PROBLEME. I. To finde a number thought upon.

BId him that hee Quadruple the Number thought upon, that is, multiply it by 4 and unto it bid him to adde, 6. 8. 10. or any Number at pleasure: and let him take the halfe of the summe, [...]en aske how much it comes to: for then if [...]ou take away halfe the Number from it which [...]ou willed him at first to adde to it, there [...]hall remaine the double of the number [...]ought upon.

Example.

PROBLEM. II. How to represent to these which are in a chamber that which is with­out, or all that which passeth by.

THis is one of the finest experiments in the Opticques, and it is done thus, chuse a Chamber or place which is towards the street, frequented with people, or which is against some faire flourishing object, that so it may be more delightfull and pleasant to the beholders, then make the Roome darke by shutting out the light, except a small hole of sixe pence broad, this done, all the Images and species of the object which are without, will be seene with­in: and you shall have pleasure to see it not one­ly upon the wall but especially upon a sheete of white paper or some

white cloth hung nere the hole: and if unto the hole you place a round Glasse, that is, a Glasse which is thicker in the mid­dle than at the edge: such as is the com­mon burning Glasses, or such which old people use, for then the Images which before did seeme dead, and [Page 7] of a darkish colour, will appeare and be seene [...]pon the paper, or white cloth, according to their naturall colours, yea more lively than their naturall; and the appearances will be so much the more beautifull, and perfect, by how much the hole is lesser, the day cleare & the sun [...]hining. It is pleasure to see the beautifull and goodly representation of the Heavens, inter­mixed with clouds in the Horizon, upon a wooddy situation, the motion of Birds in the Aire, of Men, and other Creatures upon the ground, with the trembling of Plants, tops of Trees, and such like, for every thing will be seene within even to the life, but inversed: notwithstanding this beautifull paint will so naturally represent it selfe in such a lively perspective, that hardly the most accurate Pain­ [...]er can represent the like. Now the reason why [...]he Images and objects without are inversed, is because the species doe intersect one another in the hole: so that the species of the feete ascend, and these of the head

descend.

But heere note, that they may be Repre­sented right two man­ner of wayes; first with a concave glasse, [...]econdly, by helpe of another convex glasse: disposed or placed be­ [...]weene the paper and the other Glasse: as may [Page 8] be seene here by the figure.

Now I will adde here onely by passing by, for such which affect painting, and portraiture, that this experiment may excellently helpe them, in the lively painting of things per­spective wise, as Topographicall cards, &c. and for philosophers, it is a fine secret to explaine the organ of the sight, for the hollow of the eye is taken as the close Chamber, the balle of the Aple of the eye, for the hole of the Cham­ber, the Crystalline humor at the small of the Glasse and the bottome of the eye, for the wall, or leafe of Paper.

PROBLEM. III. To tell how much waighs the blow of ones fist, of a Mallet, Hatchet or such like, or resting without giving the blow.

[...] Caliger in his 331. exercise against Cardan, [...] relates that the Mathematitians of Maxi­millian the Emperour did propose upon a day [Page 10] this Question, and promised to give the re [...] lution; notwithstanding Scaliger delivered [...] not, and I conceive it to be thus. Take a Bala [...] and let the fist, the Mallet or Hatchet rest up [...] the Scale or upon the beame of the Ballan [...] and put into the other scale, as much wei [...] as may counterpoyse it; then charging or l [...] ing more waight into the Scale, and striking [...] on the other end: you may see how much [...] blow is heavier than another, and so co [...] quently how much it may waygh: for as A [...] stotle saith; the motion that is made in strike ads great waight unto it, and so much [...] more, by how much it

is quicker: therefore in effect if there were placed a thousand mallets, or a thousand pound waight upon a stone, nay though it were exceedingly pressed downe by way of a vice, by levers or other mechanick In­gine, it would be no­thing to the rigor and violence of a blow.

Is it not evident that the edge of a knife lai [...] upon butter, and a hatchet upon a leafe of P [...] per, without striking makes no impression, o [...] at least enters not; but striking upon the wo [...] a little, you may presently see what effect it hath which is from the quicknesse of the motion which breakes and enters without resistance [Page 11] extreame quicke, as experience shewes [...]he blowes of Arrows, of Cannons, Thun­ [...]ts, and such like.

PROBLEM. IIII. How to breake a staffe which is laid upon [...] Glasses full of water, without breaking the Glasse, spilling the water, or upon two Reeds or Strawes without breaking of them.

FIrst place the Glasses which are full of n [...] upon two joynt stooles, or such like, [...] one as high as the other from the ground, [...] distant one from another by 2. or 3. foote, t [...] place the ends of the staffe upon the edges of [...] two Glasses so that they bee sharpe; this d [...] with all the force you can, with another st [...] strike the staffe which is upon the two Gla [...] in the middle, and it

will breake without breaking the Glasses or spilling the water. In like manner may you doe upon two Reeds, held in the aire without breaking them: thence Kitchin­boyes often breake bones of mutton upon their hand, or with a [Page 13] [...] without any hurt, in onely striking up­ [...] [...]e middle of the bone, with a knife.

N [...]w in this act the two ends of the staffe in [...]ing slides away from the Glasses, upon [...] they were placed; hence it commeth [...]e Glasses are no wise indangered, no more [...]he knee upon which a staffe is broken, [...] much as in breaking it presseth not: as [...]otle in his Mechanicke questions obser­ [...].

PROBLEM. V. How to make a faire Geographicall Card in Garden Plot, fit for a Prince, or great personage.

IT is usuall amongst great men to have [...] Geographicall mappes, large Cards, and g [...] Globes, that by them they may as at once h [...] a view of any place of the world, and so [...] nish themselves with a generall knowled [...] not onely of their owne kingdomes fo [...] situation, Longitude, Latitude, &c. but o [...] other places in the whole Vniverse, with t [...] Magnitudes, Positions, Climats, and distanc [...]

Now I esteeme that it is not unworthy [...] the meditations of a Prince, seing it ca [...] with it many profitable and pleasant cont [...] ments: if such a Card or Mappe by the ad [...] and direction of an able Mathematitian [...] Geographically described in a Garden [...] forme, or in some other convenient place, [...] in stead of which generall description mi [...] particularly, and Artificially be prefigured [...] whole kingdomes and dominions, the Mo [...] taines and Hils being raised like small hill [...] [...] with curfes of earth, the vallies somewhat [...] cave; which will be more agreeable & pleas [...] to the Eye, than the description in p [...] Mapps and Cards, within which may be pre [...] ted, the townes, villages, Castles, or other re [...] keable edifices in small greene Mossie ban [...] or springworke proportionall to the plat for [...] [Page 15] forrests and woods represented according to [...]ir forme and capacitie, with hearbs and [...]abs, the great rivers, lakes and ponds, to dilate [...]mselves according to their course from [...]e Artificiall fountaine made in the Garden [...]passe through Chanels; then may there bee [...]posed walkes of pleasure, Ascents, places [...] repose adorned with all varietie of de­ [...]tfull hearbs & flowers, both to please the eye, other sences. A Garden thus accommodated [...]ll farre exceede that of my Lo. of Verulams [...]cified in his Essayes; that being onely for [...]ight and plasure, this may have all the pro­ [...]ties of that, and also for singular use, by [...]hich a Prince may in little time personally [...]ite his whole kingdome, and in short time [...]ow them distinctly, and so in like manner [...]y any particular man Geographically prefi­ [...]e his owne possession, or heritage.

PROBLEM. VI. [...]w three staves, knives, or like bodies may be conceaved to hang in the Aire, without being supported by any thing, but by themselves.

TAke the first staffe A. B. raise up in the Aire, the end B. and upon him croswise [...]ce the staffe C. B. then lastly in Triangle wise [...]ce the third staffe E.F. in such manner that may be under A. B. and yet upon C. D. I [...] that these staves so disposed cannot fall, and [Page 16] the space C. B. E. is made the stronger, [...] how much the more it is pressed downe, if th [...] staves breake not, or sever themselves fro [...] the triangular forme▪

so that alwayes the Center of gravitie be in the Center of the Triangle: for A. B. is supported by E. F. and E. F. is helde up by C. D. and C. D. is kept up from falling by A. B therefore one of these staves can­not fall, and so by consequence none.

PROBLEM VII. How to dispose as many men, or other thing in such sort that rejecting, or casting away the 6. 9. 10. part, unto a certaine number, there shall remaine these which you would have.

ORdinarily the proposition is delivere [...] in this wise: 15. Christians and 15. Turke [...] being at Sea in one Shippe, an extreame tempel being risen, the Pilot of the Shippe say it is necessary to cast over board halfe of the num­ber of Persons to disburthen the Shippe, an [...] [Page 17] to save the rest: now it was agreed to bee done by lot and therefore they consent to put themselves in ranke, counting by nine and nine the ninth Person should alwayes be cast into the Sea, untill there were halfe throwne over board; Now the Pilote being a Christian in­deavoured to save the Christians, how ought hee therefore to dispose the Christians, that [...]he lot might fall alwayes upon the Turkes, [...]nd that none of the Christians be in the ninth place?

The resolution is ordinarily comprehen­ded in this verse, ‘Populeam virgam mater regina ferebat:’

For having respect unto the vowels, making [...] one, e two, i three, o foure and u five: o the [...]rst vowell in the first word sheweth that [...]ere must bee placed 4. Christians, the next [...]owell u, signifieth that next unto the 4. Christi­ [...]s must be placed 5. Turkes, and so to place [...]oth Christians and Turkes according to the [...]antitie and value of the vowels in the words [...] the verse, untill they be all placed: for then [...]unting from the first Christian that was [...]aced, unto the ninth, the lot will fall upon [...] Turke, and so proceede. And here may be [...]rther noted that this Probleme is not to bee [...]ited, seeing it extends to any number and [...]er whatsoever, and may many wayes bee [...]full for Captaines, Magistrats, or others [...]ich have divers persons to punish, and would [...]stise chiefely the unruliest of them, in taking [...] 10. 20. or 100. person, &c. as we reade was [Page 18] commonly practised amongst the ancient Ro­mans: herefore to apply a generall rule in coun­ting the third, 4. 9. 10. &c. amongst 30. 40. 50. persons, and more or lesse; this is to bee ob­served: take as many unites as there are per­sons, and dispose them in order privately: as for example, let 24. men bee proposed to have committed some outrage, 6. of them especi­ally are found accessary: and let it be agreed that counting by 8. and 8. the eight man should be alwayes punished: Take therefore first 24. unites, or upon a peece of paper write downe twenty foure Ciphars, and account from the beginning to the eighth, which eighth marke, and so continue counting al­wayes marking the eighth, untill you have markt 6. by which you may easily perceive how to place those 6. men that are to bee pu­nished, and so of others. It is supposed that Io­sephus the Author of the Iewish History, esca­ped the danger of death by helpe of this Pro­bleme; for a worthy Author of beliefe reports in his eighth Chapter of the third Booke of the destruction of Ierusalem, that the Towne of Iotapata being taken by maine force by Ve­spatian; Iosephus being governour of that Towne accompained with a troope of 40. Souldiers, hid themselves in a Cave in which they resolved rather to famish than to fall into the hands of Vespatian: and with a bloody resolution in that great distresse would have butchered one another for sustenance: had not Iosephus perswaded them to die by [Page 19] lot, and order, upon which it should fall: Now seeing that Iosephus did save himselfe by this Art: It is thought that his industrie was exercised by the helpe of this Probleme; so that of the 40. persons which hee had, the third was alwayes killed. Now by putting him­selfe in the 16. or 31. place he was saved, and one with him which hee might kill, or easily perswade to yeeld unto the Romans.

PROBLEM. VIII Three things, and three persons proposed, to finde which of them hath either of these three things.

LEt the three things bee a Ring, a peece of Gold, and a peece of Silver, or any other such like, and let them bee knowne privatly to your selfe, by these three vowels, a e i: or let there bee three persons that have different names, as Ambrose, Edmond and Iohn; which privately you may note or account to your selfe once knowne by the aforesaid vowels, which signifie for the first vowell 1. for the second vowell 2. for the third vowell 3.

Now if the sayd three persons should by the mutuall consent of each other privatly change their names, it is most facill by the course and excellencie of numbers, distinctly to declare each ones name, so interchanged: or of three persons in private, the one should take a Ring, the [Page 20] other a peece of Gold, and the third should take a peece of Silver; it is easie to finde which hath the Gold, the Silver, or the Ring, and it is thus done.

Take 30. or 40. Counters (of which there is but 24. necessary) that so you may conceale the way the better, and lay them downe before the parties, and as they sit or stand give to the first 1. Counter, which signifieth a the first vowell, to the second 2. Counters which re­presents e the second vowell, and to the third 3. Counters which stands for i. the third vowell: then leaving the other Counters upon the Ta­ble, retire apart, and bid him which hath the Ring, take as many Counters as you gave him, and hee that hath the Gold, for every one that you gave him, let him take 2. and he that hath the Silver for every one that you gave him let him take 4. this being done, consider to whom you gave one Counter, to whom two, and to whom three; and marke what number of Counters you had at the first, for there are necessarily but 24. as was sayd before, the surpluse you may privately reject. And then there will be left either 1. 2. 3. 5. 6. or 7. & no other number can remaine, w ^ch if there be, then they have failed in taking according to the di­rections delivered: but if either of these num­bers doe remaine, the resolution will bee dis­covered by one of these 6. words following, which ought to be had in memory, viz.

[Page 21]As suppose 5. did remaine, the word belong­ing unto it is semita, the vowels in the first two sillables are e and i, which sheweth accor­ding to the former directions that to whom you gave 2. Counters he hath the Ring (seeing it is the second vowell represented by two as before) and to whom you gave the 3. Counters he hath the Gould, for that i repre­sents the third vowell, or 3. in the former direction, and to whom you gave one Coun­ter, he hath the Silver, and so of the rest: the va­rietie of changes in which exercise, is layd open in the Table following.

This feat may be done also without the for­mer words by helpe of the Circle A. for ha­ving divided the Circle into 6. parts, write 1. within and 1. without, 2. within and 5. with­out, &c. the first 1. 2. 3. which are within with the numbers over them, belongs to the upper semicircle; the other numbers both within and without, to the under semicircle; [Page 22] now if in the Action there remaineth such a number which may bee found in the upper se­micircle without, then that which is opposite within shewes the first, the next is the se­cond, &c. as if 5. remaines, it shewes to whom hee gave 2. hee hath the Ring, to whom you gave 3. hee hath the Gould, &c. but if the re­mainder bee in the under semicircle, that which is opposite to it, is the first; the next back­wards towards the right hand is the second, as if 3. remaines, to whom you gave 1. he hath the Ring; he that had 3. he had the Gould, &c.

