Batman vppon Bartholome his booke De proprietatibus rerum, newly corrected, enlarged and amended: with such additions as are requisite, vnto euery seuerall booke: taken foorth of the most approued authors, the like heretofore not translated in English. Profitable for all estates, as well for the benefite of the mind as the bodie. 1582.

De tertia diuisione totius Nu∣meri. cap. 128.

NUmbers bée diuided in the thirde manner, in this wise. Some discréet and some conteined. A discreet number is conteined in discreete vnities, as three, foure, fiue, sixe, & so of other. A nūber cō∣teining is be, which ioyned with vnities is conteyned, as three is vnderstoode in greatnesse & in quantitie, and this num∣ber is diuided in Lineall, Superficiall, & in Solide The number Lineal begin∣neth at one, & is written lineally vnto endlesse. And to Alpha is written for de∣signation of lines. For among Greekes this letter betokeneth one. The number superficial is writtē not only in length, but also in breadth, and to conteyned in length and in breadth. A three cornered number, and foure cornered, & fiue corne∣red, and round, and other such, be alway written and conteined in length and in wedth: Therfore heere be figures set for ensample. For the cornered nūber is or∣deined in this wise, •• and the Qua∣drant in this wise

[illustration]

And fiue corne∣red in this wise,

[illustration]

The circle number is made thus,

[illustration]

The number Spheri∣cus and Circularis commeth of a num∣ber that is multiplyed by it selfe, and oft by the number that commeth therof, and turneth into it selfe in a circle wise, and maketh a spere all rounde, as fiue times fiue times. For this circle multiplied by it self all about, maketh a spere al round: for fiue times fiue and twentie maketh generally an hundred & fiue and twen∣tie. The number Solidus, (* 1.1 Solidus, it was among y^e Romanes diuersly taken, sometime for a coine of Brasse contey∣ning 12. smal pieces. A shilling, sometime it was taken for Dragma in siluer, as Pri. Esdrae. 8. & secundi erusdem. 7. So∣lidus aureus, in the time of Alexander, was two drams of gold. After in y^e time of Iustinian. 6. of them made an ounce, they being of the weight of our old no∣ble:) is conteined in length & bredth and deepnesse to them that be simple, propo∣sed simple to kinde, and many manner diuisions & numbers to be vnderstood & knowen, as I finde in the worse of Isid. for his words I follow at full. Heereof it followeth, & is openly knowne heere∣by, y^t vnder diuersitie of numbers be di∣uerslye his diuers vnderstandings and meanings of holy writ, the which is in∣spired by the holy Ghost. Therefore, as Boetius sayth, libro quinto, capitu. pri∣mo.

