PROBLEM I. To finde a number thought upon.

BId him that he 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 sum, then ask how much it coms to, for then if you take away half the number from it which you willed him at first to add to it, there shall remain the double of the number thought upon.

Example

The Number thought upon

5

The Quadruple of it

20

Put 8 unto it, makes

28

The halfe of it is

14

[Page 2]Take away halfe the number added from it, viz 4, the rest is

10

The double of the number thought upon, viz.

10

PROBLEM II. How to represent to those which are in a Cham [...]er that which is without, or all that which passeth by,

THis is one of the finest experiments in the Optiques, and it is done thus, chuse a Cham­ber or place which is towards the street, fre­quented with people, or which is against some fair flourishing object, that so it may be more delightfull and pleasant to the beholders, then make the Room dark by shutting out the light, except a small hole of six pence broad this done all the Images and species of the objects which are without, will be seen within, and you shall have pleasure to see it, not only upon the wall, but especially upon a sheet of white paper, or

some white cloth hung neer the hole: & if unto the hole you place a round glasse, that is, a glasse w ^ch 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 of a darkish colour, will appear and be seen upon the paper, or white cloth, ac­cording [Page 7] to their naturall colours, yea more live­ly than their naturall, and the appearances wil be so much the more beautifull and perfect, by how much the hole is lesser, the day cleere, and the sun shining

It is pleasant to see the beautifull and goodly representation of the heavens intermixed with clouds in the Horizon, upon a woody scituation, 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 seen within even to the life, but inversed: notwithstanding, this beautifull paint will so naturally represent it self in such a lively Perspective, that hardly the most accurate Painter can represent the like.

Now the reason

why the images and obj [...]cts without are inversed, is because the species doe inter­sect one another in the hole, so that the species of the feet as­cend, and these of the head descend.

But here note, that they may be represented right two manner of wayes; first, with a con­cave glasse: secondly, by help of another con­vex glasse, disposed or placed between the pa­per and the other Glasse: as may be seen here by the figure.

[Page 8] Now I will add here only by passing by, for such which affect Painting and portraiture, that this experiment may excellently help them in the lively painting of things perspectivewise, as Topographicall cards, &c. and for Philosophers, it is a fine secret to explain the Organ of the sight, for the hollow of the eye is taken as the close Chamber, the Ball of the Apple of the eye, for the hole of the Chamber, the Crystaline 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

SCaliger in his 331 exercise against Cardan, re­lates that the Mathematicians of Maximillian [Page 10] the Emperour did propose upon a day this Que­stion, and promised to give the resolution; not­withstanding [...]caliger delivered it not, and I con­ceive it to be thus. Take a Balance, and let the Fist, the Mallet, or Hatchet rest upon the scale, or upon the beam of the Balance, and put into the other Scale as much weight as may coun­terpoyse it; then charging or laying more waight into the Scale, and striking upon the other end, you may see how much one blow is heavier than another, and so consequently how much it may waigh for as Aristotle saith, The motion that is made in striking adds great waight unto it, and so much the more, by how much it is quicker: there­fore

in effect, if there were placed a thou­sand mallets, or a Thousand pounde waight upon a stone, nay, though it were exceedingly pressed down by way of a Vice, by Levers, or other Mechanick Engine, it would be nothing to the rigor and violence of a blow.

Is it not evident that the edge of a knife laid upon butter, and a hatchet upon a leafe of pa­per, without striking makes no impression, or at least enters not; but striking upon the wood a little, you may presently see what effect it hath, which is from the quicknesse of the motion, which breaks and enters without resistance, if it be extream quick, as experience shews us in the [Page 11] blows of Arrows, of Cannons, Thunder-boults, and such like.

PROBLEM IV. How to break a staffe which is laid upon two Glasses full of water, without breaking the Glasses, spilling the water, or upon two reeds or straws without breaking of them.

FIrst, place the Glasses which are full of water upon two joynt stooles, or such like, the one as high as the other from the ground, and di­stant one from another by 2 or 3 foot, then place the ends of the staffe upon the edges of the two Glasses so that they be sharp, this done, with all the force you can, with another staffe strike the staffe which is upon the two Glasses

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 with your hands in the aire without breaking them▪ thence Kitchin boyes often break bones of mutton upon their hand, or with a napkin without any hurt, in only striking upon the middle of the bone with a knife.