PROBLEM. IX. How to part a vessell which is full of wine con­taining 8. Pints, into two equall parts, by two other vessels which contained as much as the greater vessell; as the one being 5. Pints, and the other 3. Pints.

LEt the 3. vessels be represented by A. B. C. A. being full, the other two being emptie; first powre out A. into B. untill it bee full: so there will be in B. 5. Pints and in A. but 3. Pints; then powre out of B. into C. untill it bee full: so in C. shall be 3. Pints, in B 2. Pints, and in A. 3. Pints; then powre the wine which is in C. into A. so in A. will be 6. Pints, in B 2. Pints, and in C. nothing: then powre out the wine which is in B. into the pot C. so in C. [Page 23] there is now 2. Pints, in B. nothing and in A. 6. Pints. Lastly, powre out of A. into B. untill it be full, so there will

bee now in A. onely 1. Pint, in B. 5. Pints, and in C. 2. Pints: But it is now evident that if from B. you powre in unto the pot C. untill it bee full, there will remaine in B. 4. Pints, and if that which is in C. viz. 3 Pints bee pow­red into the vessell A. which before had 1. Pint, there shall bee in the vessell A. but halfe of its liquor that was in it at the first, viz 4. Pints as was required. Otherwise powre out of A. into C. untill it be full, which powre in­to B then powre out of A. into C. againe untill it bee full, so there is now in A. onely 2. Pints, in B. 3. and in C 3. then powre from C into B. untill it bee full, so in C. there is now but 1. Pint, 5. in B. and 2. in A. powre all that is in B. into A. then powre the wine which is in C. into B. so there is in C. nothing, in B. onely 1 Pint, and in A. 7. Pints: Lastly out of A. fill the pot C. so there will remaine in A. 4. Pints, or be but halfe full: then if the liquor in C. bee powred into B. it will bee the other halfe. In like manner might bee taken the halfe of a vessell which containes 12. Pints, by having but the measures 5. and 7. or 5. and 8. Now [Page 24] such others might be proposed, but wee omit many, in one and the same nature.

PROBLEM. X. To make a sticke stand upon the tipp of ones finger, without falling.

FAsten the edges of two knives or such like of equall poise, at the end of the sticke, leaning out somewhat from the sticke, so that they may counterpoise one another; the sticke being sharpe at the end and held upon the top of the finger, will there rest without suppor­ting: if it fall it must fall together, and that perpendicular or

plumbe wise, or it must fall side-wise or before one ano­ther; in the first man­ner it cannot: for the Center of gravitie is supported by the top of the finger: and see­ing that each part by the knives is counter-poysed it cannot fall sidewise, therefore it cannot fall no wise.

In like manner may great peeces of Timber as Ioists &c. be supported, if unto one of the ends be applyed convenient proportionall coun­terpoises, yea a Lance or Pike, may stand [Page 25] perpendicular in the Aire upon the top of ones finger: or placed in the midst of a Court by helpe of his Center of gravitie.

PROBLEM. XI. How a milstone or other Ponderositie, may be sup­ported by a small needle, without brea­king or any wise bowing the same.

LEt a needle be set perpendicular to the Ho­rizon, and the center of gravitie of the stone, be placed on the top of the Nedle: it is evident that the stone cannot fall, for asmuch as it hangs in aequilibra, or is counterpoysed in all parts alike; and moreover it cannot bow the Needle more on the one side, than on the other, the Needle will not therefore be eyther broken or bowed; if otherwise, then the parts of the Needle must penitrate and sinke one with ano­ther: that which is absurd and impossible to nature: therefore it shall be supported. The ex­periments which are

made upon trencher plates, or such like lesser thing doth make it most credible in greater bodies.

But here especially is to bee noted that the Needle ought to be uniforme in mat­ter and figure, and that it be erected per­pendicular to the Horizon, and lastly that the Center of gravitie be exactly found.

PROBLEM. XII. To make three knives hang and move upon the point of a Needle.

FIt the three knives in forme of a ballance, and houlding a Needle in your hand, and place the backe of

that knife which lies crossewise to the o­ther two, upon the point of the Needle: as the figure here sheweth you; for then in blowing softly up­on them, they will ea­sily turne & move up­on the point of the needle without falling.

PROBLEM. XIII. To finde the weight of smoake, which is exhaled of any combustible body whatsoever.

LEt it be supposed that a great heape of Fa­gots, or a load of straw waying 500. l should be fired, it is evident that this grosse substance will bee all inverted into smoake and Ashes: now it seemes that the smoake waighes no­thing; seeing it is of a thinne substance now delated in the Aire, notwithstanding if it were gathered and reduced into the thic­kest [Page 28] that it was at first, it would bee sensibly waighty: waigh therefore the ashes which admit 50 pound, now seeing that the rest of the matter is not lost, but is exhaled into smoke, it must necessarily bee, that the rest of the waight (to wit) 450 pound, must bee the waight of the smoke required.

PROBLEM. XIIII. Many things being disposed circular, (or other­wise) to find which of them, any one thinkes upon.

Suppose that having ranked 10 things, as A B C D E F G H I K, Circular (as the figure showeth) and that one had touched or thought upon G. which is the 7: aske the partie at what letter he would begin to account (for [Page 29] [...]count he must, otherwise it cannot bee done) which suppose, at E which is the 5 place, then [...]d secretly to this 5. 10. (which is the number of the Circle) and it makes 15, bid him ac­count 15. backward from E, beginning his account with that

number hee thought upon, so at E hee shall account to himselfe, 7, at D account 8, at C account 9 &c. So the account of 15 will exactly fall upon G, the thing or number thought upon: and so of others: but to conceale it the more, you may will the party from E to account 25. 35 &c. and it will be the same.

There are some that use this play at cards, turned up side downe, as the ten simple Cards, with the King and Queene, the King standing for 12, and the Queene for 11, and so knowing the scituation of the Cards: & thinking a cer­taine houre of the day: cause the partie to ac­count from what Card hee pleaseth: with this Proviso, that when you see where hee in­tends to account set 12. to that number, so in counting as before, the end of the account shall fall upon the Card: which shall denote or shew the houre thought upon, which being turned up will give grace to the action, and wonder to those that are ignorant in the cause.

PROBLEM. XV. How to make a dore, or a Gate, which shall open on both sides.

ALL the skill and subtiltie of this, rests in the artificiall disposer of 4 plates of Iron, two at the higher end, and two at the lower end of the Gate: so that one side may move upon the hookes or hindges of the Posts, and by the other end may be made fast to the Gate, and so moving upon these hindges, the Gate will open upon one side with the aforesayd plates, or hookes of Iron: and by helpe of the other two plates, will open upon the other side.

PROBLEM. XVI. To shew how a Ponderositie, or heavie thing, may be supported upon the end of a staffe (or such like) upon a Table, and nothing holding or touching it.

TAke a paile which hath a handle, and fill it full of water (or at pleasure:) then take a staffe or sticke which may not rowle upon the Table as E C, and place the handle of the Paile upon the staffe; then place another staffe, or [Page 31] sticke, under the staffe C E, which may reach from the bottome of

the Paile unto the former staffe C E, perpendicular wise: which suppose F.G, then shall the Paile of water hang without falling, for if it fall it must fall perpendicu­larly, or plumbe wise: and that cannot bee seeing the staffe C E, supports it, it being parallel to the Horizon and sustained by the Table, and it is a thing admi­rable that if the staffe C E, were alone from the table, and that end of the staffe which is up­on the Table were greater and heavier than the other: it would be constrained to hang in that nature.

PROBLEM. XVII. Of a deceitfull Bowle to play withall.

MAke a hole in one side of the Bowle, and cast moulten Lead therein, and then make up the hole close, that the knavery or deceit be not perceived: you will have pleasure to see, that notwithstanding the Bowle is cast directly to the play, how it will turne away side-wise: for that on that part of the Bowle which is hea­vier upon the one side than on the other, it ne­ver will goe truly right, if artificially it bee not corrected; which will hazard the game to those which know it not: but if it bee knowne that the leady side in rowling, be alwayes under or above, it may goe indifferently right; if other­wise, the weight will carry it alwayes side­wise.

PROBLEM. XVIII. To part an Apple into 2. 4. or 8 like parts, without breaking the Rind.

Passe an needle and threed under the Rind of the Apple; and then round it with diverse turnings, untill you come to the place where you began: then draw out the thred gently, and part the Apple into as many parts as you thinke convenient: and so the parts may bee taken out betweene the parting of the Rind, and the rind remaining alwayes whole.

PROBLEM. XIX. To finde a number thought upon without asking of any questions, certain opera­tions being done.

BId him adde to the number thought (as ad­mit 15) halfe of it, if it may bee, if not the greatest halfe that exceede the other but by an unite, which is 8; and it makes 23: Secondly, unto this 23. adde the halfe of it if it may bee, if not the greatest halfe viz 12. makes 35. in the meane time; note that if the number thought upon cannot be halfed at the first time as here it cannot: then for it keepe 3. in the memory, if at the second time it will not be [Page 34] equally halfed, reserve 2. in memory, but if at both times it could not be equally halved, then may you together reserve five in memory: this done, cause him from the last summe, viz. 35. subtract the double of the number thought, viz. 30. rest 5. will him to take the halfe of that if he can, if not, reject 1. and then take the halfe of the rest: which keepe in your memory: then will him to take the halfe againe if he can, if not take one from it, which reserve in your memory, and so perpetually halveing untill 1. remaine: for then marke how many halfes there were taken, for the first halfe account 2, for the second 4, for the third 8, &c. and adde unto those numbers the ones which you reserved in memory, so there being 5 remaining in this proposition, there were 2 halfeings: for which last I account 4, but because it could not exactly be halved without rejecting of 1: I adde the 1 therefore to this 4, makes 5, which halfe or summe alwayes multiplyed by 4, makes 20: from which subtract the first 3 and 2, because the halfe could not bee formerly added, leaves 15, the number thought upon.

[Page 35] Other examples.

PROBLEM. XX. How to make an uniforme, & an inflexible body, to passe through two small holes of divers formes, as one being circular, & the other square, Quadrangular, and Tri­angularwise, yet so that the holes shall be exactly filled.

THis Probleme is extracted from Geometri­call observations, and seemes at the first [Page 36] somewhat obscure, yet that which may be ex­tracted in this nature, will appeare more diffi­cult and admirable. Now in all Geometricall practices, the lesser or easier Problemes doe al­wayes make way to facilitate the greater: and the aforesaid Probleme is thus resolved. Take a Cone or round Pyramedie, and make a Circu­lar hole in some boord, or other hard materiall, which may bee equall to the bases of the Cone, and also a Triangular hole, one of whose sides may be equall to the Diamiter of the circle, and the other two sides equall to the length of the Cone: Now it is most

evident that this Co­nicall or Pyramidall body, will fill up the Circular hole, and be­ing placed side-wise will fill up the Trian­gular hole: moreover if you cause a body to be turned, which may be like to two Pyra­mides conjoyned, then if a Circular hole bee made, whose Diamiter is equall to the diamiter of the Cones conjoyned, and a Quadrangular hole, whose slopeing sides bee equall to the length of each side of the Pyramidie, and the bredth of the hole equall to the diamiter of the Circle, this conjoyned Pyramidie shall exactly fill both the Circular hole, and also Quadrangle hole.

PROBLEME. XXI. How with one uniforme body or such like to fill three severall holes: of which the one is round, the other a just square; and the third an ovall forme.

THis proposition seemes more subtill than the former, yet it may bee practised two wayes: for the first, take a Cilindricall body as great or little as you please: Now it is evident that it will fill a Circular hole, which is made equall to the basis of it: if it bee placed downe right, and will also fill a long square; whose sides are equall unto the Diamiter and length of the Cylinder, and

according to Pergeus, Archimedes, &c. in their Cylindricall de­monstrations, a true Ovall is made when a Cylinder is cut slope­wise, therefore if the Ovall have bredth e­quall unto the Diami­ter of the Basis of the Cylinder, & any length whatsoever: the Cylinder being put into his owne Ovall hole shall also exactly fill it.

The second way is thus, make a Circular hole in some board, and also a square hole, the side of which Square may bee equall to the Diamiter [Page 38] of the Circle: and lastly make a hole Ovall wise whose bredth may be equall unto the diagonall of the Square; then let a Cylindricall body bee made, whose Basis may be equall unto the Cir­cle, and the length equall also to the same: Now being placed downe right shall fall in the Circle, and flat-wise will fit the Square hole, and being placed sloping-wise will fill the O­vall.

PROBLEM XXII. To finde a number thought upon after another manner, than that which is formerly delivered.

BId him that he multiply the number thought upon, by what number he pleaseth, then bid him divide that product by any other number, and then multiply that Quotient by some other number; and that product againe divide by some other, and so as often as he will: and here note that he declare or tell you by what number he did multiply and devide. Now in the same time take a number at pleasure, and secretly multiply and divide as often as he did: then bid him devide the last number by that which hee thought upon. In like manner doe yours pri­vately, then will the Quotient of your devisor be the same with his, a thing which seemes ad­mirable to those which are ignorant of the cause. Now to have the number thought upon without seemeing to know the last Quotient, bid him adde the number thought upon to it, and aske him how much it makes: then subtract your Quotient from it, there will remaine the number thought upon. For example, suppose the number thought upon were 5, multiply it by 4 makes 20: this divided by 2, the Quotient makes 10, which multiplyed by 6 makes 60, and divided by 4 makes 15: in the same time admit you thinke upon 4, which multiplyed by 4 makes 16, this divided by 2 makes 8, which [Page 40] multiplyed by 6 makes 48, and divided by 4 makes 12; then divide 15 by the number thought which was 5, the Quotient is 3; di­vide also 12 by the number you tooke, viz. 4, the Quotient is also 3 as was declared; therefore if the Quotient 3 bee added unto the number thought viz 5, it makes 8, which being known the number thought upon is also knowne.

PROBLEM. XXIII. To finde out many numbers that sundry per­sons, or one man hath thought upon.

