thus, . The reason then of the first figurate unto the second, is the reason of 4 unto 9: or of 16 unto 36, that is, the reason of the first figurate unto the second, is Subdupla-sesquiquarta. For the consequent being divided by the antecedent, the quotient is 2¼. Therefore in Plains, where the dimensions are but two, to wit, Length and Breadth, the reason is only doubled. This then is the reason of right-angled parallelogrammes, to which Plain numbers do answer. And thus much of the first part of this proposition: The second part, of the mean proportional, followeth. Like Plains therefore, saith our Author, have but one mean proportional, comprehended of the proportional sides. The cause is understood out of the grounds of Arithmetic: For if the reason of a number unto a number be but doubled, then by the rule and nature of numbers continually proportional, there can possibly be but one mean proportional between them. The rule of this invention, is thus, laid down by Salignacus, at the 2 e 12 c ij book of his Arithmetic: If having two reasons given, the second bound of the first reason, do multiply both the bounds of the first reason: and the first bound of the second, do multiply the first bound of the first reason, the products shall be continually proportional to the four numbers given. In the two plains before mentioned, 8 and 18, the proportional sides, or the two reasons given, were 2, 3: 4, 6: The mean continually proportional, according to this rule, is thus: The mean plain then continually proportional, between the plains given, is 12: that is, as 2 is unto 3, so is 8 unto 12: and as 4 is unto 6, so is the same 12 unto 18. This mean proportional therefore now found, is made or comprehended of 4, the length of the first figurate, and of 3, the height of the second: or chose of 2, the height of the first figurate, and of 6, the length of the second: For the product, is still the same: Therefore 12 If the sides of the two like Plains be given, the mean proportional of the same Plains is also given. This is manifest by the former: For there the mean proportional was made either of the middle proportionals, or of the two extremes of the four numbers given. As for example, let the two Plains given be 12 and 48: the length of the first, let it be 4; of the second, let it be 8: The height or breadth of the first, 3; of the second, 6: The product of 3 by 8; or of 4 by 6, that is, 24, is the mean proportional desired. The example is thus: I say, as 12 is unto 24, so is the same 24 unto 48. For the reason in both is subdupla. Thus much in general of Figurate rational numbers, or Figurate plains: it followeth now in particular of their several kinds. CHAP. II. Of the Quadrate. 1 An equilater or equal-sided plain is called a Quadrate, 18 d seven,& 3 e xij R. The unequal-sided plain is called an Oblong. OF the divers sorts of Plains, handled by the Geometers, that only is a Rational plain, all whose corners are equal, and opposite sides parallel. Such an one is the Right-angled parallelogram, which alone of all the geometrical plains is comprehended of the base and height, as before is showed. Now the figurate of a rational right-angled parallelogram, is called a Plain. Admit for examples sake, that the Height of a right-angled parallelogram were 3, and the Base were 4: Here if thou shalt multiply 4 by 3, the product 12, shall be the figurate, or content of the right-angled parallelogram assigned. This product 12, is called a Plain; and 3 and 4, the numbers whereof it was made, are called Sides, by a Geometrical name. And indeed this manner of multiplication, as we have taught before, is merely Geometrical. Place 12 unities,( or, in stead of them, 12 ciphers) in equal distances one from another, as here thou seest; and it shall represent the figure, that is made of two right lines, whereof the one is 3 inches, the other 4. For if thou shalt first divide thelines given into equal parts; the one into 3 parts, the other into 4: and again shalt suppose, the Height to be erected perpendicularly upon the end of the Length: And lastly, conceive first the Height to be carried or moved all along from the one end of the Length unto the other: And in like manner, the Length to be moved upward, for all that whole plumme-line: The traces that are to be supposed those divisions will make upon the plain, shall make 12 squares, within the Oblong thus: This kind of multiplication, I say, is geometrical: For here by this means, of lines are not made lines,( as there of unities do arise unities only,) but a magnitude or bigness, exceeding a line by one dimension, to wit, a surface. Before it was observed, that the figure thus made, is a Right-angled parallelogram. Now a right-angled parallelogram, is either an Oblong, or a Quadrate. An Oblong is a right-angled parallelogram of unequal sides: A Quadrate or Square is a right-angled parallelogram of equal sides: 1& 2 e xij: The figurate of a Quadrate is called also a Quadrate: which is a figurate made of two equal numbers multiplied between themselves. This figurate above all other is to be accounted a Rational: and yet every Quadrate is not a Rational number: For that only is a Rational, whose number is a true Quadrate; that is, such a number is a rational quadrate, whose side is to be expressed by a number. Of this kind is that number only that is made by the multiplication of one number by itself: c 3 e xij: Such are these nine following, made by the multiplication of the nine single figures, or digittes, as they call them, between themselves: The Quadrates: 1 4 9 16 25 36 49 64 81 The Sides, or Roots: 1 2 3 4 5 6 7 8 9 The Quadrate or Square number, is called of the Arabians, Zensus: of the greeks Dynamis a power or valour, as Euclid, Diophantus, and Barlaam do testify. Wherefore wheresoever, in these arts, thou shalt meet with these phrases, Potentia rectae est quadratum, The power of a right line is a square, or, Potentia numeri est quadratus, The power of a number is a Quadrate: there understand that any number given, multiplied in itself, doth make a Quadrate. Item, Si basis trianguli subtendit rectum, aequè potest cruribus: 5 e xij R. If the base of a triangle be over against a right-corner, than the power of it is as much, as the power of both the other sides: that is, if of the three sides of a right-angled triangle thou shalt make three several squares, the square of that side shall be equal to the other two squares, made of those sides which do enclose or contain the rightangle. Thus in Arithmetic one number is said to be able to do as much as two other numbers, when as that number multiplied by itself, shall make as much as those two shall make, multiplied by themselves: thus 10 multiplied by 10 maketh 100 Item 6 by 6, giveth for the product 36: And 8 by 8, yieldeth 64. Now because that 36 × 64, that is, 36 added to 64, do make for the sum 100: And that 100, are equal to 100; Therefore I say that the power of 10 is as much as the powers, or possibilities of 6, and of 8. This kind of multiplication of a number by itself, is called Quadratura numeri, or Genesis quadrati, the squaring of a number, or the making of a Quadrate: And chose, the division of such a like square or quadrate, by the side whereof it was made, is called Analysis lateris quadrati, The extraction or invention of a square root. Now as every Multiplication, and Division, so also this squaring of a number, and division or extraction of a square root, is done either jointly by the whole, or severally by parts, at many operations. The first is to be done and performed by Pythagoras' table, which is set down a little before: The second is to be done by the prescript of the Rule next following. But first that Table is to be learned by heart. 2 If a number be divided into two parts, the Square of the whole number shall be equal unto the squares of the parts, with a double plain made of them both. This in Geometry, spoken of a rightline, is the 8 e xij R: and is thus laid down by Ramus: Si recta est secta in duo segmenta, quadratum totius aequatur quadratis segmentorum, cum duplici rectangulo utriusque; If a right line be cut into two portions, the square of the whole line, is equal to the squares of the portiós, with two right-angled parallelogrammes made of both the said portions. It is a consectary drawn out of the 3 c two e x book of Ramus his Geometry. Suppose the rightline given were a c: let it be in b, divided into two parts, a b, and b c. Now let the quadrate made of the whole line a c, be a c l d: and from the division b, drawn a line parallel to the side c l: Item, at the same distance, draw another parallel to a c; and thou shalt divide the quadrate a c l d, into two sorts of right-angled parallelogrammes, to wit, into c i, and d i, two diagonals: Item, into a i, and l i, two compliments, as the Geometers call them. Now that these two diagonals, with the two compliments, are equal unto a c l d, the quadrate made of the whole line a c, it is most manifest. For the whole is equal to all his parts jointly taken. But the two Diagonals, and two Compliments aforesaid, are all the parts of the Quadrate a c l d, as here thou seest. Therefore if a rightline be divided etc. This rule the Arithmeticians, speaking of numbers, do set down in these terms: If a number, etc. Which Barlaam hath arithmetically demonstrated; to whom I refer thee. Out of this rule is framed the Quadrature or Fabric of the whole square, by the two segments or portions of the side; and that in this manner: As for example, let the number given be 7, and the quadrate of the whole number let it be 49. Then divide the same into two portions, to wit, into 4 and 3. And let these two numbers multiply one another, the four products 9, 12, 12, and 16 added, shall make 49, equal unto the former product, made of 7, the whole number. Again, let the number given be 13; and the square of this whole number let it be 169. Now let 13 the whole line, be divided into two parts, to wit, into 10 and 3: let these multiply one another, and the several products shall be four, the sum of which added together, shall be equal unto the quadrate or product made of the whole line 13: the example is thus: Therefore 3 The side of the greater Quadrate, is the side of one of the Plains; and being doubled, it is the side of them both together: The other side of both the Plains jointly taken, is the side of the lesser Quadrate. It followeth necessarily of the former. This rule teacheth how to resolve a Quadrate given, or the manner of Extraction of a square or quadrate root. The former rule taught that the true squaring of a number, divided into parts, consisted of two squares and two plains. So that the true resolution of a Quadrate so made, must be the invention of the several sides of the said particular squares and plains, whereof the whole Quadrate given consisteth. For it is the same way from Cambridge to London, that it is from London to Cambridge. By the former, the side of the square 169, was 13: which side or number consisteth of two several numbers, to wit, of the article 10, and the digit 3. Here, by the former rule, the greater square is 100, and the side thereof is 10: The one Compliment or Plain is 30, made of two sides, where of the one is 10, the side of the former square: Therefore 30, is the side of 60, the sum of both the plains. The other side of the Plain( single or double) is 3, which is also the side of 9, the lesser square. These grounds thus laid, the practice is to be performed according to the direction of this rule next following. 4 If the side of some greater square be sought, first beginning at the right-hand, distinguish the number given continually by pairs, for so many particular squares: Then setting down the side of the first particular square found within the Quotient, shalt double the side found, for the base of the first Compliment. Lastly, the Compliment divided by the Base, the Quotient shall be the side of the next succeeding Square. As in 144, the example so oft mentioned, first beginning at the right-hand, or first figures, I distinguish it thus, I, 44; whereby I understand that the particular squares, whereof the whole side is comprehended, are two. This done, I seek amongst the squares of the single figures, at the 1 e, for the side of 1, the first square,& I find the side to be also 1: This side I place in the quotient. This square I subtract from 1, the last period, and there remaineth nothing. Secondarily, I double the same quotient, or side found, and I make 20( for indeed the first diagonall's side is 10; and the side of both together jointly taken is 20; as we saw at the 3e.) By this I divide 44, the whole period, or number remaining, and I find the quotient to be 2, for the other side of the Compliments, and the root of the lesser diagonal: Therefore I multiply first this quotient last found by itself, and I make 4, for the lesser diagonal or square. Again I multiply the same quotient by 20 the divisor, and I make 40, for the doubled Compliment. Lastly, I subtract 44, the sum of the said doubled compliment, and lesser diagonal, from 44, the number remaining, and there remaineth nothing: Therefore I say that, 144 the figurate given, is a true Quadrate; And the side or Root thereof is 12, which was desired. Item, Let the side of the Quadrate 9604 be sought. First, beginning at the right hand, I distinguish the quadrate given continually into pairs of degrees, thus 96, 04; and I find the number to contain two such pairs: And therefore I conclude that the whole side of the quadrate given doth consist of two single figures. This done, I first seek amongst the squares of the single figures, at the 1 e, for the square 96; which number, because I find it not amongst them, I say is no true square. Now the greatest square contained in 96 I find to be 81, and the side or Root of it to be 9 This side I place within the quotient. Then multiplying 