### CHAP. 11. What it is to extract the square roote of a number? That 25 is the number that is the roote of 666; and remarkablely opposed unto 12. Some objections answered concerning the fractions of the roote of 666.

AND thus having hitherto proved by the example of the opposite number, and by the wordes of the text, that the roote of this number ought to be extracted; I come now from quod fit, to quid fit, from pro∣ving that is to be extracted, to shew what it is to extract it.

To extract the square roote of a number gi∣ven, is to find out the greatest number, which being multiplied into it selfe and having the fractions added to the product, (if there be any fraction remaining) maketh the first number. And how this is to be performed I need not here relate; it is sufficiently declared by such as Page 64 have written of Arithmeticke. And although many learned, and worthy Divines (whose bookes I account my selfe not worthy to beare) are perhaps ignorant of it: yet is this kind of Mathematicall learning called wisedome in* the Scriptures, and in this may consist one part of that wisedome and understanding, which is in the wordes of the text required for the fin∣ding out of this mystery. Let him therefore that hath this skill in humane Arts, and Scien∣ces, and let him that hath understanding to ex∣tract the rootes of numbers, extract the roote of the Beasts number, and he shall find that fatall number to be 25, and that the fractions remai∣ning are 41: and that this is proved by multi∣plying 25 by it selfe, which makes 625, and by adding the fractions which are 41 unto 625, both which numbers added together, make the just summe 666. And although the roote of this number, not being a simple roote, as the roote of 144 is, must in strictnesse of speech be expressed by more a numbers then one, yet there can be no doubt or question which of those numbers must be the number answerable and opposite to 12. The roote of 666 may be said to be 25 41 / 51 or else, to expresse it more exactly, it may be said to be 25 25 / 31: or it may be said to be Page 65 25 806 / 1000 or 25 8069758 / 10000000 nay any number whatso∣ever may be made one of those numbers by which the fractions may be expressed. But howsoever the number of the fractions be va∣riable, yet the number •5 is alwayes constant and the same, as 12 is in the opposite roote. And as 12 is the greatest number, and the least number, and the only number of unities of the same denomination with the number 144, (which is or can be contained in) the roote of 144: so 25 is the greatest number, and the least number, and the only number of unities of the same denomination with the number 666, (which is or can be contained in) the roote of the number 666. And this sicut similitudinis is suffi∣cient to establish an evident antithesis between the two great Cardinall numbers of these two rootes, although in respect of the fractions there be no sicut aequalitatis between them. And whe∣ther the fractions be added or not added to 25, yet they can neither augment, nor diminish the roote, no not so much as by one unite, as it is sufficiently knowne to those that know what fractions are. It is no good argument to say that 25 is not opposed to 12, because 2• hath fracti∣ons appendant to it, and 12 hath not; for, Omne simile est etiam dissimile, and by the same reason Page 66 it might be said, that the 12 Apostles are not an∣swerable to the 12 Patriarches, because the A∣postles had some priviledges or defects which the Patriarches had not. Or that the Cardinals are not answerable to the Apostles in the Ro∣mish Hierarchy, because they have red hats, which Jbelieve the Apostles had not. Besides, it is often times an usuall and ordinary thing, etiam praxi mathematicâ, in many arithmeticall operations, to cast away, and not to regard the fractions of roots, because the root or Cardi∣nall number it selfe is of sufficient exactnesse to prove or effect the conclusion, which is desired; nay sometimes and in some cases, when rootes of numbers are to be extracted, they cannot make the fractions to be usefull to their purpo∣ses, though they would. For suppose a captaine haue 666 men under his command, and would reduce them to a square figure of equall sides and ranks: to effect his purpose he must extract the root of 666, which he would finde to be 25 41 / 51, and by that he would conclude that he must of necessity take the number 25 to be the number of rankes, and the number of men in euery ranke, and no other number would serve his turne. As for the 41 odd men he must reject them as unusefull, if he will have his ar∣my Page 67 exactly square. The number 50 is no equi∣laterall square number, and yet S. Augustine upon the 150 Psalme & else where maketh the mystery of this number to consist in the roote of it which is 7 without any scruple of any fraction: and it were easie to set downe many authors which interpret the same and other numbers after the same manner. Seeing there∣fore it is usuall among men in many cases, and necessary in some, not to regard the fractions, but onely the Cardinall number in the extracti∣on of rootes; why then may we not doe like∣wise in extracting the roote of 666? why may we not consider the number 25 first by it selfe, and as it is the only Cardinall number opposed to 12, by which the roote of 666 can be truely expressed; and afterwards as it hath relation to the fractions, especially being the unities of the roote of this number are sometimes to be ap∣plied to Persons, who are things indivisible in∣to parts or fractions, as are also the unities of numbers essentially and absolutely considered. And the truth is, that no number of fractions, as fractions, is properly a part of any roote ess∣entially considered: for howsoever it be true that fractions, being reduced to some certaine denomination, doe more exactly shew