IF the multitude of numbers thought upon be odde, as three numbers, five numbers, seaven &c. as for example let 5 numbers thought upon be these, 2, 3, 4, 5, 6. Bid him declare the sum of the first and second, which will be 5; the se­cond and third which makes 7, the third and fourth which makes 9, the fourth and fifth which makes 11, and so alwayes adding the two next together; aske him how much the first and last makes together, which is 8: then take these summes and place them in order, and adde all these together which were in the odde places: that is the first, third, and fifth, viz. 5. 9, 8, makes 22. In like manner adde all these numbers together which are in the even places, that is in the second and fourth places, viz. 7 and 11 makes 18, subtract this from the former 22, then there will remaine the double of the [Page 41] first number thought upon, viz. 4, which knowne the rest is easily knowne: seeing you know the summe of the first and second; but if the multitude of numbers bee even as these sixe numbers, viz. 2, 3, 4, 5, 6, 7, cause the partie to declare the summe of each two, by antece­dent and consequent, and also the summe of the second and last which will bee 5, 7, 9, 11, 13, 10, then adde the odde places together, except the first that is 9, and 13, makes 22; adde also the even places together, that is 7, 11, 10, which makes 28; subtract the one from the o­ther, there shall remaine the double of the se­cond number thought upon, which knowne all the rest are knowne.

PROBLEM. XXIIII. How is it that a man in one and the same time, may have his head upward, and his feet upward, being in one and the same place.

THe answere is very facill, for to bee so he must be supposed to be in the center of the earth: for as the heavens is above on every side, Coelum undique sursum, all that which lookes to the heavens being distant from the center is upward; and it is in this sense that Maurolyeus in his Cosmographie, and first dialogue, reported of one that thought hee was led by one of the Muses to hell, where hee saw Lucifer sitting [Page 42] in the middle of the world, and in the center of the earth as in a Throne: having his head and feete upward.

PROBLEM. XXV. Of a Ladder by which two men ascending at one time; the more they ascend the more they shall be asunder, notwithstanding one being as high as a­nother.

THis is most evident, that if there were a Ladder halfe on this side of the Center of the earth, and the other halfe on the other side: and that two at the Center of the world at one instant, being to ascend the one towards us, and the other towards our Antipodes, they should in ascending goe farther and farther, one from another; notwithstanding both of them being of like height.

PROBLEM. XXVI. How it is that a man having but a Rode or Pole of land; doth bragge that he may in a right line passe from place to place above 3000 miles.

THe opening of this is easie, forasmuch as he that possesseth a Rode of ground posses­seth [Page 43] not only the exterior surface of the earth, but is master also of that which extends even to the center of the earth, and in this wise all herita­ges and possessions are as so many Pyramides, whose summets or points meete in the center of the earth, and the basis of them are nothing else but each mans possession, field, or visible quan­titie; and therefore if there were made or ima­gined so to be made, a descent to goe to the bot­tome of the heritage, which would reach to the center of the earth; it would bee above 3000 miles in a right line as before.

PROBLEM. XXVII. How it is that a man standing upright, and looking which way he will, he loo­keth true North or South.

THis happeneth that if the partie be under either of the Poles, for if he be under the North pole, then looking any way hee looketh South, because all the Meridians concurre in the Poles of the world, and if he be under the South pole, hee lookes directly North by the same reason.

PROBLEM. XXVIII. To tell any one what number remaines after certaine operations being ended, without asking any question.

BId him to thinke upon a number, and will him to multiply it by what number you thinke convenient: and to the product bid him adde what number you please, provided that se­cretly you consider, that it may be divided by that which multiplied, and then let him divide the sum by the number which he first multiplied by, and subtract from this Quotient the number thought upon: In the same time divide apart the number which was added by that which multiplied, so then your Quotient shall bee e­quall to his remainder, wherefore without ask­ing him any thing, you shall tell him what did remaine, which will seeme strange to him that knoweth not the cause: for example, suppose he thought 7, which multiplied by 5 makes 35, to which adde 10, makes 45, which divided by 5 yeeldes 9; from which if you take away one the number thought, (because the Multiplier divided by the divisor gives the Quotient 1,) the rest will be 2, which will be also proved, if 10 the number which was added, were divided by 5, viz. 2.

PROBLEM. XXIX. Of the play with two severall things.

IT is a pleasure to see and consider how the science of numbers doth furnish us, not one­ [...]y with sports, to recreate the spirits, but al­ [...]o bring us to the knowledge of admirable [...]hings, as shall in some measure bee shewen in this ensuing progression. In the meane time to produce alwaies some of them: suppose that a man hold divers things in his hand, as Gould and Silver, and in one hand he held the Gould, and in the other hand hee held the Silver: to know subtilly, and by way of divination, or artificially in which hand the Gold or Silver si; attribute to the gould, or suppose it have a cer­taine prise, & so likewise attribute to the Silver another price, conditionally that the one be od and the other even: as for example, bid him that the Gould be valued at 4. Crownes, or shil­lings, and the Silver at 3. Crownes or 3. Shillings or any other number so that one be odde, and the other even as before: then bid him triple that which is in the right hand, and double that which is in the left hand, and bid him adde these two products together, and aske him if it be even or odde, if it be even then the Gould is in the right hand, if odde the Gould is in the left hand.

PROBLEM. XXX. Two numbers being proposed unto two seve­rall parties, to tell which of these num­bers is taken by each of them.

AS for example: admit you had proposed unto two men whose names were Peter, and Iohn, two numbers, or peeces of money, the one even, and the other odde, as 10. and 9. and let the one of them take one of the numbers, and the other partie take the other number, which they place privatly to themselves: how artificially, according to the congruitie, and ex­cellency of numbers, to finde which of them did take 10. and which 9. without asking any question: and this seemes most subtill, yet deli­vered howsoever differing little from the for­mer, and is thus performed: Take privately to your selfe also two numbers, the one even and the other odde, as 4. and 3. then bid Peter that he double the number which he tooke, and doe you privately double also your greatest num­ber; then bid Iohn to triple the number which he hath, and doe you the like upon your last number: adde your two products together, and marke if it be even or odde, then bid the two parties put their numbers together, and bid them take the halfe of it, which if they can­not doe, then immediatly tell Peter hee tooke 10. and Iohn 9. because the aggregate of the double of 4. and the triple of 3. makes odde; [Page 47] and such would be the aggregate or summe of [...]he double of Peters number and Iohns number, [...]f Peter had taken 10. if otherwise, then they might have taken halfe, and so Iohn should have taken 10. and Peter 9. as suppose Peter had taken 10. the double is 20, and the triple of [...]9. the other number is 27. which put together makes 47. odde: in like manner the double of your number conceived in minde, viz. 4. makes 8. and the triple of the 3. the other number, makes 9. which set together makes 17. odde: Now you cannot take the halfe of 17. nor 47. which argueth that Peter had the greater num­ber, for otherwise the double of 9. is 18. and the triple of 10. is 30. which set together makes 48. the halfe of it may be taken: therefore in such case Peter tooke the lesse number: and Iohn the greater, and this being done cleanly carries much grace with it.

PROBLEM. XXXI. How to describe a Circle that shall touch 3. Points placed howsoever upon a plaine, if they be not in a right line.

LEt the three points bee A. B. C. put one foot of the Compasse upon A. and describe an Arch of a Circle at pleasure: and placed at B. crosse that Arke in the two points E. and F. and placed in C. crosse the Arke in G. and H. then lay a ruler upon G.H. and draw a line, and [Page 48] place a Ruler upon E.

and F. cut the other line in K. so K. is the Center of the Cir­cumference of a Cir­cle, which will passe by the said three points A. B. C. or it may bee inverted ha­ving a Circle drawne, to finde the Center of that Circle. Make 3. points in the circumference, and then use the same way: so shall you have the Center a thing most facill, to every practitioner in the pinciples of Geometrie.

PROBLEM. XXXII. How to change a Circle into a square forme.

MAke a Circle upon pastboard or other materiall, as the Circle A. C. D. E. of which A. is the Center; then cut it into 4. quar­ters and dispose them so, that A. at the center of the Circle may alwaies be at the Angle of the square, and so the foure quarters of the [Page 49] Circle being placed

so, it will make a per­fect square, whose side A.A. is equall to the diamiter B. D. Now here is to bee noted that the square is greater than the Circle by the vacuity in the middle, viz. M.

PROBLEM. XXXIII. With one and the same compasses, and at one and the same extent, or opening, how to describe many Circles concentricall, that is, greater or lesser one than another.

IT is not without cause that many admire how this proposition is to bee resolved; yea in the judgement of some it is thought impossi­ble: who consider not the industrie of an inge­nious Geometritian, who makes it possible, and that most facill, sundry wayes; for in the first place if you make a Circle upon a fine plaine, and upon the Center of that Circle, a small pegge of wood be placed, to bee raised up and put downe at pleasure by helpe of a small hole made in the Center, then with the same ope­ning of the Compasses, you may describe Cir­cles [Page 50] Concentricall:

that is, one greater or lesser than another: for the higher the Center is lifted up, the lesser the Circle will be. Secondly, the com­passe being at that ex­tent upon a Gibus bo­dy, a Circle may bee described, which will be lesse than the for­mer, upon a plaine, and more artificially upon a Globe, or round bowle: and this againe is most obvious upon a round Pyramide, placing the Compasses upon the top of it which will be farre lesse than any of the former; and this is demon­strated by the 20. Pro. of the first of Euclids, for the Diamiter E.D. is lesse than the line A D. A E. taken together, and the lines A D. A E. being equall to the Diamiter B C. because of the same distance or extent of opening the Compasses, it followes that the Diamiter E D. and all his Circles together is much lesse than the Diamiter, and the Circle B C. which was to be performed.

PROBLEM. XXXIIII. Any numbers under 10. being thought upon, to finde what numbers they were.

LEt the first number be doubled, and unto it adde 5. and multiplyed that summe by 5. and unto it adde 10. and the next number thought upon; multiplye this same againe by 10. and adde unto it the next number, and so pro­ceede: now if he declare the last summe; marke if he thought but upon one figure, for then sub­tract onely 35. from it, and the first figure in the place of tens is the number thought upon: if he thought upon two figures, then subtract 35. also, and the 2. also the said 35. from his last summe, and the two figures which remaines are the number thought upon: if he thought upon three figures, then subtract 350. and then the first 3. figures are the numbers thought upon, &c. so if one thought upon these numbers 5.7. 9, 6. double the first, makes 10. to which adde 5. makes 15. this multiplyed by 5. makes 75. to which adde 10. makes 85. to this adde the next number, viz. 7. makes 92. this multiplyed by 10. makes 920. to which adde the next number, viz. 9. makes 929. which multiplyed by 10. makes 9290. to which adde 6. makes 9296. from which subtract 3500. resteth 5796. the foure numbers thought upon. Now because the two last figures are like the two numbers thought [Page 52] upon: to conceale this bid him take the halfe of it, or put first 12. or any other number to it, and then it will not be so open.

PROBLEM. XXXV. Of the Play with the Ring.

AMongst a company of 9. or 10. persons, one of them having a Ring, or such like: to finde out in which hand: upon which finger, joynt it is; this will cause great astonishment to ignorant spirits, which will make them be­leeve that he that doth it workes by magicke, or witchcraft: But in effect it is nothing else but an nimble act of Arithmeticke, founded upon the precedent Probleme: for first it is sup­posed that the persons stand or sit in order that one is first, the next second, &c. likewise there must be imagined that of these two hands the one is first and the other second: and also of the five fingers the one is first, the next is se­cond, and lastly of the joynts, the one is as 1. the other as 1. the other is as 2. the other as 3. &c. from whence it appeares that in perfor­ming this Play there is nothing else to be done than to thinke 4. numbers: for example if the forth person had the Ring in his left hand: and upon the fift finger and third joynt, and I would divine and finde it out thus: I would proceede as in the 35. Probleme: in causing him to dou­ble the first number: that is, the number of per­sons, [Page 53] which was 4. and it makes 8. to which ad 5. makes 13. this multiplyed by 5. makes 65. put 10. to it, makes 75. unto this put 2. for the num­ber belonging to the left hand, and so it makes 77. which multiplyed by 10. makes 770. to this adde the number of the fingers upon which the Ring is, viz. 5. makes 775. this multiplyed by 10 makes 7750. to which adde the number for the joynt upon which the Ring is viz. the third joynt, makes 7753. to which cause him to adde 14. or some other number, to conceale it the better: and it makes 7767. which being decla­red unto you, subtract 3514. and there will re­maine 4. 2. 5. 3. which figures in order declares the whole mystery of that which is to bee knowne, 4. signifieth forth person, 2. the left hand, 5. the fifth finger, and 3. the third joynt of that finger.

PROBLEM. XXXVI. The Play of 3 4. or more Dice.

THat which is said of the two precedent Problemes may be applyed to this of Dice (and many other particular things) to finde what number appeare upon each Dice being cast be some one, for the points that are upon any side of a Dice are alwayes lesse than 10. and the points of each side of a Dice may be taken for a number thought upon: therefore the Rule will be as the former: As for example, [Page 54] one having throwne three Dice & you would declare the numbers of each one, or how much they make together, bid him double the points of one of the Dice, to which bid him adde 5. then multiply that by 5. and to it adde 10. and to the summe bid him adde the number of the second Dice: and multiplie that by 10. lastly, to this bid him adde the number of the last Dice, and then let him declare the whole number: then if from it you subtract 350. there will remaine the number of the three Dice throwne.

PROBLEM. XXXVII. How to make water in a Glasse seeme to boyle and sparkle.

TAke a Glasse neare full of water or other liquor; and setting one hand upon the foote of it, to hould it fast: turne slight­ly one of the fingers of your other hand upon the brimme, or edge of the Glasse; having be­fore privatly wet your finger: and so passing softly on with your finger in pressing a little: for then first the Glasse will begin to make a noyse: secondly the parts of the Glasse will sen­cibly appeare to tremble, with notable rari­fication and condensation: thirdly the water will shake, seeme to boyle: fourthly it will cast it selfe out of the Glasse, and leape out by small drops, with great astonishment to the standers by; if they be ignorant of the cause of it, which [Page 55] is onely in the Rarifaction of the parts of the Glasse: occasioned by the motion and pressure of the finger.

PROBLEME XXXVIII. Of a fine vessell which holds wine or water, being cast into it at a certaine height, but being filled higher, it will runne out of its owne accord.