9 by itself, I make 81, for the greatest diagonal, which placed under 96, and subtracted from the same, there do remain over the head thereof 15: Now canceling 04, I place them also over the head, as high as the said 15. Secondarily, doubling 9 my quotient now found, I make 18( or, for the reason before recited, 180) for the Base of the Compliment: Then by this base I divide 1504, the number remaining, and I find the quotient 8. This therefore I likewise set down by 9, for the other side of the Plains, as also for the side of the lesser Quadrate. For the proof of this later work, I multiply first the said quotient by itself: Then I multiply the same quotient by the doubled Compliment, that is, by the divisor: Lastly I subtract the sum of these two products from the number remaining. As in this our example, I multiply 8, the quotient, by itself, and I make 64, for the second or lesser quadrate: Then I multiply 180, my devisor, or base of the compliments, by the same quotient 8, and I make 1440, for the said double Compliment: Now 1440 × 64, or the sum of 1440 and 64, is 1504 This I subduct from 1504, the number remaining, and there is left nothing: Therefore I say that 9604, the number given, is a true Quadrate, and the side or Root thereof is 98, which was desired. The example is thus: Therefore if thou shalt multiply 98 in itself, thou shalt make 9604, the number given. For the Quadrates of the Segments 90, and 8, with their Plains or Compliments, are the parts of the Quadrate of the whole number 98. The genesis or making of this Quadrate, after our prescript, is thus: 5 Having found the Quotient of two, or more figures, if yet the whole side of the Quadrate given be not found, thou shalt double the whole quotient already found: And then shalt in all things else whatsoever, observe the same order as was prescribed in the former. And in this example: The side of 15129 is desired: First, having divided it, as in the former we have taught, continually into pairs, thus, 1, 51, 29, I find 1 to be a Quadrate; and 1 also to be the side thereof. This side I write in the Quotient. The quadrate of this quotient I subtract from 1, and there remaineth nothing. Then I double the said quotient 1, and I make 2,( which in respect of the next period I call 20) for the base of the doubled Compliment: By this I divide 51, the next period, and I find the quotient 2, for the height of the said Compliment, and side of the lesser quadrate. This latter quotient therefore I multiply first in itself, and I make the quadrate 4: Then again by the same quotient I multiply my divisor 20, and I make 40, for the doubled compliment: Now 40 × 4, are 44, which I subtract from 51, and there remain 7: Therefore the doubled Compliment, with the second Quadrate, is 729. Lastly by this rule, I double the whole quotient 12, and I make 24, or in respect of the period following 240, for the base of the doubled Compliment: now the quotient of 729, by 240, is 3, for the side of the lesser quadrate, and height of the said Compliments. This done I multiply 3, the last quotient found by itself, and I make 9, for the lesser quadrate. Item I multiply by the same quotient 3, the devisor 240, and I make 720, for the doubled Compliment. Now 720 × 9, that is, 729, I subtract from 729, and there remaineth nothing. Therefore I conclude that 15129 the number given, is a quadrate: And the side thereof is 123, consisting of three figures. The whole work is thus: In greater examples there is yet greater variety to be observed. Let therefore the practioner resolve this one great Quadrate, 61929672906515252224, such as are often to be resolved in the making of those tables of Sines, Secants, and Tangents, of so great and wonderful use in many businesses, where the use of these Arts are required. From hence do follow many particular Consectaries, First the Single figures of any Quadrate given cannot exceed the double of the number of the single figures of the side thereof. The reason is from the 1 e ij: to wit, because the product of the greatest single figure by itself, consists not of more than two figures. And 2 If the number given beginning at an unity, and so increasing according to the natural order of numbers unto the middlemost, shall from the same, in the same manner decrease unto an unity, the side of the said quadrate shall consist of unities: And the same middle number shall show the number of them. As for example: Suppose the number given were 1234321, here because it beginneth at an unity, and so increaseth unto the middlemost, and from the same middlemost decreaseth in like manner continually unto an unity: And for that the same middlemost number is 4: I say that the side or root of the quadrate given consisteth altogether of unities: And that the number of them is but 4, thus 1111. Item, the side of this quadrate following, 12345678987654321, is 111111111. Therefore on the contrary, If a number to be multiplied by itself, do consist altogether of unities, under the number of ten, the product shall from an unity increase and decrease, as aforesaid. 6 The difference of two unequal quadrates given, is the quadrate of the difference of their sides, with a double plain made of the same difference, and the lesser side. This is called the Gnomon, or squire, 2 d xij; or 12 e x R. As for example, Let the two Quadrates given be 144, and 100; And their Difference let it be 44. Again, Let their Sides be 12 and 10; And their difference let it be 2. Now let the Quadrate of 2, the Difference of the sides be 4. Again, Let the Plain made of 10, the lesser side,& of 2, the Difference of the sides, be 20. Now 20 × 20, are 40. And 40 × 4, are 44, the Gnomon or difference, whereby the two Quadrates 144, and 100, do differ one from another. Therefore 7 If the number given be not a true Quadrate, the remain( to be added unto the side of the greatest Quadrate contained in the said number given) shallbe denominated of the Gnomon, or difference of that said greatest Quadrate, and that which is next above it. Let the number given be 148, which by the 5 e, is no true Quadrate: and therefore the true side thereof can never be found. But the side, somewhat near unto the true side, is thus found. The greatest Quadrate contained in 148, the number given, 144, whose side by the said 5 e, is 12. Item, the Remain or difference of 144,& 148, is 4, for the Numerator of the parts sought. Again, the Quadrate next greater than 144, is 169, and the side thereof is 13. Lastly, the Gnomon or Difference between 169, and 144, is 25, the denominator sought. Therefore 4/25, are the parts desired. The side then of 148, near unto the true side, is 12 4/25. Briefly, The Denominator of the parts sought, is the difference of the greatest square contained in the number given, and the next greater above it. Let the number given be 11: the greater Quadrate contained in it is 9, whose side is 3. Now 11— 9, is 2, for the Nummerator desired. Item, the next Square greater than 9, or 11, is 16, and the side thereof is 4. Now 16— 9, is 7, for the Denominator of the said parts. The side of 11, near unto the true side, is, 3 2/7. Item, The said Denominator is the double of the side of the greatest Quadrate increased by an unity. In the last example, the side of the greatest quadrate contained in 11, is 3; whose double is 6. Therefore 6+ 1, that is 7, is the Denominator sought, and the parts are 2/7. Again, 3 The denominator of such like parts shall be the sum of the side found, and of the side of the next greater quadrate above it. In the former example, the side of 9, the greatest Quadrate contained in 11, is 3: and the side of 16, the quadrate next above 9, is 4. Now 4 × 3, that is 7, is the Denominator desired: and the side sought is 3 2/7. 8 The product made of two like-plaines, is a Quadrate: and the side thereof is the product of the base of the one, by the height of the other. Like-plaines, are plains whose sides are proportional, 20 d seven. Let the Like-plaines given be 6, and 24: whose sides, 3, 2, 6, 4, are proportional: that is, as 3, the base of 6, is unto 2, the height of the same: so let 6, the base of 24, be unto 4, the height thereof. Here first I say, that 144, the product of 24, by 6, is a quadrate. Again I say, that 12, the product of 3, the base of 6, the first plain, by 4, the height of 24, the second plain: or chose, the product of 2, the height of 6, the first plain; by 6, the base of 24, the second plain, shall be the root or side of the said Quadrate. The example is thus: The Quadrate. The side or root thereof. 9 If three numbers be continually proportional, the product of the first and the last shallbe equal to the quadrate of the middle numbers; And chose, If the quadrate of the middle number be equal to the product of the first and the last, the three numbers given shallbe continually proportional. Let the three numbers given continually proportional, be 4, 6, 9, the product of 4, by 9, the first and the last, shall be 36: Item the square or the product of 6 by itself shall also be 36: On the contrary, if the quadrate, square, or product of the middlemost be equal unto the product of the first and the last, the three numbers given are continually proportional: As Euclid doth demonstrate and teach at the 16 p uj, and 20 p seven, of his Elements. Item, Ramus handleth this proposition in three several places; to wit at the 15 eij of his Arithmetic: At the ij consect. of the 14 ex, and again more specially at the 4 e x ij, of his Geom. Yet in deed, saith Schoner, there is something in this more than may be performed by that 15 chap. of the second book of his Arithmetic. For division cannot always find out the mean proportional: But in diverse cases there is required, to the performance of this, the extraction of a square root( of which we have spoken before at the 4 and 5 e:) by the which also the Devisour is sought. As for example, suppose the mean proportional between 16, and 64, the extremes given, were to be sought. Here the product of 64 by 16, is 1024; and the side of 1024, by the 4 e, is 32, the mean proportional desired. A proportion requireth four bounds or numbers. If the second and third bound be the same, as in this our example, it is called a Continual proportion: If they be diverse, it is named a Disjoined proportion. Now generally, whether the proportion be disjoined or continual, this rule is true: If four numbers be proportional, the product of two middle numbers shallbe equal to the product of the first and the last. As for example, As 12 is to 4, so let 6 be to 2. I say the product of 6 by 4, shallbe equal to the product of 12 by 2. For let the product of 12 by 6, be 72. Here therefore the product of 6 by 12, shallbe also 72, by the 3 c iiij c j Saligna. If two numbers be proportional to one and the same number, they are equal between themselves: 16 e v c j Salig. But 72 is proportional unto the product of 12 by 2: and again, to the product of 6 by 4 10 If one number do multiply many numbers, the products shallbe proportional unto the numbers multiplied: 5 e vi. But 12 doth multiply 6( and maketh 72 by the construction:) Item it multiplieth 2. Therefore as 72 is unto the product of 12 by 2: so is 6 unto 2: and so, by the grant, is 12 unto 4. 20 If one number do multiply many, etc. 5 e uj c j Salign. But 6 doth multiply 12, and 4. Therefore as 72( which is the product of 6 by 12, as before was manifest) is unto the product of 6 by 4, so is 12 unto 4. Therefore the products of 12 by 2, and of 6 by 4, are equal. q. e. d. This proposition is commonly called the Golden rule: and indeed truth it is that the Mathematical arts do daily reap from hence fruits in value more worth than gold. 10 If the three numbers continually proportional given, be simple between themselves, the first& the last shallbe quadrate or square numbers: Item, If of three numbers continually proportional given, the first be a quadrate, the third also shallbe a quadrate. As for example, Let the three numbers continually proportional, and simple between themselves given, be 4, 6, 9; the first and the last, to wit 4 and 9, are quadrate numbers. Again, if 16, 32, 64, be continually proportional,& 16, the first bound be a quadrate; 64 also, the third bound shallbe a quadrate. These both are demonstrated by Euclid, and his Expositors at the 2, and again at the 22 prop. of his viij book. 11 If a quadrate do multiply a quadrate( or one many continually) the product shallbe a quadrate; and the side of the quadrate so made, shallbe the product made by a continual multiplication of the sever all sides of the particular quadrates given. As for example; Let the quadrate 4, multiply the quadrate 9, and let the product thus made be 36: Here first I say, the product of 36 is also a quadrate. Item I say, that 6, the product of 2 by 3, the sides of 4& 9, the particular quadrates given, shallbe the side of 36, the compound quadrate. Item, 25401600, the product continually made of 4, 9, 16, 25, 36, 49, is a compound quadrate: And the side thereof is 5040, which is the product made of 2, 3, 4, 5, 6, 7, continually multiplied between themselves. 12 If a Quadrate do divide a number assigned by a Quadrate, the assigned shall be a Quadrate. This also ariseth from the same fountain. As for example, Let 4, a quadrate divide 64, the number assigned by 16, a quadrate. I say that 64, the number assigned is likewise a quadrate. 