[Page 13] Now in this act, the two ends of the staffe in breaking slides away from the Glasses, upon which they were placed; hence it commeth that the Glasses are no wise indangered, no more than the knee upon which a staffe is bro­ken, forasmuch as in breaking it presseth not: as Aristotle in his Mechanick Questions obser­veth.

PROBLEM V. How to make a faire Ge [...]graphic [...]ll Card in a Garden Plot, fit for a Prince, or great personage.

IT is usuall amongst great men to have faire Geographicall Maps ▪ large Cards, and great Globes, that by them they may as at once have a view of any place of the World, and so furnish themselves with a generall knowledge, not only of their own Kingdoms form, scituation, lon­gitude, latitude, &c. but of all other places in the whole Universe, with their magnitudes, po­sitions, Climats, and distances.

Now I esteem that it is not unworthy for the meditations of a Prince, seeing it carries with it many profitable and pleasant contentmen [...]s: if such a Card or Map by the advice and direction of an able Mathematician were Geographically described in a Garden plot form, or in some o­ther convenient place, and instead of which ge­nerall description might particularly and artifi­cially be prefigured his whole Kingdoms and Dominions, the Mountains and hils being raised like small hillocks with turfs of earth, the val­leys somwhat concave, which will be more a­greeable and pleasing to the eye, than the de­scription in plain Maps and Cards, within which may be presented the Towns, Villages, Castles, or other remarkable edifices in small green mo [...]e banks, or spring-work proportionall to the pl [...]t­form, [Page 15] the Forrests and Woods represented ac­cording to their form and capacity, with herbs and stoubs, the great Rivers, Lakes, and Ponds to dilate themselves according to their course from some artificiall Fountain made in the Gar­den to passe through chanels; then may there be composed walks of pleasure, ascents, places of repose, adorned with all variety of delight­full herbs and flowers, both to please the eye or other senses. A Garden thus accommodated shall farre exceed that of my Lord of Verulams specified in his [...]ssayes; that being only for de­light and pleasure, this may have all the proper­ties of that, and also for singular use, by which a Prince may in little time personally visit his whole Kingdom, and in short time know them distinctly: and so in like manner may any par­ticular man Geographically prefigure his own possession or heritage.

PROBLEM VI. How three staves, knives, or like bodies, may be conceived to hang in the aire, without being supported by any thing but by themselves.

TAke the first staffe AB, raise up in the aire the end B, and upon him cros-wise place the staffe CB, then lastly, in Triangle wise place the third staffe EF▪ in such manner that it may be under AB, and yet upon CD. I say that these staves so disposed cannot fall, and [Page 16] the space CBE is made the stronger, by how much the more it is pressed downe, if the staves break not, or sever themselves from the triangu­lar forme: so that alwayes the Center of gravi­tie

be in the Center of the Triangle: for AB is supported by EF, and EF is held up by CD, and CD is kept up from falling by AB, there­fore one of these staves cannot fall, and so by consequence none.

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

ORdinarily the proposition is delivered in this wise: 15 Christians and 15 Turkes being at Sea in one Shippe, an extreame tempest being risen, the Pilot of the Shippe saith, it is necessary to cast over board halfe of the num­ber of Persons to disburthen the Shippe, and [Page 17] to save the rest: now it was agreed to be done by lot, and therefore they consent to put themselves in rank, 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 he therefore to dispose the Christians, that the lot might fall alwayes upon the Turkes, and that none of the Christians be in the ninth place?

The resolution is ordinarily comprehended in this verse.

For having respect unto the vowels, making a one, e two, i three, o foure, and u five: o the first vowell in the first word sheweth that there must be placed 4. Christians; the next vowel u, signifieth that next unto the 4. Christi­ans must be placed 5 Turkes, and so to place both Christians and Turkes according to the quantity and value of the vowels in the words of the verse, untill they be all placed: for then counting from the first Christian that was placed, unto the ninth, the lot will fall upon a Turk, and so proceed. And here may be further noted that this Probleme is not to be limited, seeing it extends to any number and order whatsoever, and may many wayes be usefull for Captaines, Magistrates, or others which have divers persons to punish, and would chastise chiefely the unruliest of them, in taking the 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 be ob­served, take as many units as there are per­sons, and dispose them in order privately: as for example, let 24 men be 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 units, or upon a piece of paper write down 24 cyphers, and account from the beginning to the eighth, which eighth mark, 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 be punished, and so of others.