LEt there be a vessell A. B. C. D. in the mid­dle of which place a Pipe; whose ends both above at E, and below at the bottome of the vessell as at F. are open; let the end E. be some­what lower than the brimme of the Glasse: a­bout this Pipe place another Pipe as H. L, which mounts a little above E, and let it most diligently be closed at H. that no Aire enter in thereby, and this Pipe at the bottome may have a small hole to give passage unto the water: then powre in water or wine, and as long as it mounts not above E, it is safe; but if you powre in the water so that it mount above it, fare­well all: for it will not cease untill it be all gone [Page 57] out: the same may be

done in disposing any crooked Pipe in a ves­sell in the manner of a faucet or funnell, as in the figure H: for fill it, under H. at pleasure, and all will goe well; but if you fill it unto H. you will see fine sport, for then all the vessell will bee empty incontinent, and the subtiltie of this will seeme more admirable, if you conceale the Pipe by a Bird, Serpent, or such like, in the middle of the Glasse. Now the reason of this is not difficult to these which know the nature of a Cocke or Faucet; for it is a bowed Pipe one end of which is put into the water or liquor, and sucking at the other end untill the Pipe be full, then will it runne of it selfe; and it is a fine secret in nature to see, that if the end of the Pipe which is out of the water be lower than the water, it will runne out with­out ceasing: but if the mouth of the Pipe bee higher than the water or levell with it, it will not runne, although the Pipe which is without be many times bigger than that which is in the water: for it is the property of water to keepe alwayes exactly levell.

PROBLEM. XXXIX. Of a Glasse very pleasant.

SOmetimes there are Glasses which are made of a double fashion, as if one Glasse were within another, so that they seeme but one, but there is a little space betweene them. Now powre wine or other liquor betweene the two [Page 59] edges by helpe of a Tunnell, into a little hole left to this end: so will there appeare two fine delusions or fallacies; for though there be not a droppe of wine within the hollow of the Glasse, it will seeme to these which behold it that it is an ordinary Glasse full of wine, and that especi­ally to these which are sidewise of it; and if any one moove it, it will much confirme it, because of the motion of the wine: but that which will give most delight, is that if any one shall take the Glasse, and putting it to his mouth shall thinke to drinke the wine; instead of which hee shall suppe the aire: and so will cause laughter to these that stand by: who being deceived, will hold the Glasse to the light; and thereby consi­dering that the rayes or beames of the light are not reflected to the eye, as they would bee if there were a liquid substance in the Glasse: hence they have an assured proofe to conclude, that the hollow of the Glasse is totally empty.

PROBLEM. XL. If any one should hold in each hand, as many peeces of money as in the other, how to finde how much there is.

BId him that holdes the money that hee put out of one hand into the other what num­ber you thinke convenient: (provided that it may be done,) this done, bid him that out of the hand that he put the other number into, that he [Page 60] take out of it as many as remaine in the other hand, and put it into that hand: for then be as­sured that in the hand which was put the first taking away: there will be found just the dou­ble of the number taken away at the first. Ex­ample, admit there were in each hand 12 Shil­lings or Counters, and that out of the right hand you bid him take 7 and put it into the left: and then put into the right hand from the left as many as doth remaine in the right, which is 5: so there will bee in the left hand 14, which is the double of the number taken out of the right hand, to wit 7: then by some of the rules be­fore delivered, it is easie to finde how much is in the right hand, viz. 10.

PROBLEM. XLI. Many Dice being cast, how artificially to discover the number of the points that may arise.

SVppose any one had cast three Dice secret­ly, bid him that he adde the points that were upmost together: then putting one of the Dice apart, unto the former summe adde the points which are under the other two, then bid him throw these two Dice, and marke how many points a paire are upwards, which adde unto the former summe: then put one of these Dice a­way not changing the side, marke the points [Page 61] which are under the other Dice, and adde it to the former summe: lastly throw that one Dice, and whatsoever appeares upward adde it unto the former summe; and let the Dice remaine thus: this done, comming to the Table, note what points doth appeare upward upon the three Dice which adde privately together, and unto it adde 21 or 3 times 7: so this Addition or summe shall be equall to the summe which the party privately made of all the operations which hee formerly made. As if hee should throw three Dice, & there should appeare up­ward 5, 3, 2. the sum of them is 10: & setting one of them aparte as 5. unto 10, adde the points which are under 3 and 2, which is 4 and 5; and it makes 19: then casting these two Dice sup­pose there should appeare 4 and 1, this added unto 9 makes 24: and setting one of these two Dice aparte as the 4. unto the former 24, I adde the number of points which is under the other Dice, viz. under 1, that is 6, which makes 30. Last of all I throw that one Dice, and suppose there did appeare 2, which I adde to the former 30, and it makes 32: then leaving the 3 Dice thus, the points which are upward will be these, 5, 4, 2, unto which adde secretly 21, (as before was said) so have you 32 the same number which he had: and in the same manner you may pra­ctice with 4, 5, 6, or many Dice or other bodies, observing onely that you must adde the points opposite of the Dice: for upon which depends the whole demonstration or secret of the play; for alwayes that which is above and under­neath [Page 62] makes 7: but if it make another number, then must you adde as often that number.

PROBLEM. XLII. Two mettals as Gold and Silver, or of other kind weighing alike, being privately placed into two like Boxes, to finde which of them the Gold or Silver is in.

IT is said that an Emperor was requested by one of his servants after he had long time re­mained with him, to assigne him some re­ward: to which after few dayes the Emperour condescended, and caused him to come into his Treasury, where he had prepared two Boxes, one full of Gold, and the other full of Lead, both weighing, and of forme and magnitude alike: and bid him chuse which he would have. Now many thinke that in this Probleme one must be guided onely by fortune in this choise, and it is that which most makes a man happy in such a choise: but the want of knowledge causeth them so to judge which knoweth not other­wise. A Mathematician accounts it an easie proposition and will infallibly chuse the chest of gold, and leave the chest of lead, without either breaking, or opening any of the chests, and not goe by chance and fortune: for if he may bee permitted to weigh those chests first in the aire, then in the water: it is a thing cleare [Page 63] by the proportion of

mettalls, and accor­ding to the princi­ples of Archimedes, that the Gold shall be lesse weighty by his eighteenth part, and the lead by his eleventh part; where­fore there may bee gathered in which is the Gold, and in which is the lead.

But because that this experiment in water hath diverse accedents, and therefore subject to a caution; and namely because the matter of the chest, mettle, or other things may hinder:

Behold here a more subtill and certaine in­vention to finde and discover it out without weighing it in the water: Now experience and reason sheweth us that two like bodies or mag­nitudes of equall weight, and of divers mettalls, are not of equall quantity: & seeing that gold is the heaviest of all mettalls, it will occupie lesse roome or place; from which will follow that the like weight of lead in the same forme, will occupie or take up more roome or place. Now let there be therfore presented 2 globes or chests of wood or other matter alike, and equall one to the other, in one of which in the middle there is another Globe or body of lead weighing 12 pound (as C,) and in the other a Globe or like body of gold weighing 12 pound (as B.) Now [Page 64] it is supposed that the wooden globes or chests are of equall weight, forme, and magnitude [...] and to discover in which the gold or lead is in, take a broade paire of Compasses and clip one of the coffers or globes somewhat from the mid­dle as at D; then fixe in the chest or globe a small peece of iron between the feet of the compasses, as E K, at the end of which hang a weight G so that the other end may be counterpoysed, and hang in aequilibro: & doe the like to the other chest or globe. Now if that the other chest or globe being clipsed in like distant from the end, and hang­ing at the other end the same weight G. there be found no difference: then clipse them nearer towards the middle, that so the points of the Compasse may bee against some of the mettell which is inclosed: or just against the extremi­tie of the gold as in D, and suppose it hang thus in aequilibrio; it is certaine that in the other coffer is the lead; for the points of the Compasses being advanced as much as before, as at F. which takes up a part of the lead, (because it occupies a greater place than the gold) therefore that shall helpe the weight G to weigh, and so will not hang in aequilibrio except G be placed neare to F: hence we may conclude that there is the lead; and in the other chest or globe there is the gold.

PROBLEM: XLIII. Two Globes of diverse mettles, (as one gold and the other copper) yet of equall weight being put into a boxe as B. G. to finde in which end the gold or copper is.

THis is discovered by the changing of the places of the two Bowles or Globes having the same counterpoyse H. to bee houng at the other side as in N. and if the Gould which is the lesser Globe were before the nearest to the handle D.E. having now changed his place will bee farthest from the handle D. E. as in K. [Page 66] therefore the Center of gravitie of the two Globes taken together, shall bee farther separate from the midle of the

handle (under which is the Center of gra­vitie of the Box) than it was before, and see­ing that the handle is alwayes in the midle of the box, the waight N. must bee aug­mented, to keepe it in aequilibria: and by this way one may know, that if at the second time, the counter­poise bee too light, it is signe that the Gould is farthest off the handle, as at the first triall it was nearest.

PROBLEM. XLIIII. How to represent diverse sorts of Raine­bowes here below.

THe Rainebow is a thing admirable in the world, which ravisheth often the eyes and spirits of men in consideration of his rich inter­mingled colours which are seene under the cloudes, seeming as the glistering of the starres, pretious stones, & ornaments of the most beau­tious flowers: some part of it as the resplendant stars, or as a rose, or burning cole of fire: in it one [Page 67] may see dies of sundry sorts, the violet, the blew, the orion, the saphir, the jacint, and the emeraud colours, as a lively plant placed in a greene soyle: and as a most rich treasure of nature, it is a high worke of the Sunne who casteth his rayes or beames as a curious Painter drawes strokes with his pensell, and placeth his co­lours in an exquisite situation; and Salomon saith, Eccles. 43. it is a chiefe and principall worke of God. Notwithstanding there is left to industrie how to represent it from above, here below, though not in perfection yet in part, with the same intermixture of colours that is above.

Have you not seene how by Oares of a Boate it doth exceeding quickly glide upon the water with a pleasant grace? Aristotle sayes that it coloureth the water and makes a thousand a­tomes, upon which the beames of the Sunne reflecting makes a kinde of coloured Rainebow: or may we not see in houses or gardens of plea­sure artificiall fountaines, which powre forth their droppie streames of water, that being be­tweene the Sunne and the fountaine, there will be presented as a continuall Rainebow? But not to goe farther, I will shew you how you may doe it at your doore, by a fine and facill experi­ment.

Take water in your mouth, and turne your backe to the Sunne, and your face against some obscure place, then blow out the water which is in your mouth, that it may bee sprinkled in small drops and vapours: you shall see these [Page 68] atomes vapours in the beames of the Sunne to turne into a faire Rabinebow, but all the griefe is that it lasteth not but soone is vanished.

But to have one more stable and permanent in his colours, take a Glasse full of water and ex­pose it to the Sunne, so that the rayes that passe through strike upon a shadowed place, you will have pleasure to see the fine forme of a Raine­bow by this reflexion. Or take Trigonall Glasse or Cristall Glasse of diverse Angles, and looke through it, or let the beames of the Sunne passe through it, or with a candle let the appearan­ces be received upon a shadowed place: you will have the same contentment.

PROBLEM. XLV. How that if all the Powder in the world were in­closed within a bowle of paper or glasse, and being fired on all parts, it could not breake that bowle.

IF the bowle and the powder be uniforme in all his parts, thē by that means the powder would presse and move equally on each side, in which there is no possibilitie whereby it ought to be­gin by one side more than another. Now it is impossible that the bowle should bee broken in all his parts: for they are infinite.

Of like finenes or subtiltie may it be that a bowle of iron from a high place upon a plaine pavement of thin Glasse, it were impos­sible [Page 69] any wise to breake it; if the bowle were perfectly round, and the Glasse flat and uni­forme in all his parts: for the bowle would touch the Glasse but in one point, which is in the middle of infinite of parts which is about it: neither is there any cause why it ought more on one side than on another, seeing that it may not be done with all his sides together; it may bee concluded as speaking naturally, that such a bowle falling upon such a glasse will not break it. But this matter is meere Metaphysicall, and all the workemen in the world cannot ever with all their industrie make a bowle perfectly round, or a Glasse uniforme.

PROBLEM. XLVI. To finde a number which being divided by 2 there will remaine 1, being divided by 3, there will remaine 1; and so likewise being divided by 4, 5, or 6. there would still remaine 1: but be­ing divided by 7, there will remaine nothing.

IN many Authors of Arithmeticke this Pro­bleme is thus proposed: A woman carrying egges to market in a basket, mett an unruly fel­low who broake them; who was by order made to pay for them: and she being demanded what number she had, she could not tell: but she re­membred [Page 70] that counting them by 2, & 2; there remained 1: likewise by 3 and 3, by 4 and 4, by 5 and 5, by 6 and 6; there still remained 1: but when she counted them by 7 and 7, there remained nothing: Now how may the number of egges be discovered?

Finde a number which may exactly be mea­sured by 7, and being measured by 2, 3, 4, 5, and 6; there will still remaine a unity: multiply these numbers together, makes 702, to which adde 1; so have you the number, viz. 721: in like manner 301 will be measured by 2, 3, 4, 5, 6; so that 1 remaines: but being measured by 7, nothing will remaine; to which continually adde 220, and you have other numbers which will doe the same: hence it is doubtfull what number shee had, therefore not to faile it must be knowne whether they did exceed 400, 800, &c. in which it may bee conjectured that it could not exceed 4 or 5 hundred, seeing a man or woman could not carry 7 or 8 hundred egges; therefore the number was the former 301. which shee had in her basket: which being counted by 2 and 2, there will remaine 1, by 3 and 3, &c. but counted by 7 and 7, there will remaine [...].

PROBLEME XLVII. One had a certaine number of crownes, and coun­ting them by 2 and 2, there rested 1: counting them by 3 and 3, there rested 2: counting them by 4 and 4, there rested 3: counting them by 5 and 5, there rested 4: counting them by 6 and 6, there rested 5: but counting them by 7 and 7, there remained nothing: how many crownes might hee have.

THis Question hath some affinitie to the precedent, and the resolution is almost in the same manner: for here there must be found a number, which multiplied by 7, and then divi­ded by 2, 3, 4, 5, 6; there may alwayes remaine a number lesse by 1 than the divisor: Now the first number which arives in this nature is 119, unto which if 420 be added, makes 539, which also will doe the same: and so by adding 420, you may have other numbers to resolve this proposition.

PROBLEM. XLVIII. How many sorts of weights in the least manner must there be to weigh all sorts of things be­tweene 1 pound and 10 pound, and so unto 121, and 364 pound.