13 If the product of two numbers assigned be a quadrate, the side of that quadrate shall be the mean proportional between the assigned. As for example, Let the nuumber assigned, between which we desire the mean proportional, be the quadrates 144, and 64; and let the product of them be 9216: I say that 96, the side of the said compound quadrate made, shallbe the mean proportional desired, that is, as 144 is unto 96: So 96 is unto 64. The demonstration is built upon the 9 euj cj Salign. which is a special consectary drawn from the 2 c 14 e, of the x book of Ramus his Geometry: The argument is thus framed If the product of the two middlemost numbers be equal to the product of the first and the last, the four numbers given are proportional between themselves: 9 e uj cj Sal. But the product of 144 by 64, is 9216 by the grant: And the product of 96 by 96, the 2 middle numbers, is also 9216. The side is a number, which multiplied in itself, doth make a quadratc. But 96 is the side of 9216, by the construction and grant. Therefore 9216 is made of 96, multiplied by itself. Therefore 144,96,96,64, the four numbers given, are proportional between themselves: and so the mean proportional between 144, and 64, the two numbers given, is found, q. e. f. Therefore 14 If a number shall by itself divide a number made of any two numbers given, the quotient shallbe the mean proportional between the two assigned: 10 e v j c i Salig. The rule is general thus: If one number shall divide the product of two assigned, the devisor and the quotient shallbe the mean continually proportional, betweexe the assigned. As for example, let 6, multiplying 4, make 24: And let 8 divide 24 by 3: I say, as 6 is to 8, so is 3 to 4. But the special consectary is more fitter for our purpose. Therefore for example, Let the product 144 by 64, be 9216, and let 96 divide 9216, the said product, by 96: I say, that 96 is the mean proportional between 144, and 64, the two numbers given: that is, as 144 is unto 96, so is 96 unto 64. CHAP. III. Of the making and use of certain Tables and Instruments, devised for the more exact and speedy measuring of all sorts of Plains. 1 Plains, according to their divers nature and quality, are measured by divers and sundry kinds of measures: For some are measured by the Foot, others by the Yard or Elne, othersome by the Rod or Perch, and such like. Now these measures being defined by Act of Parliament, it shall not be amiss to set down the words of the Statute, so far forth as it shall concern this argument. It is ordained, saith the Statute, that three grains of barley dry and round, do make an Inch; twelves inches do make a Foot; three foot do make a Yard; five yards and an half, do make a Perch; and forty Perches in length, and four in breadth, do make an Akre. 33. Edw. ay, De terris mensurandis. Item, De compositione Vlnarum& Perticarum. See also more of this hereafter. By the Foot we measure Board and Glass. A Foot therefore of flat measure, is a right-angled square, 12 inches long, and 12 inches broad, that is, a foot of board is a plain, containing 144 square inches: For such is the product of 12 by 12. Here, as also in the other which follow, observe, That the Breadth is always given; the Length is desired. If therefore the breadth given be 12 inches, it is plain by the former definition, that 12 inches of length doth make a foot of flat measure. But if the breadth given, be either greater or lesser than 12, the length desired is not so easily found. For here some art is oft times required. This then is to be conceived and done by the 14 c of the former Chapter: For there we learned how to equal plains of divers breadths. Admit now a board to be measured were but 9 inches broad; here, as those rules did teach us, I divide 144, by 9 the breadth given, and I find in the quotient 16, for the length desired. For as 12 is to 9, so is 16 to 12; that is, 16 inches in length, of the breadth of 9, are equal to 12 inches of length, of the breadth of 12. Thus you may make a Table or Instrument to serve readily at all times, for the more speedy and exact measuring of all sorts of plains by the foot square. For, If you shall divide 144, by the sever all breadths given, the quotients shall show the length desired, answering to their sever all divisors or breadths given. Now where you do begin( I mean whether at the greater breadth, and so descending shall end at the lesser, or chose) the matter is not great. Again, the Carpenters or workman's instrument which they use in this case, being but two foot in length, it shall not be necessary to begin at any breadth greater than 24, as shall be manifest hereafter, when we come to show the use of this Instrument or ruler. Beginning then at 24, and so descending to the lesser, the Table is thus: The use of this Table, to him that understandeth the former, is plain and easy. For having any breadth given, between 24 and 1, I seek it amongst the breadths in that column upon the left hand, and in the other column upon the right hand overagainst it, I find the length desired. 1 As for example; A board to be measured is 18 inches broad, I desire to know what length of that breadth shall make a foot. R In the second column, overagainst 18, in the first column, I find 8. Therefore I say, every 8 inches in length, of that board that is 18 inches broad, shall make a foot of flat measure. 2 Again, suppose the breadth of a piece of Glass to be measured, were 5 inches broad. Here I find, that every 28 inches, and ⅘ of one inch, doth make a foot of that kind of measure. 3 But what if the breadth given be greater, or lesser than any breadth between 24 and 1? I answer, the question is answered with as great facility. For first admit the breadth given were 36, which is greater than any breadth in our Table, here if I take some such known part of 36( partem aliquotam, a known part, the Arithmeticians call it) as may be found in our Table, the length desired shall answer unto it in the like proportion: As for example, if I take 18 the ½ of 36: then it is manifest, the length desired shall be but 4 inches, the one half of 8, which is the length answering to the breadth 18. Or, if I shall amongst the Breadths take 12, which is but the third part of 36, my breadth given: then also shall 4, which is the ⅓ of 12, the length found, be the length desired, answerable to 36 the breadth given: Or which is all one, every X2 inches in length, of that board that is 36 inches broad, shall contain three foot of flat measure. 4 Again, suppose the breadth given were an 100 inches. R Here 20, is but the ⅕ part of 100 Therefore I say that 7 ● in length, of that plain that is 100 inches broad, shall contain five foot of flat measure. 5 Lastly, Admit that the breadth given were but ½ inch, which is lesser than any one of those set down in our Table. R, Here I double, treble, or quadruple the number given, until I may find it in the Table, and then it is manifest, that the length found shall be answerable to the breadth given in the like proportion. As for example, If I double ½, I make 1. Now to the breadth of 1 inch, in the Table there answereth 144 inches of length: Therefore the double of 144, that is, 288 shallbe the length desired answering to ½ inch of breadth. Or, which is all one, 144 inches of length of that breadth, that is but ½ inch broad, shallbe but ½ foot of flat measure. The reason of this is manifest, out of the 14 element of the tenth book of Ramus his Geometry. By this Table you may make a Ruler or instrument, whereby the same may be performed yet with greater speed and more facility. For this instrument is none other than that which Attificers do use in this case. It is, I say, no other but a Rule, as they call it, of two foot long( more or less, as you shall think most convenient for your use) and of what breadth you please, divided, as the manner is, into inches and parts of inches, with a thwarting or bevelling line drawn from side to side, beating the several lengths above mentioned in our Table. That, for thy better understanding, we do more particularly thus describe. First with the iage I strike two parallel lines, the one as near unto the one edge, the other unto the other, as shall be thought most convenient: These are the breadths, 24 and 6. Now to the breadth 24, in the Table, there answereth 6 inches for the length: Therefore in the upper line beginning at the right-hand, I accounted toward the left, 6 inches: There I make a mark. Again, to the breadth 6, in the same Table, there answereth 24 inches for the length: Therefore in the neither line counting as afore, I make a mark at the 24 inch. From these two marks I draw a line overthwart the ruler. In this bevelling or thwarting line, by the help of a squire, I note all the lengths noted in my Table, between 24 and 6. As for example, To the breadth 23, in the Table, there answereth for the length 6 inches, and 6/23 parts of one inch. This length, I note, beginning mine account as before, in one of the parallel lines. Then by the help of the squire I note the same upon the bevelling line. In like manner I set all the rest of the breadths from this, unto 6. This being done the ruler is perfected, and fit for use in all cases, as before is prescribed in the use of the Table. The use of this ruler is plain and easy, if either that of the table, or the manner of making this be well understood. For, If you shall seek out the breadth given in the bevelling line, the inches& parts of inches from thence unto the end of the ruler toward the right-hand, shall be the length desired. I. As for example, Suppose the breadth given were 12 inches: Here I find the breadth 12 noted at the twelfth inch from the beginning of the ruler: Therefore I say, Every 12 inches in length, of that plain that is 12 inches broad, shall be a foot of flat measure. II. Again, Admit the Breadth were 9 inches. This is noted in the thwarting line at 16 inches from the said end. Therefore I say, That 16 in length of this breadth, shall make a foot of flat measure. III. Admit the breadth given were 64 inches. Here because I have none so great, I take 16, the ¼ of it. Now the breadth 16, is noted on the bevelling line, 9 inches from the said end. Therefore I say that every 9 inches in length, of that plain that is 64 inches broad, doth contain 4 foot of flat measure. IV. If the breadth given beleffer then any upon the bevelling line, then double, treble, or quadruple, etc. the same, as afore is taught; and the length found shall be but the half, third part, or fourth part etc. of the length desired. As for example, Admit the breadth given were 2 inches: Here because upon the bevelling line I find no number less than 6: Therefore I triple 2, and I make 6. Now 6 being placed upon the line of breadths, or bevelling line, 24 inches from the said fore-end, I say, That 24 inches of length, of that plain that is but 2 inches broad, is but ⅓ part of a foot of flat measure. V. If the breadth given, besides the whole inches, do contain also some part or parts of an inch, then that is to be proportioned out between the whole number given, and the next greater above it. As for example, Admit the breadth given were 9 inches& ½: Here the length desired will fall out proportionally between 9 inches of breadth and 10, about 15 inches and 3/19, from the said end. Of the measuring of Cloth, Wainscot, Painting, Paving etc. by the yard. 2 By the Yard we do measure Cloth, Wainscot, Painting, Paving, etc. The yard, as is aforesaid, doth contain in length 3 foot, or 36 inches. Therefore the yard-square, or a yard of flat measure doth contain 9 foot, or 1296 inches. This kind of measure they commonly divide into 4 quarters: And every quarter into 4 nails. Upon the former grounds the like Table and instrument for this kind of measure, may be framed as was for that of board, if any man shall think it worth the while. But it shall not be much loss of time or labour, to show in a word or two how it may be done. In a yard-square there is 16 quarters of a yard; or 64 nails: Therefore if you shall divide 16, or 64, by the breadth given, the quotient shall be the length desired. The Table then for this kind of measure is thus: The use is plain. I. Admit a piece of Wainscotte were three quarters of a yard broad, what length of that breadth shall make a yard of flat measure? R. one yard, one quarter, one nail and half, or thereabout. The proportion, as afore, is thus: As 4 is to 3, so is 5⅓ unto 4. 2 A piece of painting is 2 quarters, that is, half a yard broad; what length shall make a yard square? R. two yards in length. 3 But the use of this is better seen in buying and selling of cloth: For here oft times the skilful Tailor, although he do well know how much cloth or stuff of this or that breadth will serve to make such a garment, is clean to seek how much shall make the like garment of stuff of any other kind of breadth: Only this advantage he hath, that he will be sure to ask enough, which shall be no loss to himself. 1 Admit 4 yards of cloth of 1 yard broad did make a garment: how much stuff of ¾ broad shall make the like? R. The Table for the breadth 1 y, giveth 1 y, for the length of 1 yard square: Therefore 4 yards of this breadth, do make 4 yards square. Secondarily, the same Table for ¾ breadth, giveth the length 1 y 1 q 1½n. Therefore four times 1 y 1 q 1½n, that is, 5 y 1 q 1 n shall make the same garment. 