It is supposed that Josephus the Author of the Jewish History escaped the danger of death by help of this Problem; for a worthy Author of beliefe reports in his eighth chapter of the third Book of the destruction of Jerusalem, that the Town of Jotapata being taken by main force by Vespatian, Josephus being Governour of that Town, accompanyed with a Troop of forty Souldiers, hid themselves in a Cave, in which they resolved rather to famish than to fall into the hands of Vespatian: and with a bloudy resolution in that great distresse would have butchered one another for sustenance, had not Josephus perswaded them to die by lot and order, upon which it should fall: Now [Page 19] seeing that Josephus did save himselfe by this Art, it is thought that his industry was exercised by the helpe of this Problem, so that of the 40 persons which he had, the third was alwayes killed. Now by putting himselfe in the 16 or 31 place he was saved, and one with him which he might kill, or easily perswade to yeild 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 be a Ring, a piece of Gold, and a piece of Silver, or any other such like, and let them be known privately to your self by these three Vowels a, e, i, or let there be three persons that have different names, as Ambrose, Edmond, and John, which privately you may note or account to your selfe once known by the aforesaid Vowels, which signifie for the first vowel 1, for the second vowell 2, for the third vowell 3.

Now if the said three persons should by the mutuall consent of each other privately change their names, it is most facill by the course and excellencie of numbers, distinctly to declare each ones name so interchanged, or if three persons in private, the one should take a Ring, the [Page 20] other a piece of Gold, and the third should take a piece 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 down 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 re­present e, the second vowel; and to the third 3. Counters, which stand 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 he 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 mark what number of Counters you had at the first, for there are necessarily but 24. as was said before, the sur­pluse you may privately reject. And then there will be left either 1.2.3.5.6 or 7. and no other number can remaine, which if there be, then they have failed in taking according to the directions delivered: but if either of these num­bers do remaine, the resolution will be 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 syllables are e and i, vvhich shevveth accord­ing to the former directions, that to vvhom you gave 2 Counters, he hath the Ring (seeing it is the second vovvell represented by tvvo as before) and to vvhom you gave the 3. Counters, he hath the Gold, for that i represents the third vovvel, or 3. in the former direction, and to vvhom you gave one Counter, he hath the Silver, and so of the rest: the variety of changes, in vvhich exercise, is laid open in the Table fol­lovving.

This feat may be done also without the for­mer words by help of the Circle A. for ha­ving divided the Circle into 6 parts, write 1. within and 1. vvithout, 2. vvithin and 5. vvith­out, &c. the first 1.2.3. vvhich are vvithin vvith the numbers over them, belongs to the upper semicircle; the other numbers both vvithin and vvithout, to the under semicircle; [Page 22] now if in the action there remaineth such a number which may be found in the upper semi­circle without, then that which is opposite with­in shews the first, the next is the second, &c. as if 5 remains, it shews to whom he gave 2, he hath the Ring; to whom you gave [...], he hath the Gold, &c. But if the remainder be 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 remains, to whom you gave 1 he hath the Ring, he that had 3 he had the Gold, &c.

PROBLEM IX. How to part a Vessel which is full of wine conteining eight pints into two equall parts, by two other vessels which conteine as much as the greater vessell; as the one being 5 pints, and the other 3 pints.

LEt the three vessels be represented by ABC, A being full, the other two being empty; first, poure out A into B until it be full, so there will be in B 5 pints, and in A but 3 pints: then poure out of B into C untill it be full: so in C shall be 3 pints, in B 2 pints, and in A 3 pints, then poure the wine which is in C into A, so in A will be 6 pints, in B 2 pints, and in C nothing: then poure out the wine which is in B into the [Page 23] pot C, so in C there is now 2 pints, in B nothing, and in A 6 pints,. Lastly, poure out of A in­to B untill it be full, so there will be now in A only 1 pint in B 5 pints, and in C 2 pints. But it is now evident, that