TO weigh things betweene 1 and 40: take numbers in triple proportion, so that their [Page 72] summe be equall, or somewhat greater than 40, as are the numbers 1. 3. 9. 27. I say that with 4 such weights, the first being of 1 pound, the second being 3 pound, the third being 9 pound, and the fourth being 27: any weight betweene 1 and 40 pound may bee weighed. As admit to weigh 21 pound, put unto the thing that is to be weighed the 9 pound weight, then in the o­ther ballance put 27 pound and 3 pound which doth counterpoise 21 pound and 9 pound: and if 20 pound were to be weighed, put to it in the ballance 9 and 1, and in the other ballance put 27 and 3, and so of others.

In the same manner take those 5 weights, 1, 3, 9, 27, 81, you may weigh with them be­tweene 1 pound, and 121 pound: and taking those 6 weights, as 1, 3, 9, 27, 81, 243, you may weigh even from 1 pound unto 364 pound: this depends upon the property of continued proportionalls, the latter of which containing twice all the former.

PROBLEM. XLIX. Of a deceitfull ballance which being empty seemes to be just, because it hangs in aequilibrio: not­withstanding putting 12 pound in one ballance, and 11 in the other, it will remaine in aequilibrio.

ARistotle maketh mention of this ballance in his mechanicke Questions, and saith that [Page 73] the Merchants of purpose in his time used them to deceive the world: the subtiltie or craft of which is thus, that one arme of the ballance is longer than another, by the same proportion, that one weight is heavier than a­nother: As if the beame were 23 inches long, and the handle placed so that 12 inches should be on one side of it,

and 11 inches on the other side: conditio­nally that the shorter end should be as hea­vy as the longer, a thing easie to bee done: then after­wards put into the ballance two unequall weights in such pro­portion as the parts of the beame have one unto another, which is 12 to 11; but so that the greater be placed in the ballance which hangs upon the shorter part of the beame, and the lesser weight in the other ballance: it is most certaine that the ballances will hang in aequili­brio, which will seeme most sincere and just; though it bee most deceitfull, abominable, and false.

The reason of this is drawne from the expe­riments of Archimedes, who shewes that two unequall weights will counterpoyse one ano­ther, when there is like proportion betweene the parts of the beame (that the handle sepa­rates) [Page 74] and the weights themselves: for in one and the same counterpoise, by how much it is farther from the Center of the handle, by so much it seemes heavier; therefore if there be a diversitie of distance that the ballances hang from the handle, there must necessarily bee an inequallity of weight in these ballances to make them hang in aequilibrio, and to discover if there be deceite, change the weight into the other ballance, for as soone as the greater weight is placed in the ballance that hangs on the longer parts of the beame: it will weigh downe the other instantly.

PROBLEM. L. To heave or lift up a bottle with a straw.

TAke a straw that is not bruised, bow it that it make an Angle, and put it into the bottle: so that the greatest

end bee in the necke; then the Reede being put in the bowed part will cast sidewise, and make an Angle as in the figure may bee seene; then may you take the end which is out of the bottle in your hand, and heave up the bottle: and it is [Page 75] so much surer, by how much the Angle is acuter or sharper; and the end which is bowed ap­proacheth to the other perpendicular parts which comes out of the bottle.

PROBLEM. LI. How in the middle of a wood or desert, without the sight of the Sunne, Starres, Shaddow or Compasse, to finde out the North or South, or the foure Cardinall points of the world, East, West, &c.

IT is the opinion of some, that the windes are to be observed in this: if it be hot, the South is found by the windes that blow that way, but this observation is uncertaine and subject to much error: nature will helpe you in some mea­sure to make it more manifest than any of the former from a tree, thus: cut a small tree off even to the ground, and marke the many circles that is about the sap or pith of the tree, which seeme nearer together in some part than in o­ther, which is by reason of the sunnes motion a­bout the tree: for that the humiditie of the parts of the tree towards the South by the heat of the Sunne is rarified and caused to extend: and the Sunne not giving such heat towards the North part of the tree, the sap is lesser rarified but condensed; by which the circles are nearer together on the North part, than on the South part: therefore if a line bee drawne from the [Page 76] widest to the narrow­est

part of the circles, it shall shew the North and South of the world: Another experiment may bee thus, take a small nee­dle such as women worke with: place it gently downe flatwise upon still water and it will not sinke, (which is against the generall tenet that iron will not swimme) which needle will by little and little turne to the North and South points. But if the needle bee great and will not swimme, thrust it through a small peece of corke or some such like thing, and then it will doe the same: for such is the propertie of iron when it is placed in aequlibrio, it strives to finde out the Poles of the word, or points of North and South: in a manner as the magnes doth.

PROBLEM. LII. Three persons having taken Counters, Cards, or other things, to finde how much each one hath taken.

CAuse the third party to take a number which may be divided by 4, & as often as he takes 4, let the second party take 7, and the [Page 78] first take 13, then cause them to put them all together and declare the summe of it: which secretly divide by 3, and the Quotient is the double of the number which the third person did take. Or cause the third to give unto the second and first, as many as each of them hath; then let the second give unto the first and third, as many as each of them hath: lastly let the third give unto the second and first, as many as each of them hath; and then aske how much one of them hath: (for they will have then all alike,) so halfe of that number is the number that the third person had at the first: which knowne all is knowne.

PROBLEME LIII. How to make a consort of musicke of many parts with one voyce, or one instrument onely.

THis Probleme is resolved, so that a singer or player upon an instrument, be neare an Echo which answereth his voice or instrument; and if the Echo answereth but once at a time, he may make a double; if twice, then a triple; if three times, then an harmonie of foure parts; for it must be such a one that is able to exercise both tune and note as occasion requires. As when he begins ut, before the Echo answere, he may be­gin sol, and pronounce it in the same tune that the Echo answereth, by which meanes you have a fift, agreeable consort of musicke: then [Page 79] in the same time that the Echo followeth, to sound the second note sol, hee may sound forth another sol higher or lower to make an eight; the most perfect consort of musicke, and so of others; if he will continue his voyce with the Echo: and sing alone with two parts. Now ex­perience sheweth this to be true, which often comes to passe in many Churches; making one to beleeve that there is many more parts in the musicke of a Quier, than in effect truly there is, because of the resounding and multiplying of the voyce, and redoubling of the Quire.

PROBLEM. LIIII. To make or describe an Ovall forme, or that which neare resembles unto it, at one turning with a paire of a common Compasses.

THere is many fine wayes in Geometricall practices, to make an Ovall figure or one neare unto it, by severall centers: any of which I will not touch upon; but shew how it may be done promptly upon one center only. In which I will say nothing of the Ovall forme which appeares, when one describeth circles with the points of a common compasses, somewhat deepe upon a skinne stretched forth hard: which con­tracting it selfe in some parts of the skinne ma­keth an Ovall forme. But it will more evident­ly appeare upon a columne or cylinder: if paper [Page 80] be placed upon it, then with a paire of Compas­ses describe as it were a circle upon it, which paper afterwards being extended, will not bee circular but ovall-wise: and a paire of Compasses may be so accommodated that it may be done al­so upon a plain thus. As let the length of the O­vall be H. K, fasten 2 pinnes or nayles neare the end of that line as

F. G, and take a threed which is double to the length of G. H, or F. K: then if you take a Compasse which may have one foot lower than another, with a spring betweene his legges: & placing one foot of this Compasse in the Center of the Ovall, and guiding the threed by the other foot of the Compasses, and so carrying it about: the spring will helpe to describe and draw the Ovall forme. But in stead of the Compasses it may be done with ones hand onely, as in the figure may appeare.

PROBLEM. LV. Of a purse difficult to be opened.

IT is made to shut and open with ring first at each side there is a strap or string, as A B [Page 81] and CD, at the end of which are 2 rings, B & D, and the string C D passeth through the ring B, so that it may not come out againe; or be parted one from another: and so that the ring B, may slide up and downe upon the string C D, then over the purse there is a peece of leather E. F. G. H. which covers the opening of the purse, and there is another

peece of leather A. E. which passeth through many ringes: which hath a slitte towards the end I. so great that the string B. C. may slide into it: Now all the cun­ning or craft is how to make fast or to o­pen the purse, which consists in making the string B.C. slide through the side at I; therefore bring downe B. to I, then make the end I. passe through the ring B: and also D. with his string to passe through the slit I, so shall the purse be fast; and then may the strings be put as before: and it will seeme diffi­cult to discover how it was done. Now to open the purse, put through the end I. through the ring B, and then through the slit I; by which you put through the string D C: by this way the purse will be opened.

PROBLEM. LVI. Whether it is more hard and admirable without Compasses to make a perfect circle, or being made to finde out the Center of it.

IT is said that upon a time past, two Mathe­maticians meete, and they would make tryall of their industry: the one made instantly a perfect circle without Compasses, and the other immediately pointed out the center thereof with the point of a needle: now which is the chiefest action: it seemes the first: for to draw the most noblest figure upon a plaine Table without other helpe than the hand, and the minde, is full of admiration: to finde the center is but to finde out only one point, but to draw a round, there must be almost infinite points, e­quidistant from the center or middle: that in conclusion it is both the circle and the center together. But contrarily it may seeme that to finde the center is more difficult, for what atten­tion, vivacitie, and subtiltie must there be in the spirit, in the eye, in the hand, which will chuse the true point amongst a thousand other points? He that makes a circle keepes alwayes the same distance, and is guided by a halfe distance to finish the rest; but he that must finde the center, must in the same time take heed to the parts a­bout it, and choose one onely point which is equall distant from an infinite of other points which are in the circumference: which is very [Page 83] difficult. Aristotle confirmes this amongst his moralls, and seemes to explaine the difficultie which is to be found in the middle of vertue; for it may want a thousand wayes and be farre separated from the true center of the end of a right mediocritie of a vertuous action: for to doe well it must touch the middle point which is but one, and there must be a true point which respects the end, and thats but one onely. Now to judge which is the most difficults as before is said, either to draw, the round or to finde the center; the round seemes to be harder than to finde the center, because that in finding of it, it is done at once, and hath an equall distant from the whole: But as before to draw a round there is a visible point imagined, about which the circle is to bee drawne. I esteeme that it is as difficult therefore if not more, to make the cir­cle without a center, as to finde the middle or center of that circle.

PROBLEM. LVI. Any one having taken 3 Cards, to finde how many points they containe.

THis is to bee exercised upon a full packe of Cards of 52, then let one choose any three at pleasure secretly from your sight: and bid him secretly account the points in each Card: and will him to take as many Cards as will make up 15 to each of the points of his Cards; [Page 84] then will him to give you the rest of the Cards, for 4 of them being rejected, the rest shewes the number of points that his three Cards which he tooke at the first did containe. As if the 3 Cards were 7, 10, and 4; now 7 wants of 15, 8: take 8 Cards therefore for your first Card: the 10 wants of 15. 5, take 5 Cards for your second Card: lastly 4 wants of 15, 11, take 11 Cards for your third Card, and giving him the rest of the Cards, there will be 25; from which take 4, there remaines 21: the number of the three Cards taken, viz. 7, 10, and 4.

Whosoever would practise this play with 4, 5, 6, or more Cards, and that the whole num­ber of Cards be more or lesse than 52; and that the terme be 15, 14, 12, &c. this generall rule ensuing may serve: multiply the terme by the number of Cards taken at first: to the product adde the number of Cards taken, then subtract this summe from the whole number of Cards; the remainder is the number which must bee subtracted from the Cards, which remaines to make up the game: if there remaine nothing after the substraction, then the number of cards remaining doth justly shew the number of points which were in the Cards chosen. If the substraction cannot be made, then subtract the number of Cards from that number, and the re­mainder added unto the Cards that did remaine, the summe will be the number of points in the Cards taken, as if the Cards were 7, 10, 5, 8, and the terme given were 12; so the first wants 5, the second wants 2, the third wants 7, and [Page 85] the fourth wants 4 Cards, which taken, the party gives you the rest of the Cards: then se­cretly multiply 12 by 4, makes 48; to which adde 4 the number of Cards taken, makes 52, from which 52 should bee taken, rest nothing: therefore according to the direction of the re­mainder of the Cards which are 30, is equall to the points of the foure Cards taken, viz. 7, 10, 5, 8. Againe let these five Cards bee supposed to be taken, 8, 6, 10, 3, 7; their differences to 15, the termes are 7, 9, 5, 12, 8, which number of Cards taken, there will remaine but 6 Cards: then privatly multiply 15 by 5, makes 75, to which adde 5 makes 80, from this take 52 the number of Cards, rest 28, to which adde the re­mainder of Cards, make 34: the summe with 8, 6, 10, 3, 7.

PROBLEM. LVII. Many Cards placed in diverse rankes, to finde which of these Cards any one hath thought.

TAke 15 Cards and place them in 3 heapes in ranke-wise, 5 in a heape: now suppose any one had thought one of these Cards in any one of the heapes, it is easie to finde which of the Cards it is, and it is done thus: aske him in which of the heapes it is, which place in the middle of the other two: then throw downe the Cards by 1 and 1 into three severall heapes in ranke-wise, untill all be cast downe; then aske [Page 86] him in which of the rankes his Card is: which heape place in the middle of the other two heapes alwayes; and this doe foure times at least, so in putting the Cards altogether, looke upon the Cards, or let their backe bee towards you, and throw out the eight Card: for that was the Card thought upon without faile.

PROBLEM. LVIII. Many Cards being offered to sundry persons, to finde which of those Cards any one thinketh upon.

ADmit there were 4 persons, then take 4 Cards and shew them to the first: bid him think one of them, & put these 4 away; then take 4 other Cards and shew them in like manner to the second person, and bid him thinke any one of these Cards: and so doe to the third person, and so the fourth, &c. Then take the 4 Cards of the first person, and dispose them in 4 rankes: and upon them the 4 Cards of the second per­son, upon them also these of the third person, & lastly upon them these of the fourth person: then shew unto each of these parties each of these rankes, and aske him if his Card be in it which he thought: for infallibly that which the first partie thought upon will bee in the first ranke, and at the bottome; the Card of the se­cond person will bee in the second ranke: the Card of the third thought upon will be in the [Page 87] third ranke, and the fourth mans Card will be in the fourth ranke; and so of others: if there bee more persons use the same methode. This may be practised by other things, ranking them by certaine numbers: allotted to peeces of mo­ney, or such like things.

PROBLEM. LIX. How to make an instrument to helpe hea­ring, as Gallileus made to helpe the sight.