the side Page 68 of a •quare figure as it is quantitas continua, yet it cannot be proved that they are any proper essentiall part of the roote it selfe as it is quanti∣tas discreta. For the fractions of a roote doe sup∣pose every unitie in the roote to be divided in∣to many parts, and the number it selfe, whose roote is to be extracted, to be resolved into ano∣ther number farre greater then it selfe. And the fractions (if it be well considered) are rather part of the roote of the second number into which the first is supposed to be resolved, then of the roote of the first number which was to be extracted. As for example the roote of 666 is 25 806 / 1000 which fractions doe suppose every u∣nitie of 666, to be multiplied into one million; and every unite of the roote 25 to be multiplied into one thousand: for if the figures of the roote and of the fractions be joyned together they doe make 25806. which number is the true roote of 666000000 so that 806, (being now unities of the same denomination with the number 666000000,) are more proper∣ly a part of the roote of 666 millions, then of the roote of 666 unities. And by this it may be observed and understood, that while w• doe •e•ke after Ordinall numbers, more exactly to expresse that roote whose Cardinall number Page 69 we have already found out, we doe nothing els in effect (although many times we consi∣der it not) but seeke after the Cardinall num∣ber of another roote whose fractions, being now the fractions of a greater number, are not at all, or not so much to be regarded. By these considerations it may sufficiently appeare, that that Cardinall number which is the ex∣acte roote of the greatest square number con∣tained in any number given, whose roote is to be extracted, is the number which is most re∣markeable and chiefly sought after in the ex∣tr•ction of every roote; for this number is, and is to be reputed, not only the roote of the grea∣test square number contayned in the number given, but al•o of divers other numbers which doe exceede it, but yet with this difference, that it is the roote of the square number without fractions, and o• o•he• numbers with fracti∣ons added to it. A•d that the same Cardinall number with a little difference of fractious, should be the ro•t• o•••• numbers then one and of •ny nu•be•s▪ 〈◊〉 i• a thingth ••ose which are not 〈◊〉 ver••• in the ••t•action of rootes, doe neith•• co•••••r nor well under∣stand: and this •••s them 〈◊〉, that be∣cause 25 is the •oo•• o•〈◊〉 th•t therefore it i•Page 70 not the roote of 666. But such should consider that one reason why these kinds of numbers are called rootes, is, because every such num∣ber, is in this respect like unto the roote of a tree; for as one roote hath many branches growing upon it, and issuing from it, although some grow nearer the roote then others: so the same number may be the root of divers other numbers, which have all a reall, and yet a diff∣ering dependance upon it. And although as∣cending upward, there be no infallible directi∣on from the roote to any one particular branch, yet descending from the top of any one branch, there is certaine and infallible di∣rection to the same roote: and so whosoever shall goe to extract the roote of any number greater then 624 and lesse then 676, according to such rules of art as are, and have hitherto been commonly taught, and generally received, shall be infallibly directed, not to the number 26, but to the number 25, & to that number only, as unto the only Cardinall number first sought after in the extraction of all rootes; & this num∣ber either by it selfe, or having some fractions appendant to it, is the true root of all such num∣bers as are included between those two num∣bers above mentioned Neither is it usual or pos∣sible Page 71 truly to expresse any root that hath fracti∣ons, by any other Cardinall number, but only by that number which being multiplied into it selfe produceth the greatest square number contained in that number, whose roote is to be extracted. And although there be divers o∣ther numbers besides this number 666, by any one of which we might have been infallibly directed to this number 25, as unto the only Cardinall number by which their roots could have been expressed: yet there is no one of those numbers but only the number 666, whose most perfect figure doth represent the figure of Rome, as the most perfect figure of the number 144 doth represent the figure of Hieru∣salem: and for this reason chiefly, and for di∣vers others (as shall be shewed abundantly in due place) it was both convenient, and neces∣sary, that this number 666 should be chosen ra∣ther then any other. But if it be objected, that the roote of 666 is nearer to 26, then 25; yet I answere that it cannot be truely said to be 26, but is truely said to be 25: and that not only be∣cause 25 is contained in this roote, as are also all other numbers lesse then 25, but because 25 is the greatest number of unities of the same denomination with the number 666, contain∣ed Page 72 in this roote. And that I may expresse this the more clearly I will make it plaine by an in∣stance, and because there is only a threefold ambiguity incident to the expression of such rootes as have fractions, I will therefore sup∣pose the same question to be proposed to three severall men concerning the length of one side of an exactly square figure containing pre∣cisely 666 foote of square measure. The fi•st being asked how many foote in length one side of this figure must be, would perhaps say 25, because that is the greatest number of feet contained in it. The second being asked how many foote in length one side of this figure must be, would perhaps say 26, because the exact length is nee•est unto it. The third being asked the same question, would perhaps say, that it was neither 25 foote long nor 26, but that it was nine or ten inches