2 Again, 12 yards of stuff of ¾ broad, did make a gown; how much cloth of 1 y 3 q broad shall make the like? R. The Table for ¾ breath, doth give the length 1 y 1 q 1½n: This length I seek how many times I may find in 12 yards: that is, I seek how many square yards 12 yards of ¾ broad doth contain. Again, the same Table for 1 y 3 q breadth, giveth 2 q ½n. This length so many times taken as you found the former length in 12, is the number of yards required. Of measuring of Land by the Acre. 3 BY the Rod we measure Land, Mcdowes, Wood, grass, Corn, etc. The Rod, lug, Perch, or Pole doth contain, as we have heard, 16½ foot, or 5½ yards. This kind of measure did vary according to their customs with the country: But now by Act of Parliament that variety is taken away. The words of which Statute, entitled, An Act for restrains of new buildings, etc. in and near to the cities of London and Westminster, made in the xxv year of the reign of Queen Elizabeth, are these: Be it enacted by the authority aforesaid, that a Mile shall be taken and reckoned in this manner, and no otherwise: that is to say, a Mile to contain eight Furlongs; and every Furlong to contain forty Lugges or Poles; and every Lug or Pole to contain sixteen foot and an half. Thus far that Statute. Now an Acre of land, as we have showed before, is a plot of ground containing 40 Rods in length, and 4 in breadth; or, which is all one, an Acre of land containeth 160 square Rods or Poles of ground. If therefore you shall divide 160 by all the sever all breadths between 40 and 1, the products shall show the lengths desired, answering to those sever all breadths given. The Table then of Land-measure is thus: The use of this Table is in all respects like unto that of boord-measure: Therefore one or two examples shallbe sufficient to make it known unto the simplest. I Suppose a piece of ground to be measured were 32 pole in Length: I demand how much in breadth is required to make an Acre. R. To 32, the length in the first column answereth in the next column 5: Therefore I say, every 5 rods in breadth, shall be equal to that field or plot which is 40 pole long, and 4 pole broad. two Admit the field to be measured were 23 pole long. R. To 23 in the first column I find answering in the second column 6 22/33, that is, 6 pole, and 22 parts of a pole divided into 23 parts. Therefore I say every 6 22/33 poles, or 7 pole ferè, is equal unto that which is 40 long, and 4 broad, that is, to 160 square poles. III But what if the length be greater than any number found in the first column? R. Here as before, I take some part aliquotan, as one half, one third, one fourth etc. As for example, suppose the field to be measured were 100 poles in length, here because I find no number so great as 100, therefore I take 20, the fifth part of 100: To which 20 I find 8 pole for the breadth. Wherefore the fifth part of 8 rods, that is, 1 rod and ⅗ parts of one rod of that length shall be equal to that plot which is 40 long, and 4 pole broad. Or, which is the same in effect, Every 8 pole in breadth, of that field which is 100 pole in length shall contain 5 Acres of land. IV Admit it were 84 pole in length. R. Here the fourth part is 21. Now to 21 pole of length, there do answer 7 poles and 13/21 parts of one pole in breadth. Therefore every 7 poles, and about two third parts of a pole in breadth, of that length, shall make 4 Acres. For if it were but 21 pole long, than every 7 pole and 13/21 of breadth, shall make one Acre. But 84 containeth the length 21, four times: Therefore 7 13/21 of the whole length 84, shall contain 4 Acres. Having finished this Treatise, and it being altogether ready for the Press, turning over the Statutes, to see what I might find more for this our purpose, I light upon the very like Table, made many years since, by Act of Parliament, in the time, as I take it, of Edward the first, entitled De terris mensurandis. See the latter Abridgement of the Statutes of Rastall or Poulton, in the title of Weights and Measures. A very speedy and ready way or manner of Division in this case of surveying. 4 SVrueyours, which are often employed in measuring of whole Farms, Lordships and Towns, are at last to give an estimate of the gross sum or number of Acres contained in some very great sum or number of Rods, after multiplication of the sides, and addition of many several parts, etc. This usually they perform and find by dividing of the product or sum by 160, the number of poles that one Acre doth contain. Now here because the Divisor consisteth of many several or single figures, if especially the product or dividend be any thing great, the work, as every one meanly practised in Arishmeticke doth know, must needs be long, and peradventure( except great heed be taken) not without some error. And again because, Faciliùs& expeditiùs numer amus numeros paruos, quam magnos, we do more readily, and with greater facility number small numbers, then great: If any man shall teach how this Divisor may be reduced to a small number, consisting of one single figure, he shall much shorten the work, and deserve much thank of all Surveyors or others delighted in these studies. Ingenui pudoris est, saith Pliny, fateri per quos profeceris. I confess the invention is not mine: but I am the first, if I be not deceived, that hath made it publici juris, common to all. Thus than this matter is to be done. If you shall divide the number given, and likewise the quotient now found, continually by 20 four times, the last quotlent( with the parts, if any do remain) shall show the Akens, and parts of an Acre contained in the number given: or, if you shall like better, shall divide the number given, by 40 twice. Let the examples be those following. I. Suppose a field being measured and cast up by multiplication, the product or number of perches were 1280, I would know how many Acres it doth contain. Here note out of an abridgement of division, because my Divisor 20, hath for his last figure a cipher or naught: therefore I cut off the last figure of my Dividend, thus 128( 0. Or thus, R. I answer by the last quotient of both forms, that that field which containeth 1280 square perches, doth contain 8 Akres. II. Item, Suppose the product or gross sum of some piece of land surveyed were 102400 perches. Or thus, Here I say, that 102400 perches, do make 640 acres of land. These examples may suffice in such cases where no fraction shall remain: but if in any of the works of division, any parts shall remain, the value of those parts shall be esteemed by these rules following. 1 If the first figure( the last, some peradventure would call it) of the number given be a significant figure, than it is manifest, that over and above certain acres, etc. there are some odd perches. 2 Secondarily, if dividing by 20, there do remain aught in the first division, that remainder shall be ¼ of 2 Rood or 10 perches: if in the second, ½ Rood or 20 perches: if in the third, ¼, of an Acre, or 40 perches: if in the fourth, ½ of an acre, or 80 perches. 3 Lastly, if dividing by 40, there shall aught remain at the first division, those parts shall be parts of a Rood: if at the second, they shall be parts of an acre. The examples following, shall make all plain and easy. Or thus by 40: Therefore that field that containeth 939 perches, doth contain 5 Acres, 3 roods, and 19 Perches. two Or thus: The number 13593, containeth in land-measure, 84A, 3R, 33P. or 84 Acres, 3 roods, and 33 Perches. The like may be done in Boord-measure, and Timber-measure, if any man shall find it of any such use, as it is in this kind. CHAP. four Of the Extraction of a Cubic root. 1 A Cube is a right-angled solid, comprehended of equal sides: 25 d xj E. HItherto we have spoken of the Quadrate, and of such proprieties and corrollaries as did belong unto this our purpose. Now it remaineth that we in like manner do handle the Cube and Cubic number, and that with as few words, and as briefly as we may. For as the uses of the Quadrate, and of the extraction of his Root were many: So the Cube, being of a more excellent nature, cannot yield lesser, if not more, and those also of more and greater worth. For as vitrvuius, in the preface to his fifth book of Architecture, writeth, Pythagoras was so much delighted with the Cube, that he wrote all his precepts in cubical numbers. His words are these: Etiamque Pythagora, hisque qui eius haeresim fuerunt secuti, placuit Cubisis rationibus pracepta in voluminibus scribere, constitueruntque Cubum 216 versuum, cosque non plus quam tres in una conscriptione oportere esse pusarunt, etc. Moreover also Pythagoras, and those which followed his faction, were much delighted to write their precepts and rules of Philosophy in a kind of cubical proportion, making a Cube of 216 verses; deeming that there ought not to be above three in one staff. Now a Cube is a square body, consisting of six sides, plain and equal one to another. This kind of body, when it is cast at an adventure out of the hand, upon which side soever it pitcheth, no man touching it, standeth firm and constant. Such are the dice, which gamesters that play at Tables do use. The Greek comical Poets also, which in the midst of their plays do cause the Choristers to sing a song, have so divided the pauses of their comedies, that making the parts in a kind of cubical proportion, they much, by such rest, do ease and help the pronunciation of their actors. Thus far vitrvuius. This Cube or cubical number of 216, hath for his side 6. For 6 times 6, do make 36, for the one side, or square plain including the Cube. And 6 times 36, are 216, the Cubic number of the Cube here mentioned. The mystery hereof conceived by Pythagoras and his scholars, I leave to others to unfold. That pertaineth not to the Mathematician: that other also belongeth to the Poets, whereof our age doth afford plenty. That which the same author in another place hath, of the answer of the oracle of Apollo: Item, that of Eratosthenes unto Ptolomey king of Egypt, of Glaucus his tomb, do more concern our business: And therefore hereafter, in their place, we shall handle them at full, if God permit. Hereyou see how the definition of the Cube doth answer to the definition of the Quadrate: for each of them is a right-angled and straight-bounded figure: And as the general differences of bodies or solids, were drawn from the general differences of surfaces, plain and oblique: So here the particular differences are taken from the special differences of the same surfaces. A plain solid is that which is comprehended of plain surfaces: this is general of what kind soever those plains are of: but a Cube, or cubical body, is that which is comprehended of square or quadrate surfaces. The nature then of those surfaces which do comprehend them, is that they must be plain, not oblique or uneven: Secondarily, they must be squares or Quadrata, that is, Right-angled, and of equal sides. This is Euclids definition, and meaning. The number of these surfaces vitrvuius doth tell: Cubus est corpus ex sex lateribus aequali letitudine planitierum quadratum: that is, A Cube is a square body comprehended of six plain sides of equal breadth. Item, Martianus Capella in these words: Solida figura Quadrati sex superficies habet: The solid square( so he calleth the Cube) hath six surfaces. From hence is it that they attribute stability or constancy unto this kind of body. For 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, corpus quadratum, is such a body, if I mistake not, that consisteth of such an equal temperature of the humours,( eucrasia,) that it is not subject to that alteration and change that others are. And Aristotle calleth a good man 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, virum quadratum, that is, Cubicum, as I understand him, meaning such a man as is constant, and not easily moved upon every chance and misfortune, that in this world doth happen to mortal men. Such an one'as they report Bias, one of the seven wise men of Greece to have been. For Cubus cum est iactus, saith vitrvuius, quam in partem incubuit, dum est intactus, immotam habet stabilitatem: uti sunt etiam tessera, quas in aluso ludentes iaciunt. A Cube when it is cast out of the hand at an adventure, look upon what side soever it lighteth, it standeth fast and stable; like as those dice also do, which gamesters use that play at Tables. Therefore, If six equal squares be joined with solid corners, they shall comprehend a Cube. Item, the sides of a Cube( hedrae) are six: the edges( latera) are twelve: the plaine-angles, twenty four: the solid-angles are eight; as the author of the Scholium upon the xv book of Euclids elements, hath taught. 2 The power of the diagonal line of the Cube, is thrice so much as the power of the side of the Square comprehending it. Harmonia, saith our Author, est musica literaetura obscura& difficilis; maximè quidem quibus Graecae literae non sunt notae, quam si volumus explicare-necesse est etiam Graecis verbis uti, quod nonnulla eorum Latinas non habent adpellationes: that is, The theory of Music is very hard and difficult: especially to those which are ignorant of the Greek tongue: For whosoever shall take upon him to write of this argument, in what language soever, he shall be forced to use many Greek words, because many of them have no terms answerable to them in other languages. The same may I say of Geometry, where the most terms or words of art are merely Greek, or at leastwise feigned in imitation of, and that oft times not very fitly. For surely I doubt not but at the first those words seemed harsh enough, even to the first inventors of them: but time and use have made them familiar and pleasing. Of this sort is 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, Potentia, power, we call it: and posse, to be able, to be of value and power. Both which do signify in this place or this art, naught else but a Geometrical multiplication, as they are oft used by Euclid the great Geometer, and Diophantus the ancient Algebraist, and that by the testimony of Aristotle the Prince of Philosophers. For of this rule, Potentia rectae est quadratum; or as Diophantus speaketh, Appellatur quadratus Facultas, that is, the power of a rightline is a square: the meaning is, that if a right line be multipled in itself geometrically, it shall make a Quadratum or square surface. Likewise then in this place this proposition, Diagonius Cubi potest triplum lateris, The power of a diagonal line of a Cube, is thrice so much as is the power of the side; is thus to be understood: If the diagonal of a Cube be multiplied by itself geometrically; and the side of that square which comprehendeth the same Cube, be also multiplied by itself, the square that is made of the diagonal line, shall be three times so great as is that which is made of the quadrate including the same Cube. Item, again here observe, that that right line which crossing a plain from side to side by the centre, is called the Diameter: in a solid, the Axis or axle-tree; in both, if it pass from corner to corner, is properly termed the Diagonius, or the diagonal line. The power of this line, saith our author, is thrice so great as is the power of the said side or edge, as I call it. That we thus demonstrate: The power of the Diagonius of a Cube, is three times so much as the power of the side. That, whose power is as much as is the power of any thing single, and of that whose power is doubled to the power of the same single, jointly: is of treble power unto the single. Axio. Logicum. But the power of the diagonal line in a Cube, is as much as is the power of the single side of the square: and the diagonal of the same square, which is by the 365e xij R, double to the side. The power of that side in a right-angled triangle, which is opposite to the rightangle, is equal to the single powers of the two other sides. 