THinke not that the Mathematickes (which hath furnished us with such admirable helpes for seeing) is wanting for that of hearing: its well knowne that long trunkes or pipes makes one heare well farre off, and experi­ence shewes us that in certaine places of the Arcades in a hollow vault, that a man speaking but softly at one corner thereof, may be audi­bly understood at the other end: notwithstan­ding these which are betweene the parties can­not heare him speake at all: And it is a generall principle, that pipes doe greatly helpe to strengthen the activitie of naturall causes: we see that fire contracted in a pipe, burnes 4. or 5. foot high, which would scarce heat, being in the open aire: the rupture or violence of water issu­ing out of a fountaine, shewes us that water be­ing contracted into a pipe, causeth a violence in its passage. The Glasses of Gallileus makes us [Page 88] see how usefull pipes or trunkes are to make the light and species more visible, and propor­tionable to our eye. It is said that a Prince of Italy hath a faire hall, in which he can with fa­cility heare distinctly the discourses of these which walke in the adjacent gardens, which is by certaine vessels and pipes that answere from the garden to the hall. Vitruvius makes mention also of such vessels and pipes, to strengthen the voyce and action of Comedians: and in these times amongst many noble personages, the new kinde of trunkes are used to helpe the hearing, being made of silver, copper, or other resounding materiall; in funnell-wise putting the widest end to him which speaketh, to the end to contract the voyce, that so by the pipe applyed to the eare it may be more uniforme and lesse in dan­ger to dissipate the voyce, and so consequent­ly more fortified.

PROBLEM. LX. Of a fine lampe which goes not out, though one carry it in ones pocket: or being rouled upon the ground will still burne.

IT must be observed that the vessell in which the oyle is put into, have two pinnes on the sides of it one against another, being inclu­ded within a circle: this circle ought to have two other pinnes, to enter into another circle of [Page 89] brasse, or other sollid matter: lastly this second circle hath two pinnes which may hang within some box to containe the whole lampe, in such manner, that there be 6 pinnes in different po­sition: Now by the aide of these pegges or pinnes, the lampe that is in the middle will bee alwayes well scituated according to his Center of gravity, though it

bee turned any way: though if you endea­vour to turne it up­side downe, it will lie levell: which is plea­sant and admirable to behold to these which know not the cause: And it is facill from this to make a place to rest quiet in, though there bee great agitation in the outward parts.

PROBLEM. LXI. Any one having thought a Card amongst many Cards, how artificially to discover it out.

TAke any number of Cards as 10, 12, &c. and open some 4 or 5 to the parties sight, and bid him thinke one of them, but let him note whether it be the first, second, third, &c. then with promptnesse learne what number of Cards [Page 90] you had in your hands, and take the other part of the Cards, & place thē on the top of these you hold in your hand; and having done so, aske him whether his Card were the first, second, &c. then before knowing the number of Cards that were at the bottome, account backwards untill you come to it: so shall you easily take out the Card that he thought upon.

PROBLEM. LXII. Three women A. B. C. carryed apples to a mar­ket to sell, A. had 20, B. 30, and C. 40; they sold as many for a penny, the one as the other: and brought home one as much money as another, how could this be.

THe answere to the Probleme is easie, as suppose at the beginning of the market: A. sold her apples at a

penny an apple: and sold but 2. which was 2 pence, and so shee had 18 left: but B. sold 17. which was 17 pence, and so had 13. left: C. sold 32 which was 32 pence, and so had 8 apples left: then A. said she would not sell her apples so [Page 91] cheape, but would sell them for 3 pence the [...]eece, which shee did: and so her apples came [...]o 54 pence, and B. having left but 13 apples sold them at the same rate which came to 39 [...]ence: and lastly C. had but 8 apples, which at the same rate came to 24 pence: these summes of money which each others before received come to 56 pence, and so much each one recei­ved; and so consequently brought home one as much as another.

PROBLEM. LXIII. Of the properties of some numbers.

FIrst, any two numbers is just the summe of an number that have equall distance from the halfe of that number: the one augmenting, and the other diminishing, as 7 and 7, of 8 and 6, of 9 and 5, of 10 and 4, of 11 and 3, of 12 and [...], of 13 and 1: as the one is more than the halfe the other is lesse.

Secondly, it is difficult to finde two numbers whose summe and product is alike, (that is) if the numbers be multiplyed one by another, and added together, will be equall: which two num­bers are 2 and 2, for to multiply 2 by 2 makes 4, and adding 2 unto 2 makes the same: this property is in no other two whole numbers, [...]ut in broken numbers there are infinite, whose summe and product will bee equall one to ano­ther. As Clavius shewes upon the 36 Pro. of the 9 ^th booke of Euclide.

[Page 92] Thirdly, the numbers 5 and 6 are called cir­cular numbers, because the circle turnes to the point from whence it begins: so these numbers multiplyed by themselves, doe end alwayes in 5 and 6, as 5 times 5 makes 25, that againe by 5 makes 125; so 6 times 6 makes 36, and that by 6 makes 216, &c.

Fourthly, the number 6, is the first which A­rithmeticians call a perfect number, that is, whose parts are equall unto it, so the 6 part of it is 1, the third part is 2, the halfe is 3, which are all his parts: now 1, 2, and 3, is equall to 6. It is wonderfull to conceive that there is so few of them, and how rare these numbers are, so of perfect men: for betwixt 1 & 1000000000000 numbers there is but ten, that is; 6, 28, 486. 8128. 120816. 2096128. 33550336. 536854528.8589869056, & 137438691328: with this admirable property, that alternately they end all in 6 and 8, & the twentieth perfect number is 151115727451553768931328.

Fiftly, the number 9 amongst other privi­ledges carries with it an excellent property, for take what number you will, either in grosse or in part: the nines of the whole or in its parts rejected, and taken simply will be the same, as 27 it makes 3 times 9, so whether the nines bee rejected of 27, or of the summe of 2 and 7, it is all one: so if the nines were taken away of 240. it is all one, if the nines were taken away of 2, 4, and 0; for there would remaine 6 in either: and so of others.

Sixtly, 11 being multiplyed by 2, 4, 5, 6, 7, [Page 93] 8, or 9, will end and begin with like numbers; [...]0 11 multiplyed by 5 makes 55: if multiplyed by 8, it makes 88, &c.

Seventhly, the numbers 220 and 284 being unequall, notwithstanding the parts of the one number doth alwayes equallise the other num­ber: so the alliquot parts of 220 are 110, 54, 44, 22, 20, 11, 10, 5, 4, 2, 1, which together makes 284: the alliquot parts of 284 are 142, 71, 4, 2, 1, which together makes 220, a thing rare and admirable, and difficult to finde in other numbers.

Eightly, the numbers 3, 4, 5, (found out by Pythagoras) have an excellent property in ma­king of rectangle Triangles: upon which the [...]47 Pro: of the first booke of Euclide, was grounded, that the square of the Hypotenusae in any such Triangle, is

equall to the square of the other two sides: that is 5, the Hypotenusae multiply­ed in 5 makes 25, and 4 multiplyed in 4 makes 16, and 3 multiplied in 3, makes 9: but 9 and 16 is e­quall to 25: or if these numbers 3, 4, 5, bee doubled, viz. 6, 8, 10: the square of 10 is equall to the square of 8 and 6, viz. 10 times 10 makes 100, and 8 times 8 makes 64, and 6 times 6 is 36; which [Page 94] 36 and 64, put together makes 100 as before: and so may they be Tripled, Quadrupled, &c.

The use of these numbers 3, 4, 5, are mani­fold, but it may bee applied thus, for the helpe of such which plot out gardens, houses, en­campe horse or foote, &c. Example, take 3 cords: one of 5 yards, another of 4 yards, and another of 3 yards; or the dou­ble,

triple, decuple, &c. or all in one line: and make knots at the tearmes of these mea­sures; so these three parts will make a right angled Triangle, as A. B. C. and it is easie with this Triangular cord to plot out a gar­den plot: a square buil­ding plot, or other long square. As suppose there is a figure E. D. G. F. to bee plotted: E. D. of 60 yardes broad, and D. G. 100 yardes long. First measure out E. D. 60 yardes, and at E. and D. place two pinnes or pegges; then at E. place the angle of your Triangular cord B, and let the line of the Triangles A. B. be in the line E. D: which suppose at A: make the cord A. B. fast in E. and A, then put the other two cords of the Triangle untill they meere, which will be in C, and place a pegge at C: take after­wards a long cord, and by the points E. and C, augment it unto F. 100 yards from E, and at F, place a pegge: then at F, apply your Triangular [Page 95] cord as you did at E, and so may you draw the line F. G. as long as E. D, viz. 60 yards. Lastly it is easie to draw the line G. D, and so the re­ctanguled figure or long square shall be plotted, whose bredth is 60 yards, and length 100 yards as was required: and to examine this, measure E. G, then if F. D. be as long, the figure is true: otherwise it is defective and may easily bee a­mended.

If one bee taken from any square number 9 which is odde, the square of halfe of it being added to the first square, will make a square number.

The square of halfe any even number ✚. 1 10 being added to that even number makes a square number, and the even number taken from it leaves a square number.

If odde numbers bee continually added from 11 the unitie successively, there will bee made all square numbers, and if cubicke numbers bee ad­ded successively from the unitie, there will bee likewise made square numbers.

PROBLEM. LXV. Of an excellent lampe which serves or fur­nisheth it selfe with oyle, and burnes a long time.

I Speake not here of a common lampe which Cardanus writes upon in his book de subtili­tate, for thats a little vessell in collumne-wise, [Page 96] which is full of oyle, and because there is but one little hole at the bottome neare the weeke or match; the oyle runnes not, for feare that there be emptinesse above: when the match is kindled it begins to heat the lampe, and rarify­ing the oyle it issueth by this occasion: and so sends his more airie parts above to avoide va­cuitie.

But that which I here deliver is more ingenious, the princi­pall peece of which is a vessell as C. D: which hath neare the bot­tome a hole, and a fun­nell or pipe C: & then a bigger funnell which passeth through the middle of the ves­sell, having an opening at D. neare the E top, and another at the bot­tome as at E, neare the vessell under it, so that the pipe touch it not: the vessell being thus made, fill it with oyle, and opening the hole C. the oyle running out will stop the hole at E: or throwing in oyle into the vessell underneath, un­till E. bee stopped; then the oyle at C. will not runne: because no aire can come into the pipe D. E. Now as the oyle burneth and consumeth in the vessell A. B. the hole at E, will begin to be open, then immediatly will C. begin to runne to fill up A B: and E being stopped with the oyle, the oyle at C. ceaseth to run.

[Page 97]It is certaine that such a lampe the Athenians [...]ed, which lasted a whole yeare without be­ing touched: which was placed before the sta­tue of Minerva, for they might put a certaine quantitie of oyle in the lampe C. D, and a match to burne without being consumed: such as the naturallists write of, by which the lampe will furnish it selfe and so continue in burning: and here may be noted that the oyle may be powred in at the top of the vessell at a little hole, & then made fast againe that the aire get not in.

PROBLEM. LXV. Of the play at Keyles or nine Pinnes.

YOu will scarce beleeve that which one bowle and at one blow playing freely, one may strike downe all the Keyles at once: yet from Mathematicall principles it is easie to bee de­monstrated, that if the hand of him that playes was so well assured by experience, as reason in­duceth one thereto: one might at one blow strike downe all the Keyles, or at least 7 or 8, or such a number as one pleaseth.

For they are but 9 in all disposed or placed in a perfect square, having three every way. Let us suppose then that a good player begin­ning to play at 1 somewhat low, should so [Page 98] strike it, that it should

strike down the Keyles 2 and 5, and these might in their violence strike down the Keyles 3, 6, and 9: and the bowle being in motion may strike downe the Keyle 4, and 7; which 4 Keyle may strike the Keyle 8: and so all the 9 Keyles may bee stri­ken downe at once.

PROBLEM. LXVI. Of Spectacles of pleasure.

SImple Spectacles of blew, yeallow, red or greene colour, are proper to recreate the sight, and will present the objects died in like colour that the Glasses are, onely those of the greene doe somewhat degenerate; instead of shewing a lively colour it will represent a pale dead colour, and it is because they are not dyed greene enough, or receive not light enough for greene: and colour these images that passe through these Glasses unto the bottome of the eye.

PROBLEME LXVII. Of the Adamant or Magnes, and the needles touched therewith.

VVHo would beleeve if he saw not with his eyes, that a needle of steel being once touched with the magnes, turnes not once, not a yeare, but as long as the world lasteth; his end towards the North and South, yea though one remove it, and turne it from his position, it will come againe to his points of North and South. Who would have ever thought that a brute stone blacke and ill formed, touching a ring of iron, should hang it in the aire: and that ring support a second, that to support a third, and so unto 10, 12, or more, according to the strength of the magnes; making as it were a chaine without a line, without souldering toge­ther, or without any other thing to support them onely; but a most occult and hidden vertue, yet most evident in this effect: which penitra­teth insencibly from the first to the second, from the second to the third, &c.

Is it not a wonder to see that a needle touch­ed once will draw other needles; and so a nayle, the point of a knife, or other peeces of iron. Is it not a pleasure to see how the magnes will turne file dust, or move needles, or nayles being upon a Table, or upon a peece of paper; for as soone as the magnes turnes, or it moves over, it moues also: who is it that would not bee ra­vished [Page 104] as it were, to

see a hand of iron, write upon a planke, without seeing the Magnes which cau­seth that motion be­hinde the planke, or to make an image of iron to runne up and downe a Turrot: now infinite of such inven­tions is proper to be extracted from the properties of the magnes.

What is there in the world that is more capa­ble to cast a deeper astonishment in our mindes, than a great massie substance of iron to hang in the aire in the midest of a building without any thing in the world touching it, only but the aire? And histories assure us, that by the aide of a magnes or adamant, placed at the roofe of one of the Turkish Synagogues in Meca: the sepulcher of that infamous Mahomet rests sus­pended in the aire; and Plinie in his naturall historie writes that the Architecter Dinocrates did begin to vault the Temple of Arsinoe in Alexandria, with store of magnes to produce the like deceit, to hang the sepulcher of that Goddesse likewise in the aire.