more then 25, •nd two or three inches lesse then 26. The first of these three answers is clearly and evidently true, for 25 is the greatest number and the least number, and the only number of feet by which that length can be expressed. The second an∣swere is clearly and evidently false, for if that length, which wanteth above two inches of 26 foote, had lacked but one inch or one small Page 73 part of an inch, it could not then have been •ru∣ly said to have been 26 foot long, and who•oe∣ver taketh proximum vero, pro vero, in this kinde, 'tis plain that he taketh falsum pro vero, falshood for truth. The third answer is impertinent, and not to the purpose, for the question propoun∣ded, was not how many inches, but how many foot long one side of that figure was. Neither ought the question to have been any other waies propounded; because in the extraction of all roots, the first number sought after, is a Cardinall number, and not an ordinall number, a number of such parts of which every one may be said to be an integrum, and not a number of fractions, which result of themselves without seeking after, when this first number is found out. And as •for the fractions, I have already shewed, that they cannot darken the remarka∣blenesse of the number 25, nor disanull that an∣tithesis, which is and ought to be between this number and that number, which is opposite un∣to it. Yet if any number, by which the fractions of this root may be expressed, be more remark∣able and rather to be chosen then another: then without question it is that number, by which the fractions may be most exactly expressed by fewest figures, and by such numbers as doe Page 74 leave fewest fractions of fractions remaining. And certainly it seemeth strange and wonder∣full to me, neither doe J think it meerely acci∣dentall, that the number 25, should so exactly expresse the fractions of the root of 666, as that no other number lesse then it or neere unto it, can so perfectly expresse them. For neither 41 / •1 nor 806 / 1000 doe so exactly expresse the fractions of the root of 666 as 25. 25 / 31 nay although those* numbers be infinite by which the fractions of this root may be expressed, yet I believe there is not any one of them which leaveth so small a number of fractions, as this number doth. And although numbers and their roots be infinite in number, yet that there should be any other number besides this number 666, the fractions of whose root may be so exactly ex∣pressed by the Cardinal number of its own root with any denominator whatsoever, as the frac∣tions of the root of 666, are by 25; this is such a thing as I conceive to be 〈 in non-Latin alphabet 〉, a thing that cannot happen to be found out, although I will not say 'tis absolutly unpossible. But in the mean time untill some such number shall be produ∣ced, whose root may be after the same manner, and with the like exactuesse expressed, shall be farre from thinking that this happeneth ca•ual∣ly Page 75 and accidentally; but shall believe rather, that as the doubling of Pharaohs dreame was an ar∣gument of the certainty of that which was sig∣nified by it: so because this number 25 is in a double respect remarkable in the root of 666, (first, in that it is the onely Cardinall number of the prime or Cardinall unities: and secondly, in that it is the onely number of ordinall unities or fractions, by which that root can be by fewest figures most exactly expressed) I doe there∣fore conclude, that it is a certaine and esta∣blished truth, that this number twenty five is that fatall and unfortunate number of Anti∣christ opposed to the number 12, and that in an higher nature, & in a greater degree of oppositi∣on then 666 is opposed to 144, it being that very number which as it is most apparently and re∣markably applicable to the City and Hierarchy of Antichrist, so is it also chiefly intended by the number 666: although it pleased the wis∣dome of God to seale it up in a mystery, and as it were to lock it up in the cabinet of a greater number, untill that time came which God had appointed for mans reason to unlock this cabi∣net, by the key of computation, and so to take out this so long hidden number, by which An∣tichrist is (as it shall be shewed) most evident∣ly, Page 76 and miraculously described. For if this num∣ber had been expressely named in this place to have been the number of the Beast, or if that mysticall Babylon, in which Antichrist raign∣eth, had been measured in the Scriptures by this number 25, as the new Jerusalem is by the number 12, then there had been no mystery at all contained in it; then it had been so plainely set downe, that Antichrist would have preven∣ted it. For as it is not probable that ever any Pope will now chuse such a name, whose nu∣merall letters shall make the just •umme 666, (because some men suppose that this number is so to be applied:) so neither is it likely that An∣tichrist would ever have chosen and affected this number 25 above, and before any other number, to be the only conspicuous, and remar∣kable number in the foundation of his Hierar∣chy, except the wisdome of God, who taketh the wise in their own craftinesse, had sealed it up in a mystery in such sort that they should not understand it, as long as they had any possibility either to alter it or to deny it. For even so hath it come to passe in the Hierarchy of the Romish Clergy, that their ancesters have fatally, and un∣wittingly laid the foundation of the Papacy up∣on this number 25, and have made this number Page 77 so particularly applicable to their City, and to themselves in all those materiall circumstances, in which the number 12 is applicable either li∣terally to the city Jerusalem, or spiritually to the Church of God, and Hierarchy thereof; that no pollicy is now sufficient to cover it, nor their own impudence (with any shew of probabili∣ty) to deny it.