5 e xij R. But the diagonal in the Cube, the side of the square, with the Diagonal of the same square, do make a right-angled triangle; and this diagonal of the Cube, is opposite to the rightangle, ex thesi. Therefore the power of the diagonal of the Cube, is as much as is the power of the side, and diagonal of the square. Therefore the power of the diagonal line of the Cube, is thrice so much as is the power of the side of the square. q. e. d. 3 If of four right-lines continually proportional, the First be the half of the Fourth; the Cube of the first, shallbe the half of the Cube of the second. This proposition is a consequent or corrollary drawn out of the 15 e iiij R, which teacheth that, if certain right-lines be proportional( to wit, more in number by one then are the dimensions of the like figures, alike situated unto the First and Second) it shall be as the First rightline is unto the Last, so the First figure shall be unto the Second: and chose. Now a Cube, by the I e xxj R, is a figure of three dimensions; because it is a body, which hath length, breadth and thickness. Therefore here the lines compared in the proportion must be four. The truth of this rule will easily appear by an example in numbers. But here observe, that you must not expect that our example shall altogether directly answer to this rule of doubling the Cube; but to some other. For truth it is, that Arithmetic, as hereafter shall more plainly appear, cannot double the Cube; that is, Arithmetic although it can tell what shall be the double of the Cube given, yet it can by no means tell thee in numbers, what the Latus or side of that Cube shall be. Let the four numbers continually proportional given, be 2, 4, 8, 16: that is, as 2 is unto 4, so let 4 be to 8: and as 4 is to 8, so let 8 be to 16. Here I say by that 15 e iiij R, as 2 the first number is unto 16 the last: so shall the Cube of 2 the first number, be unto the Cube of 4 the second. But 2 is but the eighth part of 16: Therefore the Cube of 2 shall be but the eighth part of the Cube of 4. The Cube of 2, is 8. The Cube of 4 is 64. Now and Here you see, that the quotient of 16, the last number, by 2, the first: Item, the quotient of 64, the Cube of the second, by 8, the Cube of the first, to be alike, or the same: And therefore the proportion is the same. q. e. o. By this rule an answer was made unto that great and strange question, moved by the oracle of Apollo at Delphos. The invention of this proposition, that is, of the first answer to that problem, by some is attributed to Plato the divine Philosopher, a though certain it is, that Eratosthenes, in an epistle of this argument written to Ptolomey king of Egypt, doth ascribe it to Hypocrates Chius. The history is briefly touched by vitrvuius in these words: Alius enim alia ratione explicare curavit, quod Delo imperaverat responsis Apollo, uti arae eius quantum haberet quadratorum id duplicaretur,& it afore, ut hi qui essent in ea insula tunc religione liber arentur. For each of them( meaning Architas Tarentinus, and Eratosthenes Cyrenaeus) laboured by sundry ways to perform that which Apollo at Delos had given in charge in his answer, whereby he commanded, that look how much soever the altar that was before him should contain in square feet, it should be doubled, and so it should come to pass, that those in that Island, which were sick of the plague, should be freed from that curse. But Eratosthenes before mentioned, layeth down the history more largely and plainly, thus: To king Ptolomaeus, Eratosthenes sendeth greeting; It is reported that one of the ancient tragedians bringeth in Minos, purposing to re-edify Glaucus his tomb. And being given to understand, that it was every way an hundred foot, said, That is too little for the sepulchre of so great a king. Let it be therefore doubled. It seemeth that Minos understood not what he said: For if you shall double the sides, the plain shall be four times so great, and the solid eight times. It was demanded therefore of the Geometricians, How the solid given, might be made just as big again as it was at the first, and yet the form and nature of the figure to remain the same still. This proposition is called, The doubling of the Cube. For making the question of the Cube, they laboured to make that double or as big again. All men therefore doubting and studying a long time how this should be done, at length Hypocrates Chius found that, If unto two right-lines given, whereof the greater should be double unto the lesser, two mean proportionals might be found, than the Cube might be doubled. But thus this doubt was to be resolved by another thing, as difficult and as hard to be done as that other. Within a while after, they report that the citizens of Delos being commanded by the Oracle to double a certain altar, and falling into the same doubt and difficulty, were constrained to seek unto the Geometricians which were with Plato in the University, that they would at their request, find out some way how to perform the same. Now these students diligently conferring upon the matter, and laying their heads together to find out how, unto two right-lines given, two mean proportionals might be found, it is said that Architas Tarentinus found them by semicylinders, and Eudoxus by those lines that are called crooked lines, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, curuae, etc. Thus far Eratosthenes. How then to satisfy the demand of the citizens of Delos, to perform, I say, the command of Apollo and Minos, was demonstrated by Plato, or Hypocrates Chius, to wit, by the finding out of two mean proportionals, between two right-lines given, whereof the one is double unto the other. But now again here ariseth their question, or as Eratosthenes saith, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, that this doubt of Apollo was to be resolved by another doubt, no less difficult than itself. For how this was to be done, had not as then by any before that time been taught. But upon these occasions given( moved also partly with glory) many fine wits in sundry ages have found out divers and sundry ways how to perform the same, as Eutocius Askalonita, Archimedes his learned interpreter, hath at large set down in his commentary upon his second book De cylindro. These authors, as there you shall find, are thus and in this order named by him: Plato, Heron, Philo Bizantius, Apollonius, Diocles, Pappus, Sporus, Menechmus, Architas Tarentinus, Eratosthenes, Nicomedes, and Eudoxus Cnidius, whose invention he rejecteth, as not answering to the purpose of his own proposition. Of all these P. Ramus, at the 8 e xiij of his Geometry, describeth that of Heron only, neglecting all the rest. Hinc existit, faith he, Mesographus Heronis Mechanici seu Mesolabus, dictus ab inuentione duarum continue mediarum proportionalium inter duas datas: unde existit problema Deliacum quod Apollinem ipsum exercuit. Mesographus autem Heronis est infinita regula, quae sistitur cochleato unco per cawm mobili. Est verò, ut Pappus ait initio libritertij, Mesographus iste architectis aptissimus, multoque promptior Platonis Mesographo. Mesographi mechanica est apud Eutocium secundt de sphaera, sed paulò faciliùs à nobis ita proponctur: Si duas dat as rectas comprehendentes rectangulum,& infinitè continuat as, mesographus tangens oppositum angulum angulo datarum intersecet aequidistantèr à centro, intersegmenta erunt media continuè proportionalia datis: that is, From hence( he meaneth out of the 7 e of that book) ariseth the Mesographus of Heron the engineer, which otherwise is called a Mesolabus: For the use of it is to find out two lines continually proportional between two other lines given. By this means was found how to answer that problem of Delos, which did much trouble Apollo himself. Now this Mesographus of Heron, is an infinite tight-line which is fastened with a screw-pin, that is to be slid up and down in a riddie. This Mesographus, as Pappus Alexandrinus in the beginning of his third book writeth, is a marvelous commodious instrument for Architects and Carpenters, and is much more convenient and ready than the Mesographus of Plato. A mechanical description and use of this Mesographus, is set down by Eutocius in his commentary upon the ij book of Archimedes De sphaera& cylindro, but is somewhat more plainly set out by us in this manner: 8 If( a right-angled parallelogram being made of the two lines given, the same also being continued as far as need shall require) the Mesographus touching the angle, that is opposite unto the angle contained of the lines given, shall cut those continued lines equally distant from the centre, the portions of those continuations thus cut, shall be the middle lines continually proportional between the two lines given. Thus far Ramus, unto whom I refer thee for further satisfaction. But whereas P. Ramus, as we have showed before, doth affirm that the Mesographus of Heron, as Pappus Alexandrinus writeth in the beginning of his third book, is a marvelous commodious instrument for architects, and is much more ready and convenient than the Mesographus of Plato; it is so to be understood, that those words, Multoque promptior Platonis mesographe, be not supposed to be spoken by Pappus: For indeed in that place he doth not once name Plato's mesographus. His words at the 4 p iij, as Commandinus, his interpreter hath expressed them, are these: Exponemus igitur quatuor eius constructiones, unà cum quadam nostra tractatione; quarum prima quidem est Eratosthenis, secunda Nicomedis, tertia Heronis maximè ad manuum operationem accommodata, iis qui architecti esse volunt. Of those many and sundry instruments, that have been found out by divers men for the invention of two proportionals, etc. we will describe the use and making of four, adding thereto a certain treatise of our own. The first is Eratosthenes his way, the second is Nicomedes, the third is Herons, most commodious for all handicrafts, or mechanics, and such as are desirous to be architects. Here you see is no mention of Plato's invention. But howsoever, that of Plato's, in mine opinion, is more proper to this place and purpose of ours; because it is to be done with the Carpenters tools, which are always at his hand: And therefore we will here also, out of the same Eutocius, describe that more fully. This Mesographus, or instrument devised by Plato, is described by Eutocius to have been a right-angled parallelogram, consisting of four strait rulars, so contrived and put together, that they might be put nearer or farther off, as occasion should require. We have used two Carpenters squires, so placing the side of the one squire upon the side of the other, that the other two sides might continually be parallel, and contain rightangles between them. Or if any man so please, one squire may serve the turn with the rule which they use in measuring. The instrument now according to these directions, is thus to be used: If( the two lines given comprchending a rightangle, and from the angle infinitely continued) the instrument be so applied, that when in each corner the infinite 〈◊〉 do fall, the sides of the same instrument do touch the ends of the lines given; the portions of the continued lines intercepted between the said corner, and the corners of the instrument, shall be the two middle lines continually proportional between the two lines given. Concerning the Diagramme, I must refer thee either to Eutocius upon Archimedes, or Daniel Barbarus vpon vitrvuius: Let the instrument be f m, the squire: The ruler to be moved up and down, let it be n o. The use according to our rule is thus: Let the two lines, whereof the one is double to the other, be e b, and b g; the one falling upon the other perpendicularly, that is, they both containing the rightangle e b g. Again, let e b, be drawn out infinitely toward c: and g b, in like manner toward d. Now apply the instrument to this figure in this manner, that when by moving n oh, the movable side, up and down, the two continued lines falling precisely in the corners m, n; the two parallels sides m, and n oh, may at the same instant touch e and g, the ends of the lines given. I say that b d, and b e, the portions of the continued lines, cut or intercepted between the corner b, and n, m, the corners of the instrument, are the two middle lines continually proportional between e b, and b g, the two lines given. The words of Daniel Barbarus at the 3 chap. of the ix book, are these: Coniungantur in b, ad rectum angulum duae rectae, inter quas duae comparabiles, ac ratione pendentes medias vis invenire. Esto b g, mayor: e b, minor: utrag, verò extra angulum b producatur: Maior, ad d: minor, ad c:& ad duos rectos angulos, unum in c; alterum in d: in suis respondentibus lineis. Estoque angulus unus, g c d; alter c de: Aio inter duas datas c b,& b g, esse duas medias comparabiles inventas: bd, scilicet,& b c. Quoniam posuimus angulum e d c rectum: etc. d aequedistantem ipsi c g; Ideo sequitur, ex 29 e j, angulum g c d, rectum esse:& aequalem