I should passe the bounds of my counter­poise, if I should divulge all the secrets of this stone, and should expose my selfe to the laugh­ter of the world: if I should brag to shew other the cause how this appeareth, than in its owne [Page 105] naturall sympathy, for why is it that a magnes with one end will cast the iron away, and attract it with the other; from whence cometh it that all the magnes is not proper to give a true touch to the needle, but onely in the two Poles of the stone: which is knowne by hanging the stone by a threed in the aire untill it be quiet, or placed upon a peece of Corke in a dish of water, or upon some [...]hinne board, for the Pole of the stone will then turne towards the Poles of the world, and point out the North and South; and so shew by which of these ends the needle is to be tou­ched.

From whence comes it that there is a varia­tion in the needle, and pointeth not out truly the North and South of the world, but onely in some place of the earth.

How is it that the needle made with pegges and inclosed within two Glasses, sheweth the height of the Pole, being elevated as many de­grees as the Pole is above the Horizon.

Whats the cause that fire and Garlicke takes away the propertie of the magnes: There are many great hidden misteries in this stone, which have troubled the heads of the most learned in all ages; and to this time the world remaines ig­norant of declaring the true cause thereof.

Some sayes that by helpe of the Magnes persons which are absent may know each others minde, as if one being here at London, and a­nother at Prage in Germany: if each of them had a needle touched with one magnes, then the vertue is such that in the same time that the [Page 106] needle which is at Prage shall moove, this that is at London shall also; provided that the parties have like secret notes or alphabets, and the ob­servation be at a set houre of the day or night; and when the one party will declare unto the other, then let that party move the needle to these letters which will declare the matter to the other, and the mooving of the other par­ties needle shall open his intention.

The invention is subtile, but I doubt whe­ther in the world there can be found so great a stone, or such a Magnes which carries with it such vertue: neither is it expedient, for treasons would be then too frequent and open.

PROBLEM. LXVIII. Of the properties of Aeolipiles or bowles to blow the fire.

THese are concave vessels of brasse or copper, or other materiall which may indure the fire: having a small hole very narrow by which it is filled with water, then placing it to the fire, before it bee hot there is no effect seene; but assoone as the heate doth penitrate it, the water begins to rarifie and issueth forth with a hide­ous and marvellous force; it is pleasure to see how it blowes the fire with great noyse.

Vitruvious in his

first booke of Archi­tecture, Cap. 8. ap­proves from these In­gines, that the winde is no other thing than a quantitie of vapours and exhalations agita­ted with the aire by rarifaction and co [...] densation, and wee may draw a conse­quence from it, to shew that a little water may ingender a very great quantitie of vapours and aire: for a Glasse of water throwne into an Aeolipile will keep blowing neare a hole houre, sending forth his vapoures a thousand times greater than it is extended.

[Page 109]Now touching the forme of these vessels, they are not made of one like fashion: some makes them like a bowle, some like a head pain­ted representing the winde, some makes them like a peare: as though one would put it to rost at the fire, when one would have it to blow, for the tale of it is hollow, in forme of a funnell, having at the toppe a very little hole no greater than the head of a pinne.

Some doe accustome to put within the Aeo­lipile a crooked funnell of many foldings, to the end that the winde that impetuously rowles to and fro within, may imitate the noyse of thunder.

Others content themselves with a simple funnell placed right upward, somewhat wider at the toppe than else where like a Cone, whose basis is the mouth of the funnell: and there may be placed a bowle of iron or brasse, which by the vapoures that are cast out will cause it to leape up, and dance over the mouth of the Aeo­lipile.

Lastly, some apply neare to hole small wind-mills, or such like, which easily turne by reason of the vapours; or by help of two or more bow­ed funnells, a bowle may be made to turne: these Aelopililes are of excellent use for the melting of mettalls and such like.

Now it is cunning and subtiltie to fill one of these Aeolipiles with water at so little a hole, and therefore requires the knowledge of a Phi­losopher to finde it out: and the way is thus.

Heate the Aeolipiles being empty, and the [Page 110] aire which is within it will become extreamely rarified; then being thus hot throw it into wa­ter, and the aire will begin to bee condensed: by which meanes it will occupie lesse roome, therefore the water will immediately enter in at the hole to avoide vacuitie: thus you have some practicall speculation upon the Aeolipile.

PROBLEM. LXIX. Of the Thermometer: or an instrument to measure the degrees of heat and cold in the aire.

THis Instrument is like a Cylindricall pipe of Glasse, which hath a little ball or bowle at the toppe: the small end of which is placed into a vessell of water below, as by the figure may be seene.

Then put some coloured liquor into the Cy­lindricall glasse, as blew, red, yealow, greene, or such like: such as is not thicke. This being done the use may he thus.

First, I say that as the aire inclosed in the Thermometer is rarifyed or condensed, the water will evidently ascend or descend in the Cylinder: which you may try easily by carrying the Themometer from a place that is hot unto a place that is cold, or without removing of it; if you softly apply the palme of the hand upon the balle of the Thermometer: the Glasse being so thinne, and the aire so capable of rarifaction, [Page 111] that at the very instant you may see the water descend: and your hand being taken away, it will softly ascend to his former place againe. This is yet more sencible when one heates the ball at the toppe with

his breath, as if one would say a word in his eare to make the water to descend by command: and the reason of this motion is, that the aire hea­ted in the Thermome­ter, doth rarifie and dilate, requiring a greater place; hence presseth the water and causeth it to descend: contrariwise when the aire cooleth and con­denseth, it occupieth lesse roome; now nature abhorring vacuity, the water naturally ascen­deth. In the second place I say, that by this meanes one may know the degrees of heate and cold; which are in the aire each houre of the day; for asmuch as the exterior aire is either hot or cold, the aire which is inclosed in the Thermometer doth likewise either rarifie or condense, and therefore the water ascends or descends; so you shall see that the water in the morning is mounted high, afterward by little and little it will descend towards noone or mid­day; and towards evening it will againe ascend: so in winter it will mount so high, that all the Cylinder of the Thermometer will bee full, but [Page 112] in Summer, it will descend so low that scarce there will be perceaved in it any water at all.

These that will determine this change by numbers and degrees, may draw a line upon the Cylinder of the Thermometer; and divide it into 4 degrees, according to the ancient Philosophers, or into 4 degrees according to the Physitians, dividing each of these 8 into 8 others to have in all 64 divisions, and by this way they may not onely distinguish upon what degree the water ascendeth in the morning, at midday, and at any other houre: but also one may know how much one day is hotter or colder than another: by marking how many degrees the water ascen­deth or descendeth, one may compare the hot­test and coldest dayes in a whole yeare together with these of another yeare: againe one may know how much hotter one roome is than a­nother, by which also one might keepe a cham­ber, a furnis, a stove, &c. alwayes in an equalitie of heate, by making the water of the Thermo­meter rest alwayes upon one and the same de­gree: in briefe, one may judge in some measure the burning of fevers, and neare unto what ex­tension the aire can bee rarified by the greatest heate.

Many make use of these glasses to judge of the weather, for it is observed that if the water fall in 3 or 4 houres a degree or thereabout, that raine insueth; and the water will stand at that stay, untill the weather change: marke the wa­ter at your going to bed, for if in the morning it hath descended raine followeth, but if it bee [Page 113] mounted higher, it argueth faire-weather: so in very cold weather, if it fall suddainly, it is snow or some sleekey weather that will insue.

PROBLEM. LXX. Of the proportion of humaine bodies of sta­tues, of Colossus or huge images, and of monstrous Giants.

PYthagoras had reason to say that man is the measure of all things.

First, because he is the most perfect amongst all bodily creatures, and according to the Maxi­me of Philosophers, that which is most perfect, and the first in ranke, measureth all the rest.

Secondly, because in effect the ordinary mea­sure of a foote, the intch, the cubit, the pace, have taken their names and greatnesse from humaine bodies.

Thirdly, because the simmetrie and concor­dancie of the parts is so admirable, that all workes which are well proportionable, as namely the building of Temples, of Shippes, of Pillars, and such like peeces of Architecture, are in some measure fashioned and composed after his proportion. And we know that the Arke of Noah built by the commandement of God, was in length 300 cubits, in bredth 50 cubits, in height or depth 30 cubits, so that the length containes the bredth 6 times, and 10 times the depth: now a man being measured [Page 114] you will finde him to have the same propor­tion in length, breadth, and depth.

Vilalpandus treating of the Temple of Salo­mon, that chiefetaine of workes was modula­ted all of good Architecture, and curiously to be observed in many peeces to keepe the same proportion as the body to his parts: so that by the greatnesse of the worke and proportionable symmetrie, some dare assure themselves that by knowledge of one onely part of that building, one might know all the measures of that goodly structure.

Some Architects say that the foundation of houses, and basis of columnes, are as the foote; the tôp, and roofe as the head; the rest as the body: those which have beene somewhat more curi­ous, have noted that as in humaine bodies, the parts are uniforme as the nose, the mouth, &c. these which are double are put on one side or other, with a perfect equallitie in the same Ar­chitecture.

In like manner, some have beene yet more curious than solid; comparing all the orna­ments of a Corinth to the parts of the face, as the brow, the eyes, the nose, the mouth; the rounding of pillars, to the writhing of haire, the chanells of columnes, to the fouldings of womens robes, &c.

Now building being a worke of the best Ar­tist, there is much reason why man ought to make his imitation from the chiefe worke of nature; which is man.

Hence it is that Vitruvious in his third book, [Page 115] and all the best Architectures, treateth of the proportion of man; and amongst others Albert Dureus hath made a whole booke of the mea­sures of mans body, from the foot to the head; let them reade it who will, they may have a perfect knowledge therof: but I will content my selfe, and it may satisfie some with that which followeth.

First, the length of a man well made, which commonly is called height, is equall to the di­stance from one end of his finger to the other: when the armes are extended as wide as they may be.

Secondly, if a man have his feet and hands extended or stretched in forme of S. Andrewes Crosse, placing one foote of a paire of Compasses upon his navill, one may describe a circle which will passe by the ends of his hands and feet, and drawing lines by the termes of the hands and feet, you have a square within a circle.

Thirdly, the breadth of man, or the space which is from one side to another; the breast, the head, and the necke, makes the 6 part of all the body taken in length or height.

Fourthly, the length of the face is equall to the length of the hand, taken from the small of the arme, unto the extremity of the longest finger.

Fiftly, the thickenesse of the body taken from the belly to the backe; the one or the other is the tenth part of the whole body, or as some will have, it the ninth part, little lesse.

Sixtly, the height of the brow, the length of [Page 116] the nose, the space betweene the nose and the chinne, the length of the eares, the greatnesse of the thumbe, are perfectly equall one to the other.

What would you say to make an admirable report of the other parts, if I should reckon them in their least; but in that I desire to be ex­cused, and will rather extract some conclusion upon that which is delivered.

In the first place knowing the proportion of a man, it is easie to Painters, image-makers, &c. perfectly to proportionate their worke; and by the same is made most evident, that which is related of the images and statues of Greece, that upon a day diverse workemen having enterpri­sed to make the face of a man, being severed one from another in sundry places, all the parts be­ing made and put together, the face was found in a most lively and true proportion.

Secondly, it is a thing most cleare that by the helpe of proportion, the body of Hercules was measured by the knowledge of his foote onely; a Lyon by his claw, the Gyant by his thumbe, and a Man by any parte of his body. For so it was that Pythagoras having measured the length of Hercules foote, by the steps which was left upon the ground, found out all his height: and so it was that Phydias having onely the claw of a Lyon, did figure and draw out all the beast ac­cording to his true type or forme; so the exqui­site Painter Timantus, having painted a Pigmey or Dwarfe, which he measured with a fadome made with the intch of a Gyant; it was suffici­ent [Page 117] to know the greatnesse of that Gyant.

To be short, we may by like methode come easily to the knowledge of many fine antiquities touching Statues, Colosses, and monstrous Gy­ants, onely supposing one had found but one on­ly part of them, as the head, the hand, the foote, or some bone mentioned in ancient Histories.

PROBLEM. LXXI. Of the game at the Palme, at Trappe, at Bowles, Paile-maile and others.

THe Mathematickes often findeth place in sundry Games to aide and assist the Game­sters, though not unknowne unto them: hence by Mathematicall principles, the games at Tennis may be assisted; for all the moving in it is by right lines and reflections. From whence comes it, that from the appearances of flat or convex Glasses, the production and reflection of the species are explained, is it not by right lines? in the same proportion one might suffici­ently deliver the motion of a balle or bowle by Geometricall lines and Angles.

But the exercise, experience, and dexteritie of the player seemes more in this action than any any other precepts: notwithstanding I will deliver here some maximes, which being redu­ced to practice, and joyned to experience, will [Page 123] give a great advan­tage

to those which would make use of them in such ga­mings.

And the first maxi­me is thus: When a Bowle toucheth ano­ther Bowle, or when a trapsticke striketh the Balle, the moving of the Balle is made in a right line, which is drawn from the Center of the Bowle by the point of contingencie.

Secondly, in all kinde of such motion; when a Balle or Bowle rebounds, be it either against wood, a wall, upon a Drumme, a pavement, or upon a Racket; the incident Angle is alwayes equall to the Angle of reflection.

Now following these maximes it is easie to conclude, first in what part of the wood or wall, one may make the Bowle or Balle goe to reflect or rebound, to such a place as one would: Secondly, how one may cast a Bowle upon another, in such sort that the first or the second shall goe and meete with the third, kee­ping the reflection or Angle of incidence equall. Thirdly, how one may touch a Bowle to send it to what part one pleaseth: such and many other practices may bee done. At the exercises at Keyles there must be taken heed that the moti­on, slacke or diminisheth by little and little, and [Page 124] may bee noted that the Maximes of reflections cannot be exactly observed by locall motion, as in the beames of light and of other quallities, whereof it is necessary to supply it by industry or by strength, otherwise one may be frustrated in that respect.

PROBLEM. LXXII. Of the Game of square formes.

NVmbers have an admirable secrecie, di­versly applyed, as before in part is shew­ed, and here I will say some thing by way of transmutation of numbers.

It is reported that at a certaine passage of a square forme, there were 4 gates opposite one to another, that is, one in the middle of each side, and that there was appointed 9 men to defend each front thereof, some at the gates, and the other at each corner or Angle, so that each Angle served to assist two faces of the square if neede required: Now this square passage be­ing thus manned to have each side 9, it hapned that 4 Souldiers comming by, desired of the Governour of the passage, that they might bee entertained into service, who told them hee could not admit of more then 9, upon each side of the square: then one of the Souldiers being versed in the Art of numbers, said that if he would take them into pay, they would ea­sily place themselves amongst the rest, and yet [Page 125] keepe still the order

[...]f 9, for each face of [...]e square to defend the Angles and Gates, [...] which the Gover­nours agreed, & these Souldiers being there some few weekes li­ [...]ed not their service, [...]ut indeavoured to re­move themselves, and so laboured with some of the rest; that each of these foure Souldiers tooke away his Cumrade with him, and so de­parted: yet left to defend each side of the pas­sage, and how may this be.