if from B you poure in unto the pot C untill it be full, there wil remain in B 4 pints, and if that which is in C, viz. 3 pints be poured into the vessell A, which be­fore had 1 pint, there shall be in the vessel A, but halfe of its liquor that was in it at the first, viz. 4 pints as was re­quired. Otherwise poure out of A into C untill it be full, which pour into B, then poure out of A into C again untill it be full, so there is now in A onely 2 pints, in B 3, and in C 3, then pour from C into B untill it be full, so in C there is now but 1 pint, 5 in B, and 2 in A: poure all that is in B into A, then poure the wine which is in C into B, so there is in C nothing, in B one­ly 1 pint, and in 7 A 7 pints: Lastly, out of A fill the pot C, so there will remain in A 4 pints, or be but halfe full: then if the liquor in C be poured into B, it will be the other half. In like manner might be taken the half of a vessell which conteins 12 pints, by having but the mea­sures 5 and 7, or 5 and 8. Now such others might be proposed, but we omit many, in one and the same nature.

PROBLEM. X. To make a stick stand upon the tip of ones finger, without falling.

FAsten the edges of tvvo knives or such like of equall poise, at the end of the stick, leaning out somevvhat from the stick, so that they may counterpoise one another; the stick being sharp at the end, and held upon the top of the finger, vvill there rest vvithout support­ing: if it fall, it must fall together, and that perpendicular or plumb-wise,

or it must fall side-wise or before one ano­ther; in the first man­ner it cannot: for the Centre of gravitie is sup­ported by the top of the finger: and seeing that each part by the knives is counterpoised, it cannot fall sidevvise, therefore it can fall no vvise.

In like manner may great pieces of Timber, as Joists, &c be supported, if unto one of the ends be applied convenient proportionall coun­terpoises, yea a Lance or Pike, may stand per­pendicular in the Aire upon the top of ones finger: or placed in the midst of a Court by help of his Centre of gravitie.

PROBLEM XI. How a milstone or other Ponderosity, may be sup­ported by a small needle, without break­ing o [...] 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 needle: it is evident that the stone cannot fall, forasmuch as it hangs in aequilibra, or is counterpoysed in all parts a­like; and moreover it cannot bow the needle more on the one side then on the other, the nee­dle will not therefore be either broken or bow­ed; if otherwise then the parts of the needle must penetrate 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 be 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 gravity be exactly found.

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

FIt the three Knives in form of a Ballance, and holding a Needle in your hand, and place the back of that, Knife

which lyes cross-wise 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 and move upon the point of the Needle with [...]ou falling.

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

LEt it be supposed that a great heape of Fa­gots, or a load of straw weighing 500 pound should be fired, it is evident that this grosse sub­stance will be all inverted into smoak and ashes: now it seems that the smoak weighs nothing; seeing it is of a thin substance now dilated in the Aire, notwithstanding if it were gathered and reduced into the thickest that it was at first, it would be sensibly weighty: weigh therefore the ashes which admit 50 pound, now seeing that [Page 28] the rest of the matter is not lost, but is exhaled into smoake, it must necessarily be, that the rest of the weight (to wit) 450 pound, must be the weight of the smoak required.

PROBLEM XVI. Many things being disposed circular, (or otherwise) to finde which of them, any one thinks upon▪

SUppose that having ranked 10 things, as ABCDEFGHIK, Circular (as the figure sheweth) and that one had touched or thought upon G, which is the 7: ask the party at what [Page 29] letter he would begin to account (for account he must, otherwise it cannot be done) which sup­pose, at E which is the 5 place, then add 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 he shal

account to himself 7, at D account 8, at C account 9, &c. So the account of 15 wil ex­actly fall upon G, the thing or number thought upon: and so of others: but to con­ceal 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 upside downe, as the ten simple Cards, with the King and Queen, the King standing for 12, and the Queene for 11, and so knowing the situation of the Cards: and thinking a cer­tain houre of the day: cause the party to ac­count from what Card he pleaseth: with this Proviso, that when you see where he 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 Door or Gate, which shall open on both sides.

ALL the skill and subtilty of this, rests in the artificiall disposer of foure 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 hooks or hinges of the Posts, and by the o­ther end may be made fast to the Gate, and so moving upon these hinges, the Gate will open upon one side with the aforesaid plates, or hooks of Iron: and by help of the other two plates, will open upon the other side.