Among the science Mathematike, wise

men shall most take héede of the science of numbers. For the lore of Arethmetik passeth all other to helpe to knowe all thinges of kinde, of the which Philoso∣phie must treate: For without number is not a letter ioyned to a letter, nor sila∣ble to silable in right order, neither Sub∣iectum knowne from the Predicatum, nor the conclusion in Silogisinus is di∣stinguished from the premises, nor the first meane and lesse, nor of the third and fourth. Therefore (as Boetius sayeth) the science of numbers passeth all other sci∣ences. For without thrée is no Trian∣gle, nor Quadrangle without foure, and so of other. And so it fareth in Musicke, for accords Musick hath names of num∣bers, as Boetius sayeth. As it fareth in Diatesseron, in Diapente, and in Dia∣pason, and in other Consonants and accords of Musicke, y^t haue no name with¦out number y^t commeth before. And the course of starres is not knowen, and ri∣sing nor passing, nor diuersitie of time ruled, but by helpe of number. Also all y^t is made is shaped by reason of num∣bers, as he sayeth. Also the ensample in the wit and thought of the maker, was reason of number: And by certaine num∣ber thrice three orders of Angell•• be di∣stinguished. By three & seauen, vertues & might of all reasonable things & of spiri∣tuall wits be distinguished. And the E∣lementes be fastened by vertue and sci∣ence of numbers. And so for to speak, all thing vsed coniunction of numbers both spirituall and corporall, both of heauen & earth. And numbers haue composition among themselues, as Boetius sayeth. For in y^e substance of numbers is found euen and odde, that maketh all number by certaine might of God, for they bee diuers & contrary, and commeth neuer∣thelesse of one gendering and well, that is one, and bee ioyned in one compositi∣on without meane, and in lykenesse of proportion. And so it appeareth well that euery number is odde or euen. The euen number may be dealed euen a twaine, and leaueth not one, but the odde num∣ber is it which may not be dealed euen a two, without one odde. Or else by Pi∣thagoras lore, the euen number may be dealed vnder the same dimension, and in least and in most. In least diuision & most greatest number, as if thou dealed ••n hundred in fiftie, & fiftie is y^e most part, and fiftie is the least diuision, for it is di∣uided but once, and there maye bee no lesse diuision then in two parts. For the more an euen number is diuided in ma∣ny parts, so much the greatnesse is dimi∣nished. As it fareth of a tree y^t is hewen in many parts: but the number of diuisi∣ons is alway more. And the cause is (as he saith) for a great quantitie may be di∣minished, diuided in••••tly. But a num∣ber increaseth & waxeth endlesse. There∣fore y^e diuision of an euē number is most in continuall quantity, & lesse in number & discréet quantitie. The odde number is kindly diuided in two partes, more and lesse. The euen number is sometime dea∣led in two euen parts, and sometimes in vneuen more & lesse. And when the num∣ber is dealed euen in two, of the one part bee odde, the other is odde also, and if the one part be euen, the other is euen, as when eight be dealed in foure and foure, & twelue in sixe & sixe, and so of other. And if one of euen diuision bee odde, the other is odde also, as when was dealed in three & three, and ten in fiue and fiue, & foureteene in seauen & seuen. And so in euen diuision is not euennesse meddeled with oddenesse, nor oddenesse with euen∣nesse, but onely in the number of two, that is prince of euennesse, and taketh not euen diuision For it is compow••••d of twice one, and of the first euennesse of two. And if y^e euē number be dealed in two parts, more and lesse, if the one part be euen, the other is euen, and if ten bee dealed in eight and two or in sixe and foure. Also if sixe be diuided in foure and two, and eight in sixe and two, and so of other. But if the one part be odde, neede the other is odde. And if ten be dealed in seauen and three, and eight in three and fiue, & so of other And it maye neuer be that one part of such a diuision to odde, and the other euen, nor one euen and the other odde.

And alway where the odde number is diuided in two parts, more & lesse. y^e one parte is euen, and the other odde, as if

thou deniest seuen in three and foure, the one parte is euen, and the other is odde, & that is generally found in all odde nū∣bers. And one is mother of pluralitie, and cause of euen & odde, for if thou put one to an odde number, ••••••de then makest an euen number: And if thou takest one out of an euen number, euen thou makest an odde number. Also of all numbers set in kinde disposition about one, and ioyned togethers, and is the middle: As if thou sayest, one, two, thrée, one put to one ma∣keth two in the middle betwéene one and thrée. Also if thou sayest, two, thrée, foure, one put to twaine maketh thrée, the middle betwéene thrée and foure. Al∣so if thou saiest, three, foure, fiue, one put to thrée maketh foure, the middle between thrée and fiue. And of other passing vp∣ward of partes, and speciall kindes of euen number and odde, it is treated be∣fore. To make processe of all the gende∣ringes and proportions, accorde and di∣uersitie of these numbers, it were right long. Therfore of properties of numbers is 〈…〉〈…〉 for this time. Onely we shall wit, y^t in numbers it is hard to finde the middle, as Isid. saith. For it is most cer∣taine, y^t numbers be endlesse 〈…〉〈…〉, for tel thou neuer so long til thou think to make an end, yet one put to the number ma∣keth the number more, and odde or euen. The reason & property of y^e 〈…〉〈…〉 th•••• 〈…〉〈…〉 know in this wise: First put togethers the lesse number and the more, & depart euen in 〈…〉〈…〉, and thou shalt finde the middle ••n this wise: Take sixe for the lesse, & twelue for the more, & put them together, and sixe & twelue ma∣keth eight 〈…〉〈…〉. Deale them euen in two that is nine, and so it is generall in A∣rethmetike, that by as many as the mid∣dle passeth the least, by so many y^t most passeth the middle. 〈…〉〈…〉 passeth sixe by thrée, and twelue passeth nine by thrée, as Isidore sayth, libro. 2. Heereof excepti∣on is set before.

Notes

• * 1.1

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