angulo c d e, quem similiter rectum esse posuimus. Sed a b, ex constructione supra c b e, ad rectos cadit, similiter c b perpendicularis est ipsid bg. Ex corollario itaque 8 e uj, b d est comparabilis illi quae cadit inter c b,& b c: Pari quoque ratione b e, est media inter b d,& bg. etc. Vide pag. 276& 277. That d b, and b e, are continually proportional to c b, and b g. Continua proportio est, in qua idem terminus pro secundo& tertio sumitur. A continual proportion is that where the same bound, is taken for the second and third: 8 e 2 c ij Salig. But here d b: and c b, the several mean bounds, are each of them taken for two: 10 A plumline falling from a rightangle upon the base, is the mean proportional between the portions so made: 1 c 4 c viij R. But e d c, being a rightangle ( ex thesi& fabrica) and d b falling plum upon e c, cutteth the portions e b, and b c. Therefore d b is the mean proportional between e b, and b c: that is, as e b, is to db, so is bd, to bc. 20 A plumline falling, etc. 1 c 4 e viij R. But d c g being a rightangle; and c b falling upon d g, doth make the portions d b, and b g. Therefore c b, is the mean proportional between d b, and b g: that is, as d b, is to b e: so is b e, unto b g. Therefore d b, and b e, are continually proportional between c b, and b g, the two lines given. That is, as c b, is to d b; so is d b, unto b e: And as b d, is unto b c; so is b c, unto b g. quod erat demonstrandum. Thus you see, that that which was then so difficult, how easily it is now to be done, and that by many and sundry manner of ways; and yet neither of these do satisfy the Geometrician: The reason is, for that, as Pappus Alexandrinus saith, those authors quod natura solidum est, Geometrica ratione innixi, construere non potuerunt. Instrumentis enim tantùm ipsum in operationem manualem,& commodam, aptamque constructionem mirabilitèr traduxerunt. Because, saith he, those authors, although they have all showed how to do this wonderfully fitly and well; yet all of them have done it but mechanically only, by the help of certain instruments, none of them yet having found how it may be done geometrically. 4 The Solid number of a Cube, is called a Cubic. 5 A Cubic number is that which is made by multiplying of any number given into his quadrate. A Quadrate number is a number that is made by the multiplication of a number in itself. Therefore a Cube is a number which is made by the multiplication of three equal numbers; or, a Cubic number is that which is made by the multiplication of any number by itself, and again, by the multiplication of himself into the said product. As for example, 2 by 2, do make the quadrate 4: Now 2 by the said 4, do make 8, a Cubic. Of this number, Martianus Capella thus writeth: Octonarius numerus, primus cubus est,& perfectus, Vulcano dicatus: Nam ex primo motu, id est, Diade, quae juro est, constat. Nam dias per diadem, facit tetradem: At bis ducta, facit octadem. Perfectus item, quòd à septenario tegitur. Omnis enim Cubus sex superficies habet. Item, Ex imparibus consecutis impletur. Nam primus imparium trias, fecundum pentas, ambo octadem faciunt. Item, Cubum, qui à triade venit, id est, 27, sequentes impares reddunt, id est, heptas, enneas,& 11: qui omnes faciunt 27. Item, Tertius Cubus, qui à tetrade venit, id est, 64. Nam quater quaterni sunt sedecim. Hoc quater, 64 fit. Et hic ex imparibus quatuor, qui supcriores sequuntur, id est, 13, 15, 17, 19, fiunt simul 64. Et sic omnes Cubi per imparium incrementa inveniuntur, sui duntaxat numeri. Sanè hic octonarius Cubus, it a omnium Cuborum primus est, ut monas omnium numerorum. Cubus autem omnis etiam matri Deum tribuitur: Nam ideò Cybele nominatur. Thus far Capella. This rule is conceived in few words thus: Digestis à ternario imparibus, si duo priores; postea tres; deinde quatuor, etc. coniungantur, Cubos proferunt: that is, If odd numbers digested according to their natural order, be added,( an unity only, which is of itself a Cube, excepted) first two; then three; then four, etc. they shall make the Cubes naturally following one another, thus: 1Therfore Therefore look of how many unities the radix or side of any Cube doth consist, of so many odd numbers is the same Cube composed and made. Item, If the first figure( I mean that on the right hand) of a Cube given, be an odd number, the number of odd numbers whereof it was made, is odd: If it be an even number, the number of odd numbers whereof it consisteth, is even. Item, 3 If a Cube do multiply a Cube, the product shall be a Cube; and the side of that Cube shall be the product made by the multiplication of the sides of the Cubes given. As for example, Let the two Cubes given be 8 and 27; whose sides let them be 2 and 3: Now let 8 multiply 27, and let the product be 216: I say the side of the Cube 216, is 6, which is the product of 2 by 3. Thus have you an easy way, by the help of the xiv Chapter of the second book of Salignacus his Arithmetic, how to find out the cube of any place, and the fide thereof, without any farther ado, only by the help of one multiplication. Now the sides of cubes less than 1000, being single figures, to wit, what cube is made of every single figure multiplied in itself cubically, and what the side of every such cube is, must first be known before we go any farther: That is done by this table: 6 If a rightline be cut into two portions, the Cube of the whole line shallbe equal to the Cubes of the segments, and a double solid thrice comprehended of the quadrate of his portion, and the other portion. The general invention of a Cube, both Geometrical and Arithmetical, was showed at the former proposition: The special or particular invention of the same is divers and manifold by numbers continually proportional and out of the cubes themselves, as Euclid teacheth, of whom thou mayst learn, if thou shalt think the fruit of that knowledge may countervail thy travel. Analysis quadrati lateris, The extraction of a square root, as they call it, hath in Euclids Geometry a proper element, theorem, or rule, teaching how to perform it; but the extraction of a cubic root hath not any at all: Notwithstanding according to that of the square by him laid down, it is not difficult by analogy to make one, whereby the root also of the Cube may be found. The proposition therefore according to that analogy, teaching this skill, is thus laid down by Ramus: If a rightline be cut into, etc. As for example, Let the side or root 12, be cut into two portions, 10 and 2: I say the Cube 1728, made of 12, the whole line, is equal to 1000, and 8, the cubes made of 10 and 2, the said portions: and two divers solids, whereof the first is 600, comprehended thrice of 100, the square of 10, his segment; and of 2, the other segment: The second, 120, which is thrice comprehended of 4, the square of his segment 2, and of 10 the other segment. But the frame and making of the whole cube, will make the matter more plain and easy in one example, to wit, how the outer and mean solids are made. Let therefore a cube be made of 12, 12, and 12, three equal sides: and first let the second side be multiplied by the first, thus: Let not the products 24, and 120, be added together; but let the other side multiply them severally, and then add the several degrees by themselves, thus: Or thus, This is the making the Cube of the whole line: Now the making of the same according to the former division of the said line, is thus: But in this manner which followeth, of our invention, the particular solids, in this kind of making the cube of the several segments, do more plainly appear, then in that practised by Ramus: as these examples do show. Now that we may apply this example unto our rule, add the solids of the same kind and quality together, that is, 40, 40 and 40, together: Item, 200, 200 and 200 together, and add the sums of them, 120, and 600, unto 1000, and 8, the cubes of the segments, and the sum 1728, shall be equal unto 1728, the cube of the whole line. If any man shall think this rule to be true only in this case where the line is divided according to the nature of the number, thus consisting of two digits, as they call them, he is deceived: For the same effect will fall out, howsoever the line shall be divided. Let therefore the same number first be divided into two other segments unequal between themselves, to wit, into 8 and 4. Secondarily let the fame line be cut into two equal segments, to wit, 6, and 6. Therefore, if the rightline given be cut into two equal segments, the several solids of the segments shall be equal unto the Cubes that are also equal between themselves, that is, there shall be eight solids all equal one unto another: or, that which is all one, the solids are the same with the cubes. Therefore 7 The side of the first particular Cube, is the one side of the second solid, and the square of the same side is the other side of the first solid, whose other side is the side of the scond Cube; and the square of the same other side, is the other side of the second solid. In this equalling then of four solids with one solid, there is to be observed a singular kind of frame and composition: First, that the last cube be made of 2, the last segment: Then, that the second solid made of 4, the square of 2 his segment, and of 10, the other segment, be thrice taken. Again, that the first solid made of 100, the square of 10 his own segment, and of 2 the other segment, be also thrice taken. Lastly, that the cube 1000, be made of 10, the greater segment. Out of this frame or making of a Cube, the contrary Analysis or resolution of the same is derived, out of the mutual combination of the Cubes with the solids, such as we have before showed in the Analysis of a square. For here, although a solid be named only, yet there are two sides to be considered: because that the one is compound and plain. Therefore 10, the side of 1000, the first cube, is the key for the opening of the two solids and cube following. For it is the one side of the second solid, to wit, thrice comprehended of it and of the square of the second segment. Again, 100, the square of the same side 10, is the one side of 600, the first solid, to wit, thrice comprehended of this quadrate 100, and the other segment 2. Item 2, the other side of the first solid, is the side of the next cube. And lastly, the square of 2, the same side, is the other side of the second solid. By this means then, the great variety and difficulty of this business is unfolded, like as was done in the square. For when you have found the sides of the several cubes, then have you withal also found the side of the whole cube: For although the whole Cube be greater than the cubes of the parts; yet the whole side is equal to the sides of the several cubes: For we use the solids that are considered between the cubes, only as a means to find out the side of the following cube. Thus much concerning the true form of analysing or resolving of a Cube: But because this may seem somewhat difficult and hard unto a learner, out of this proposition we have framed another, which doth as it were more distinctly express and point at every particular in this practice. 8 If the side of any greater Cube shall be sought, thou shalt from the right-hand toward the left distinguish the number given into perfect periods, for so many particular Cubes: Then having found the side of the first particular Cube, thou shalt set it down within the quotient: Again, having squared this side now found, thou shalt triple the product for the base of the first solid; but for the height or other side of the second solid, thou shalt only square it. Lastly, the first solid divided by his base, the quotient shall be the side of the next following Cube. As for example; Let the side of the cube 1728, be sought. I. Here first beginning at the right-hand, I distinguish it into perfect periods, that is, into three degrees thus: 1, 728. Now because after this distinction it appeareth, that our number given is a compound period, composed of two single periods; therefore by the former rule I say that the whole consisteth of two particular cubes. II. Secondarily, the first particular cube being 1, I seek his side amongst the cubes of single figures, at the 5 e, and I find the side to be 1: that I set down therefore in my quotient. Now, as the manner is in division, I set down 1, the cube of this quotient or side, underneath the first figure of the number given: Item, subtracting the one out of the other, I cancel all the figures of the number given, setting the remain, or the whole next period, above the head right over their degrees. III. Thirdly, I square 1, the side found, and I make 1, which I treble for the base of the first solid, and I make 3: by this base, or treble 3, I divide 7, the first solid or first compliment; the quotient 2, I place in the quotient for the side of the succedent cube: By this quotient 2, I multiply 3, the Divisour, and I make 6 for the first solid: Then by 4, the square of the same quotient, I multiply 3, the treble before reserved; and I make 12 for the second solid: Thirdly, I multiply 2, the quotient cubically, and I make 8: Fourthly, placing all these products in their true places, just one degree behind another, so that the first solid be placed in the first degree; the second solid one degree farther toward the right-hand: the cube one degree farther than that, or next of all to the right-hand, in this manner: Again, they being thus placed, I add them, and I find the sum 728: Now this sum I subtract from the remain 728: Lastly, because after subtraction there remaineth nothing, I say, that 1728, the number given, is a cubic, and the side of it is 12. The example of this practice is thus: But here observe, for that the first solid( or compliment) is the third degree from the second cube; and the second solid, the second or next unto the same; therefore unto the first solid I add two