Its answered thus, in the first forme the men were as the figure A, then each of these 4 Souldiers placed themselves at each Gate, and removing one man from each Angle to each Gate, then would they be also 9 in each side ac­cording to the figure B. Lastly, these 4 Soul­diers at the Gates take away each one his Cum­ [...]rade, and placing two of these men which are at each Gate to each Angle, there will bee still 9 for each side of the square, according to the fi­gure C. In like manner if there were 12 men, how might they be placed about a square that the first side shall have 3 every way, then dis­ordered, so that they might be 4 every way; and lastly being transported might make 5 every way, and this is according to the figures, F. G. H.

PROBLEM. LXXIII. How to make the string of a Viole sencibly shake, without any one touching it.

THis is a miracle in musicke, yet easie to bee experimented; take a Viole or other Instru­ment, and choose two strings, so that there bee one betweene them; make these two strings a­gree in one and the same tune: then move the Viole bow upon the greater string, and you shall see a wonder: for in the same time that that shakes which you play upon, the other will likewise sencibly shake without any one tou­ching it; and it is more admirable that the string which is betweene them will not shake at all: and if you put the first string to another tune or note, and loosing the pin of the string, or stop­ping it with your finger in any fret, the other string will not shake: and the same will happen if you take two Violes, and strike upon a string of the one, the string of the other will sencibly shake.

Now it may bee demanded how comes this shaking, is it in the [...]ult sympathie, or is it in the strings being wound up to like notes o [...] tunes, that so ea [...]y the other may receive the impression of the [...], which is agitated or mo­ved by the shaking or the trembling of the o­ther: and whence is it that the Viole bow moved upon the first string, doth instantly in the same time move the third string and not the second; [Page 127] if the cause be not either in the first or second: I leave to others to discant on.

PROBLEM. LXXIIII. Of a vessell which containes three severall kindes of liquor, all put in at one bung-hole, and drawne out at one tappe severally without mixture.

THe vessell is thus made, it must be divided into three sells for to containe the three li­quors, which admit to be Sacke, Clarret, and Whitewine: Now in the bung-hole there is an Ingine with three pipes, each extending to his proper sell, into which there is put a broach or funnell piersed in three places, in such sort that placing one of the holes right against the pipe which answereth unto him, the other two pipes are stopped; then when it is full, turne the fun­nell, and then the former hole will be stopped and another open, to cast in other wine with­out mixing it with the other.

Now to draw out also without mixture, at the bottome of the vessell there must be placed a pipe or broach which may have three pipes, and a cocke piersed with three holes so artifici­ally done, that turning the cocke, the hole which answereth to such of the pipes that is placed at the bottome, may issue forth such wine as be­longeth to that pipe, and turning the Cocke to another pipe, the former hole will bee stopped: [Page 129] and so there will issue

forth another kinde of wine without any mixtures; but the Cocke may bee so or­dered that there may come out by it two wines together, or all three kindes at once: but it seemes best when that in one ves­sell and at one Cocke, a man may draw severall kindes of wine, and which he pleaseth to drinke.

PROBLEM. LXXV. Of Burning-glasses.

IN this insuing discourse I will shew the in­vention of Prometheus how to steale fire from Heaven, and bring it downe to the Earth; this is done by a little round Glasse, or made of steele, by which one may light a Candle and make it flame, kindle Fire-brans to make them burne, melt Lead, Tinne, Gould, and Silver, in little time: with as great ease as though it had beene put into a Cruzet over a great fire.

Have you not read of Archimedes of Syracu­ses, who when he could not come to the Ships of Marcellus, which asseiged that place, to hin­der and impeach their aproach, hee flung huge [Page 130] stones by his Ingines to sinke them into the Sea and transformed himselfe into Iupiter; thunde­ring downe from the highest Towers of the Towne, his thunderbolts of lightning into the Shippes, causing a ter­rible

burning, in des­pite of Neptune and his watery region: Zo­naras witnesseth that Proclus a brave Ma­thematician, burned in the same manner the Shippes of Vitalian, which was come to asseige Constantinople; and dayly experience may let you see great effects of burning, for a Bowle of Crystall polished, or a Glasse thicker in the middle than at the edges, will burne ex­ceedingly; nay a bottle full of water exposed to the Sunne will burne when the Sunne shineth hot, and children use with a Glasse to burne Flies which are against the walles, and their fellowes cloathes.

But this is nothing to the burning of those Glasses which are hollow, namely these which are of steele well polished, according to a para­bolicall or ovall section: A sphaericall Glasse, or that which is according to the segment of a Sphaere, burnes very effectually about the fourth part of the Diamitor; notwithstanding the Pa­rabolie and Eclipticke sections have a great ef­fect: by which Glasses there is also diverse [Page 131] figures represented forth to the eye.

The cause of this burning is the uniting of the beames of the Sunne, which heates migh­tily in the point of concourse or inflammation, which is either by transmission or reflection: Now it is pleasant to behold when one breath­eth in the point of concourse, or throweth small dust there, or sprinkles vapours of hot water in that place; by which the pyramidall point, or point of inflammation is knowne. Now some Authors promiseth to make Glasses which shall burne a great distance off, but yet not seene vul­garly produced, of which if they were made, the Parabolie makes the greatest effect, and is generally held to bee the invention of Archi­medes or Proclus.

Maginus in the 5. Chap. of his Treatise of sphericall Glasses, shewes how one may serve himselfe with a concave Glasse, to light fire in the shaddow, or neare such a place where the Sunne shines not, which is by helpe of a flat Glasse, by which may be made a percussion of the beames of the Sunne into the concave Glasse, adding unto it that it serves to good use to put fire to a Mine, provided that the combustible matter bee well applyed before the concave Glasse; in which hee sayes true: but because all the effect of the practice depends upon the placing of the Glasse and the Powder which he speakes not of: I will deliver here a rule more generall.

How one may place a Burning-glasse with his combustible matter in such sort, that at a con­venient [Page 132] houre of the day, the Sunne shining, it shall take fire and burne: Now it is certaine that the point of inflammation or burning, is changed as the Sunne changeth place, and no more nor lesse, then the shaddow turnes about the stile of a Dyall; therefore have regard to the Suns motion, and his height and place: a Bowle of Crystall in the same place that the toppe of the stile is, and the Powder or other combu­stible matter under the meridian, or houre of 12, 1, 2, 3, &c. or any other houre, and under the Sunnes arch for that day: now the Sunne comming to the houre of 12, to 1, 2, 3, &c. the Sunne casting his beames through the Crystall Bowle, will fire the materiall or combustible thing, which meets in the point of burning: the like may be observed of other Burning-glasses.

PROBLEM. LXXVI. Containing many pleasant Questions by way of Arithmeticke.

I Will not insert in this Probleme that which is drawne from the Greeke Epigrams, but proposing the Question immediatly will give the answere also, without staying to shew the manner how they are answered; in this I will not be tyed to the Greeke tearmes, which I ac­count not proper to this place, neither to my purpose; let these reade that will Diophanta Scheubelius upon Euclide and others, and they may be satisfied.

PROBLEM. LXXVII. Divers excellent and admirable experiments upon Glasses.

THere is nothing in the world so beautifull as light: and nothing more recreative to the sight, than Glasses which reflect: therefore I will now produce some experiments upon them, not that I will dive into their depth (that were to lay open a misterious thing) but that which may delight and recreate the spirits: Let us suppose therefore these principles, upon which is built the demonstration of the appa­rances which is made in all sort of Glasses.

First, that the rayes or beames, which reflect­eth upon a Glasse, maketh the Angle of Incident equall to the Angle of Reflection, by the first Theo. of the Catoptick of Euc.

Secondly, that in all plaine Glasses, the Images are seene in the perpendicular line to the Glasse [Page 142] as farre within the Glasse as it is without it.

Thirdly, in Concave, or Convex Glasses, the Images are seene in the right line which passeth from the object and through the Center in the Glasse Theo. 17. and 18.

And here you are to understand that there is not meant onely these which are simple Glasses or Glasses of steele, but all other bodies, which may represent the visible Image of things by reason of their reflection, as water, marble, mut­tle, or such like. Now take a Glasse in your hand and make experiment upon that which follow­eth.

PROBLEM: LXXVIII. 1. How to shew to one that is suspitious, what is done in another Chamber or Roome: not­withstanding the interposition of the wall.

FOr the performance of this, there must bee placed three Glasses in the two Chambers, of which one of them shall bee tyed to the plan­ching or seeling, that it may be common to com­municate the Species to each Glasse by reflexion, there being left some hole at the top of the wall against the Glasse to this end: the two other Glasses must be placed against the two walls at right Angles, as the figure here sheweth at B. and C.

[Page 161]Then the sight at E. by the line of incidence F E, shall fall upon the Glasse B A, and reflect upon the superficies of the Glasse B C, in the point G; so that if the

eye be at G, it should see E, and E, would reflect upon the third Glasse in the point H, and the eye that is at L, will see the Image that is at E. in the point of the Catheti: which Image shall come to the eye of the suspicious, viz. at L. by helpe of the third Glasse, upon which is made the second reflexion, and so brings unto the eye the object, though a wall be betweene it.

PROBLEM. LXXIX. How with a Musket to strike a marke, not looking towards it, as exact as one aymed at it.

AS let the eye be at O, and the marke C; place a plaine Glasse perpendicular as A B: so the marke C shall hee seene in Catheti C A, [Page 163] viz. in D, and the line

of reflexion is D: now let the Musket F E, upon a rest, bee mo­ved to and fro untill it be seene in the line O D, which admit to be H G: so giving fire to the Musket, it shall undoubtedly strike the marke.

PROBLEM. LXXX. How to make an Image to be seene hanging in the aire, having his head downeward.

TAke two Glasses, and place them at right Angles one unto the other, as admit A B, and C B, of which admit C B Horizontall, & let the eye be at H, & the object or image to be D E; so D will bee refle­cted

at F, so to N, so to H, E: then at G, so to M and then to H; and by a double re­flexion ED will seeme in Q R, the highest point D in R, and the point E in Q inver­sed as was said, ta­king D for the head, and E for the feete; so it will be a man inversed, which will seeme to be flying in the aire: if the Image had wings unto it, and had secretly some motion: and if the Glasse were bigge enough to receive many reflexions, it would deceive the sight the more by admiring the changing of colours that would be seene by that motion.

PROBLEM. LXXXI. How to make a company of representative Soul­diers seeme to be a Regiment, or how few in number may bee multiplyed to seeme to be many in number.

TO make the experiment upon men, there must be prepared two great Glasses; but in stead of it we will suppose two lesser, as G H. and F I, one placed right against another per­pendicular to the Ho­rizon,

upon a plaine levell Table: between which Glasses let there bee ranged in Battalion-wife upon the same Table an number of small men, according to the square G, H, I, F, or in any other forme or posture: then may you evidently see how the said battle will bee multiplyed and seene farre bigger in the appea­rance than it is in effect.

PROBLEM. LXXXII. Of fine and pleasant Dyalls.

COuld you choose a more ridiculous one than the naturall Dyall written amongst the Greeke Epigrams, upon which some sound Poet made verses; shewing that a man carryeth about him alwaies a Dyall in his face by meanes of the nose and teeth: and is not this a jolly Dyall, for he neede not but open the mouth, the lines shall bee all the teeth, and the nose shall serve for the stile.

PROBLEM. LXXXIII. Of Cannons or great Artillery. Souldiers, and others would willingly see this Probleme, which containes three or foure sub­tile questions: The first is how to charge a Cannon without Powder.

THis may be done with aire and water only, having throwne cold water into the Can­non, which might be squirted forceably in by the closure of the mouth of the Peece, that so by this pressure the aire might more condence; then having a round peeee of wood very just, and oyled well for the better to slide, and thrust the Bullet when it shall be time: This peece of wood may bee held fast with some Pole, for feare it be not thrust out before his time: then let fire bee made about the Trunion or hinder part of the Peece to heate the aire and water, and then when one would shoote it, let the pole be quickly loosened: for then the aire search­ing a greater place, and having way now offe­red, will thrust out the wood and the bullet very quicke: the experimence which wee have in long trunkes shooting out pellats with aire on­ly, sheweth the verity of this Probleme.

PROBLEM. LXXXIIII. Of prodigious progression and multiplication, of Creatures, Plants, Fruites, Numbers, Gold, Silver, &c. when they are al­wayes augmented by certaine proportion.

HEre we shall shew things no lesse admira­ble, as recreative, and yet so certaine and easie to be demonstrated, that there needes not but Multiplication only, to try each particular: and first,

PROBLEM. LXXXV. Of Fountaines, Hydriatiques, Machinecke, and other experiments upon water, or other liquor.

PROBLEM. LXXXVI. Of sundry Questions of Arithmeticke, and first of the number of sands.

IT may be said incontinent, that to undertake this were impossible, either to number the sands of Libya, or the sands of the Sea; and it was this that the Poets sung, and that which the vulgar beleeves; nay, that which long a­goe certaine Philosophers to Gelon King of Sici­ly [Page 209] reported, that the graines of sand were in­numerable: But I answere with Archimedes, that not onely one may number these which are at the border and about the Sea; but these which are able to fill the whole world: if there were nothing else but sand, and the graines of sands admitted to bee so small, that 10 may make but one graine of Poppy: for at the end of the account there neede not to expresse them, but this number 30840979456, and 35 Ci­phers at the end of it. Clavius and Archimedes makes it somewhat more; because they make a greater firmament than Ticho Brahe doth; and if they augment the Vniverse, it is easie for us to augment the number, and declare assuredly how many graines of sand there is requisite to fill another world, in comparison that our visi­ble world were but as one graine of sand, an atome or a point; for there is nothing to doe but to multiply the number by it selfe, which will amount to ninety places, whereof twenty are these, 95143798134910955936, and 70 Ciphers at the end of it: which amounts to a most prodigious number, and is easily supputa­ted: for supposing that a graine of Poppy doth containe 10 graines of sand, there is nothing but to compare that little bowle of a graine of Poppy, with a bowle of an inch or of a foote, and that to be compared with that of the earth, and then that of the earth with that of the firma­ment; and so of the rest.