siphers, and unto the second but one: For these figures do so guide the practitioner, that he cannot easily err in the addition of them. Or thus, as Salignacus an excellent artist, setteth it down. Let the side of the cubic 389017, be found. Here first, because that 389 is not found amongst the cubes at the 5 e, that is, because that 389 is not a cubick number, therefore I take 343, the next lesser cubic before it, whose side 7 I place within the quotient. II. Then taking 343 out of 389, there do remain 46: Therefore here the two compliments with the second cube, are 46017. III. This done, I square 7 the side found, and I make it 49. IV. This square I treble, and I make 147, for the base of the first compliment. V. Again, I triple the side found, and I make 21 for the height of the second compliment. VI Lastly, dividing 460, the first compliment by 147 his base, I find the quotient 3, which I set down in the quotient, for the side of the second cubic. For the proof of this work or practise, First we multiply the quotient in itself cubically: Secondly, we multiply the height of the cube by the base of the first compliment: Thirdly, we multiply the base of the same cube, by the height of the second compliment: Fourthly, we add all these products together: Lastly, we subtract the sum found, out of the upper number. As for example: ay cube 3, or multiply it in itself cubically, and I make 27. Now the height of this cube is 3; the base is 9: Therefore I multiply 3, this height of the cube, by 147, the base of the first compliment, and I make 441. Again, I multiply 9, the base of the said cube, by 21, the height of the second compliment, and I make 189. Now I add all these products, and the sum is 46017: which sum subtracted out of 46017, the upper number, nothing remaineth. Therefore if you shall square 73, the quotient; and then shall multiply this square by the same 73, thou shalt make the cubick 389017. For the cubes of the segments 70, and 3, with the two compliments now found, are the parts of the cube that is made of the whole 73; as may easily be tried by that which we have taught in the former. Briefly then, and in this order is this whole work to be done. Let the example be the cube 1728. I. By the 5 e, I find the side of the cubick 1, to be also 1, which I set down in the quotient. II. This quotient now found, I treble, and I make 3. III. Again, this treble I multiply by the said quotient, and I find 3 also for the Divisor. By this Divisor 3, I divide 7, the first compliment or solid: and I find the quotient 2 for the side. IV. Now by this quotient I multiply the Divisor 3, and I make 6, for the first compliment: Therefore to it I add two ciphers, and I make 600. V. Again, I square the quotient 2, that is, I multiply it by itself, and I make 4: This 4, I multiply by the treble 3, and I make 12, for the second compliment: Therefore I add to it one cipher, and I make 120. VI This done, I multiply the same quotient 2, in itself cubically, and I make 8. VII. Lastly, placing all these orderly one under another, I add them, and I find the sum 728, which subtracted from 728 the number remaining, nothing is left: Therefore I say that 1728 is a cubic. 9 If, having found the quotient of two or more figures, as yet the whole side of the Cubic given be not found, then for the finding out of the quotient following, thou shalt square the whole quotient now found for the base of the next compliment; and shalt triple the same square; and then shalt in all things follow the prescript of the former rule, the whole side shall be found. Suppose the side of 34, 012, 224, were to be sought: Item of 320, 013, 504: The examples would be thus: Sometimes after that you have found the cubical side, I mean, having subtracted the cube of the first period, there will remain in the next places neither solid nor cube. Therefore in this case unto the side found, you shall adjoin a cipher, as in this cube 8120601( 201. If the second side of the first compliment be greater than the side of the cube following, than the solid divided did contain a part of the second compliment: And therefore that side must be diminished, as in the cube 17, 576, the first side shall be 2; then if thou shalt make the second side 7, as it seemeth at the first view it should be, thou shalt make the second side of the first compliment greater than the side of the cube following. Therefore that side must be taken less, and for 7 I take 6, and so thou shalt find the true side of the cubic given to be 26. The example is thus: This is the general and common way of finding of the side of any cubick whatsoever, though never so great. Thus also parts, whose bounds are cubical numbers, may be resolved into lesser bounds, by finding out of the sides of the same bounds; As 8/27 by this means are resolved into ⅔, And 1782/2197, into 12/13. In parts( fractions they commonly call them) there is also a kind of cube. For if foe be the number given be not a cubick, than it hath no side that may be expressed in numbers; and yet the true side of the greatest cube contained in any number given, may be found: As in this number 17, 616, which is not a cubic, the greatest cubic is 17, 576, and the side of it is 26, and there do remain, over and above the cubic, 40. Therefore, there cannot possibly any side of a number that is not a cubic, be found so near, but it is possible to find one more near; as is also before taught of the Quadrate. There are also here two manner of ways of finding out a cubical side in such like numbers very near to the true side, such as were before showed in the quadrate. The first subducteth the two compliments and the last cubick, contained in the next greater cubic above it; whereby may be understood the difference of two continual cubickes, like as before the difference of two quadrates was understood. For here there is to be understood a certain cubical Gnomon or squire, which you may conceive to be made of three plains or sides of the cube, as before you conceived the squire or gnomon of the quadrate to be made of two sides of the quadrate. So in this example 17, 616, where the side is 26, and there do remain 40, of the cube next following, thou shalt divide 27, the next greater cubic, into 26 and 1, and, as before thou hast learned, thou shalt make two compliments and one solid of them. The first compliment 676, made of the square of 26, the one segment; and of 1 the other segment; which being thrice taken, shall be 2028. The second compliment is 26, comprehended of the square of 1, the one segment; and of 26, the other segment: which being thrice taken, do make 78. Now the cube of 1, is 1. This done, add all these, and the sum 2107, shall be the denominator of the fraction sought. The parts therefore to be added unto 26, the former side, shall be 40/2107. So that the whole side of 17, 616, the number given, somewhat near unto the true side, is 26 40/2107. Whereby it is to be understood as above in the quadrate, that look how much the numerator 40, doth want of 2107, so much the cubic given doth differ from the next cubick above it. And therefore if you subtract 40 the numerator, from 2107 the denominator; and shall add the remain 2067, unto 17,616, the cubic given, the whole shall be 19,686, the cubic of the side 27. The second way is by parts of some great denomination, so as it be understood that they be cubical parts, that their side may be certainly known before. As for example, the same number 17,616, reduced unto one hundred cubical parts, that is, unto 1,000,000, do make 17,616,000,000, for the numerator. The parts than are thus; 17,616,000,000/1,000000. Now the side of the numerator is 2,601, for the numerator of the one hundred parts given:( For the former denominator being made by the multiplication of 100, by itself cubically; the root or side of it must needs be 100, for the denominator of the parts sought.) Therefore the parts desired are 2,601/100, that is, by reduction 26 1/100, and besides that, there do remain 19,712,199, which cannot add so much as 1/100 part unto the side found; because that the difference of this cubic from the next greater above it, is greater than this remain: And therefore that remain is neglected, as not of any moment. Thus far of the extraction or finding out of a cubic side or root. 10 The two compliments are the two means continually proportional between the two cubes: that is, as the greater cube is unto the greater compliment; so is the greater compliment unto the lesser: and so is the lesser compliment unto the lesser cube. As for example, Let the whole cubic be 1728. Here the two several cubes, let them be 1000, and 8. The greater compliment let it be 200, and the lesser 40, as we have showed at the 6. p. Here I say 200 and 40 are the mean proportionals between the two cubes 1000 and 8: That is, as 1000 are to 200, so are 200 to 40; and so are 40 to 8. The cause of this is manifest by the 5 e xxij R, which teacheth, that, Solids that are alike, have a trebled reason of their correspondent sides; And also they have two mean proportionals comprehended of the cross multiplication of the base and height of the extremes: 19 p viij E. But the two compliments here are contained of the base and height of the extremes crossly multiplied. Therefore the two compliments are the mean proportionals between the two cubes, that is, the two extremes. 11 If four numbers be continually proportional, the products of the extremes by the each other squares, shall be the cubes of the middle numbers: to wit, the greater of the greater; and the lesser of the lesser. jordanus 57 p 6, of his Arithmetic. Let 16, 24, 36, and 54, be four numbers continually proportional: And let the quadrate of 16, be 256; of 54, be 2916. Again, let the product of 16, the lesser extreme, by 2916, the square of 54, the greater number, be 46,656. Item, let the product of 54 the greater extreme, by 256, the square of 16 the lesser extreme, be 13,824. Here I say, the side of 46,656, the greater cubic, shall be 36: and the side of 13,824, the lesser, shall be 24, as may be proved for experience, by the former. The example is thus: The four numbers given, 16 24 36 54 The squares of the extremes: The product of the greater's square by the lesser extreme, The product of the lessers square by the greater extreme: Now the side of 46, 656, the greater, is 36; of 13, 824, the lesser, it is 24. But 24, and 36, are the middle numbers of the four number: Therefore, If four numbers be continually proportional, the products of the extremes by their each other squares, are the cubes of the mean. This rule then, as you see, is a kind of Mesolabium, that is, a way to find out the middle numbers, or mean proportionals: whereby I mean, the extremes being given, the mean proportionals are easily found. Any two numbers are thus in the manner of solids alike between themselves, conceived, as oft as between them two mean continually proportional, are sought. But many times these middle numbers, the cubical sides of the products of the extremes by the each other quadrates or squares, are furred numbers, as they call them, that is, such as cannot be expressed by arithmetical numbers, but by such characters as are devised by the Algebraists. Many other rules might hither be added, but we especially in this place regard such as may any kind of way help, either to the understanding of this present argument of extraction of the square and cubic root; or may be of use for the making of our Mesolabium architectonicum, or Carpenters ruler. 12 The product made continual of three numbers continually proportional, shall be the cube of the mean or middlemost number. 36 p 11 E. Let 4, 6, 9, be three numbers continually proportional: And let them be multiplied continually between themselves: 216, the product so made, shall be the cube of 6, the middle number. This proposition, saith Schoner, is a kind of golden rule in solids, by which having the one of the extremes of the three proportionals given, with the solid made of them all multiplied between themselves continually, the rest are also given. As for example, Let 216 be the solid given, made after this order of three proportional numbers, and let 4 be the one extreme given. Here I say, the middle number by this rule shall be 6. For the Cubic root of 216, is 6. Now the product of 4 by 6, is 24; and the quotient of 216, by 24, is 9, the third or greatest extreme. Or, the product of 6 by itself, is 36. Now 36 divided by 4, doth yield 9, for the other extreme. For, if three numbers be continually proportional, the product of the middlemost by itself, shall be equal to the product of the two extremes by themselves, by the rule of proportion. Therefore, if one of the extremes given shall divide the product of the middlemost by itself, the quotient shall be the other extreme that was desired. 13 Having found the one of the mean proportionals, the other also shall be found, if the mean proportional given be multiplied by the extreme that is farthest from it, the root of the product shall be the mean proportional desired. As for example, Let 32, 16, 8, 4, be four proportionals given; and suppose that the third bound were not known. Here therefore I multiply 16 by 4, the extreme that is farthest from it, and I make 64: Now the square side of 64, by the 5 ej, is 8, the third proportional sought. Item, suppose that 16, the first of the two mean proportionals were unknown; Here 8 shall multiply 32, the first bound; not 4, the last; because that is farthest off from it: The root of 256, the product, by the ejs, is 16. This root is the first mean proportional of the two, between 32 and 4, the extremes given. CHAP. V. Of the measuring of Timber by the Foot. BY the Foot we do also measure Timber: but Timber being a solid body of three dimensions, to wit, length, thickness and breadth; by a foot of timber we understand here a cube of 12 inches square, as they call it:( For here they abuse the word, as also some even of the learned have done, as before we have showed:) That is, a foot of timber doth contain 1728 square inches. Here therefore commonly two dimensions are given, to wit, breadth and thickness: the length is sought. If then the square timber to be measured be 12 inches thick, and 12 inches broad; there is no question of the length: For every 12 inches of that piece shall make a foot of timber. But if the breadth and thickness do vary never so little from these two cases nominated by the statute, although they vary no whit one from another, that is, although the breadth be equal to the thickness, here presently riseth a question, what the length should be( according to that breadth and thickness) that must make a foot square, or that must be equal to that piece which is 12 in length, 12 in breadth, and 12 in thickness. Yet this every one can do, that knoweth aught in this business: For the Carpenters have upon their Rulars, or upon some piece of paper or parchment, all the square measures set down, from 1 inch square, unto 24, 36, or 40 inches square. But if the breadth and thickness be different one from another whether little or much, this is not only troublesome and unready, as the former, but very false and erroneous, as some of their own company have truly noted. For that of the squares, was made, as seemeth, by some skilful in the Mathematics; but their practice in the latter case, is deceitful, false, and wholly against the rules of Geometry, and can no way be justified Now because that all men that have occasion to use this skill of measuring, do not understand how these tables or rules are made: as also for that the same being copied out oft times by unskilful men; I think it not amiss here to set down by the former grounds, as we have done before for boord-measure and others, the manner of calculating and making of the same: and that especially because that these tables tend directly to the making of our ruler, which is the chief cause that first moved me to undertake this labour, or to write of this argument. The Rule therefore whereby this is performed, is thus: If by the product of the breadth and thickness given, thou shalt divide the cube of 12,( that is, 1728) the quotient shall show the length required to make a foot of timber. The forms of timber which commonly are to be measured of Carpenters, are long squares( such as the Geometers call Parallellepipida oblonga) or Rounds, otherways called by them Cylinders, which the vulgar sort of carpenters do also accounted, as we have showed, amongst the number of squares, whose breadth and thickness are the same. A foot square( a cubical foot I mean) saith our statute, as we have before taught, must contain 1728 square inches. If therefore the breadth and thickness be greater or less than 12, the length shall be found by the prescript of the former rule. Suppose, for example, that a piece of squared timber were to be measured, whose breadth and thickness are equal, yet greater than 12 as namely 16 inches: Here I multiply the breadth by the thickness, that is, 16 by 16; and I make 256. By this product I divide 1728, and I find the quotient 6¾: Therefore I say, that every 6 inches, and three quarters of one inch, shall, of that breadth and thickness, make a foot of timber, according to the intention of that Statute. Item, suppose the piece to be measured, whose breadth and thickness is equal, were 6 inches square, and it were demanded, how much of that in length would make a foot of timber? R. The product of 6 by 6, is 36: Now the quotient of 1728 by 36, is 48. Therefore I say, that 48 inches of that stick are required to make a foot of timber. Now suppose the breadth and thickness were unequal; as for example, first let the breadth be 18, and the thickness 12: here the product of 18 by 12, is 216; and the quotient of 1728 by 216, is 8: Therefore I say, every 8 inches of that piece of timber shall be equal to a foot of solid measure. Admit a plank or table were to be measured after the manner of timber measure, whose breadth is 36 inches, and thickness 4: how much in length fhall make a foot of solid timber? R. The product of 36 by 4, is 144. And the quotient of 1728, by 144, is 12, the desired length. Thus than a Table for solid or timber measure is to be made, like unto those which we have before showed for board and land-measure. Only this is to be observed, that this table is only of such pieces as are square, that is, of such whose breadth and thickness are equal; although it may also be extended& made for all, if need so require. If beginning at an unity, and so ascending upward unto the greatest, thou shalt divide 1728 by the square of any number asigned, and so forth, the several quotients placed against the number given, shall make the table of timber measure. As for example, The square of 1 is 13 and the quotient of 1728 by 1, is 1728. Item, the square of 2 is 4; and the quotient of 1728 by 4, is 432. Lastly, the square of 12 is 144, and the quotient of 1728 by 144, is 12, for the length desired. The Table then for square measure, that is, for the measuring of such timber where the breadth and thickness are equal, is thus. The use of this Table is easy out of the former: only this difference is to be observed, that because plains have but two dimensions, therefore one dimension given there was sufficient for the finding out of the other unknown. But here, for that solids have three dimensions, two( to wit, breadth and thickness) are required for the finding out of the third desired. Yet now again, seeing that here the said two dimensions given, are equal one to another; if you shall with either of the dimensions given, enter the first column of the Table, the column on the right-hand shall yield the length desired. As for example, Admit a timber stick to be measured were 4 inches square, that is, were 4 inches thick, and 4 inches broad; I demand, how much of that stick in length, shall be required to make a foot of solid measure. R. Here because 4 is equal unto 4, that is, for that the breadth and thickness are equal, I enter the column on the lefthand with 4, and I find answering to it, in the column on the right-hand, 9 : Therefore I say, every 9 foot in length, of that breadth and thickness, shall make a foot of solid measure; The product, I say, made continually of 4, 4, and 108, shall be equal to the product continually made of 12, 12, and 12; that is, the product continually made of 4, 4, and 108, shall be 1728, thus: Item, suppose a timber stick to be measured were 16 inches square. The column on the right-hand answering to 16, giveth 6¾ ⁱ: Therefore I say, every 6 inches, and 3 quarters of an inch in length, of that breadth and thickness, shall make a foot of solid measure. Again, admit it were 48 inches square: what length shall be required to make a foot of solid measure: R. Here 48 is greater than any number contained in the column of breadth, on the lefthand: Therefore I take 24, the half of 48, and with that entering the Table, I find 3 j to answer for the length desired. But here observe, 24 as it is but the half of 48 i, the breadth: so is it also but the half of the thickness, which is supposed likewise to be 48i. Therefore 24 is but the one quarter of both the assigned: Wherefore I say that 3 i in length, of that breadth and thickness, shall contain 4 foot of solid measure. Lastly, admit a pillar were but half an inch square: Here I find no number amongst the breadths of the lefthand so little as ½: Therefore I enter the Table with 1 i, the double of ½,& I find for the length desired 144f. But here because 1 i, the breadth, is twice so much as ½ ⁱ, the breadth given; and is also twice so great as the thickness, which is supposed to be equal to the breadth. Therefore 144 , or 1728 i in length, shall contain but ¼ of a foot of solid measure: or every 576 in length, of that breadth and thickness, shall be equal to the cube of 12i. Having finished the former Table for timber-measure,& knowing it to be serviceable in those cases only where the two dimensions given are equal one to another, I bethought me of a Table more general, for the measuring of all sorts of squared timber whatsoever. That this might be done, I doubted not; but that it might be done in few words, or within so little a compass as here thou seest, I knew not until I had made trial: And having contrived it, in the manner of Pythagoras tables, into a triangular form, like unto that musical instrument which Swidas, as I remember, calleth Trigonum musicum, I thought good to name it of the form and use, Trigonum architectonicum, The Carpenter's squire. Now that by this Table all those foul and gross errors are avoided, which in measuring of timber are by the common sort of workmen committed, that is, that by this our Table the Carpenter may not only measure all sorts of squared timber most exactly and truly, but also more easily and speedily then by any way commonly practised or published; besides that, it is demonstrable out of the grounds of these arts, every man of mean understanding, that shall please to compare them, shall be able to testify with me. The making of this Table is in a manner the same with that of the former: the difference only is this, that there 1728, the cube of 12, was divided only by the square of every several number given: Here the same cube is to be divided, not only by the square, that is, by the product of every such number by itself, but also by the products of every such several number by any other assigned whatsoever. Now it is manifest, that the quotient thus found( which answereth unto the third dimension sought) is to be placed in the common angle or meeting of the columns of the two dimensions given, that is, of those two numbers, by whose product this quotient was found. First therefore having made a right-angled triangle, I place the numbers given upon the sides without the same: These two ranks of numbers answering( as we have said) unto the two dimensions given, the one of them at the top beginneth with the greatest of the numbers given, and endeth at the right-corner with an unity: The other running along underneath the triangle, beginneth at the same right corner with the greatest, and endeth with the least. These numbers also, for the better help and guide unto the eye, may also be placed upon the hypotenusa, or slanting side of the triangle which is opposite unto the said right-corner. This being done, I begin either at the greatest or least numbers, it skilleth not whether, and I find the numbers for the third dimension, as before is taught, which I place within the triangle in their several columns, as the factors shall appoint. As for example, beginning at the greatest, I multiply 24 by 24; and with 576 the product, I divide 1728, and I find 3, which I place within the triangle in the first column against 24, etc. Again, I multiply the same 24 by 23, and with the product 552, I divide 1728, and the quotient 3 72/552( or 3 3/13, being reduced unto parts of the least denomination) I set in the same first column against 23. In like manner I multiply 23 by 23; and by the product 519, I divide 1728, and I find 3 ●, which I place in the second column from the right-hand, against the same 23 on the outside. Item, thus I multiply 22, first by 24; then by 23; and lastly by 22; by the products still dividing the same 1728, I set the quotients in their several places. Thus dealing with all the numbers on the right-hand of the Triangle, until thou comest unto 1, the lowest on that side, the Table shall be made. The use of the Table in Timber-measure. This Table, as the Title showeth, serveth generally for the measuring both of Plains and Solids by the foot. Now Solids, such as Timber and Stone are, have three dimensions, to wit, breadth, thickness and length. Of these, two, breadth and thickness, are given: The third, I mean length, is sought. Here therefore, Seek the one of the dimensions given amongst those numbers on the outside of the Triangle, which at the top begin at 24, and so descend to 1: The other seek amongst those at the base underneath the same: The space within the Triangle, to which these two do point, shall show the length desired. As for example, Suppose a piece of squared timber were 12 inches broad, and 12 inches thick: In the common angle, or meeting of these two numbers, next to the bevelling or slanting line of the Triangle, I find 12 for the length desired. Again, suppose the breadth of some stone to be measured were 18 inches, and the thickness 6. These numbers do show the length desired to be 16 inches: Not 12, as the vulgar rules do teach. In Boord-measure. In board: measure, only breadth and length are considered: Therefore here, The breadth given, is sought sometimes amongst those numbers on the outside upon the right-hand: Sometimes amongst those underneath the same, as occasion shall serve: For it must so be taken, that it may meet continually with 12. An example or two shallmake all plain: Suppose a board were 9 inches broad; here I seek 9 amongst those numbers on the right hand: The common angle or meeting of this with 12, of those numbers at the bottom, doth show that 16 inches is the length desired. Again, suppose it were 18 inches broad; here no number on the right hand, greater than 12, can meet with 12, in the under-ranke: Therefore I seek 18 amongst those of the lower rank; And I find the common angle of 18 of this rank, and of 12, of the numbers amongst those on the right hand, to give 18 inches for the length desired. If any man shall desire that the fractions should be made more serviceable for the use of practitioners, it may easily be done if any man will take the pains. FINIS. TRIGONUM ARCHITECTONICUM: OR, THE CARPENTER'S SQVIRE: that is, A Table serving for the measuring of Board, Glass, Stone, Timber, and such like Plains and Solids by the Foot.