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

TAke a pale which hath a handle, and fill it full of water (or at pleasure:) then take a staffe or stick which may not rowle upon the Table as EC, and place the handle of the Pale upon the staffe; then place another staffe, or stick, under the staffe CE, which may reach from the bottom of the Pale unto the former staffe CE, perpendicular wise: which suppose FG, then shall the Pale of water hang without [Page 31] falling, for if it fall it

must fall perpendicu­larly, or plumbe wise: and that cannot be seeing the staffe CE supports it, it being parallel to the Hori­zon and susteined by the Table, and it is a thing admirable that if the staffe CE 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 molten 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 wil turn away side-wise: for that on that part of the Bowle which is heavier upon the one side then on the other, it never will go truly right, if artificially it be not cor­rected; which will hazard the game to those which know it not: but if it be known that the leady side in rolling be always under or above, it may go indifferently right; if otherwise, the weight will carry it always side-wise.

PROBLEM. XVIII. To part an Apple into 2.4. or 8. like parts, without breaking the Rinde.

PAsse a needle and threed under the kinde of the Apple, and then round it with divers turnings, untill you come to the place where you began: then draw out the threed gently, and part the Apple into as many parts as you think convenient: and so the parts may be ta­ken out between the parting of the Rind, and the rind remaining alwayes whole.

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

BId him adde to the number thought (as ad­mit 15) halfe of it, if it may be, if not the greatest halfe that exceeds 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 be, 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 keep 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. to 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 keep 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 mark 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 halfings: for which last! 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 multiplied by 4, makes 20. from which subtract the first 3 and 2, because the halfe could not be formerly added, leaves 15, the number thought upon.

[Page 35] Other Examples.

The number thought

12

The halfe of it

6

The summe

18

The halfe of it

9

The summe of it

27

The double of the number,

24

Which taken away, rests

3

The halfe of it

1

For which account

2

and 1 put to it because the

3

could not be halfed, makes

3

this multiplied by 4 makes

12

The number thought

79

The greatest halfe

40 3

The summe

119

The greatest halfe of which is

60 2

The summe of it is

179

The double of 79 is

158

Which taken from it, rests

21

The lesser half 10. w ^ch halve:

 

The halfe of this is 5 which makes

 

The half of this is 2 w ^ch is 10

 

The half of this is 1, with 10 and 11 is 21.

 

this 21 which is the double of the last halfe with the re­mainder being multiplied by 4. makes 84, from which take the aforesaid 3 and 2, [...]st 79, the number thought upon.

 

PROBLEM. XX. How to make an uniforme, & an inflexible body, to passe through two small holes of divers formes, as one being circular, and the other square, Quadrangular, and Triangular-wise, 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 practises, the lesser or easier Problemes do al­wayes make way to facilitate the greater: and the aforesaid Probleme is thus resolved. Take a Cone or round Pyramide, and make a Circu­lar hole in some board, or other hard material, which may be equall to the bases of the Cone, and also a Triangular hole, one of whose sides may be equall to the Diameter 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 be made, whose Diameter is equal to the Diameter of the Cones conjoyned, and a Quadrangular hole, whose sloping sides be equall to the length of each side of the Pyramide, and the breadth of the hol equal to the Diameter of the Circle, this conjoyned Pyramide shall exactly fill both the Circular hole, and also the Qua­drangle hole.

PROBLEM. 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 then the former, yet it may be practised two wayes: for the first, take a Cylindricall 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 be placed downe right, and will also fill a long square; whose sides are equall unto the Diameter and length of the Cylinder, and

acording to Pergeus, Archimedes, &c. in their Cylindricall de­monstrations, a true Ovall is made when a Cylinder is cut slopewise, therefore if the oval have breadth equall unto the Dia­meter 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, & also a square hole, the side of which Square may be equall to the Diameter [Page 38] of the Circle: and lastly, make a hole Oval-wise, whose breadth may be equal unto the diagonall of the Square; then let a Cylindricall body be 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 [...]fter another manner, then what 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 & divide Now in the same time take a number at pleasure, and secretly multiply and divide as often as he did: then bid him divide the last number by that which he thought upon. In like manner do yours pri­vately, then will the Quotient of your divisor 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 seeming 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 1 [...]. in the same time admit you think upon 4, which multiplied by 4, makes 16, this divided by 2, makes 8, which [Page 40] multiplied by 6 makes 48, and divided by 4 makes 1 [...]; then divide 1 [...] by the number thought, which was 5, the Quotient is [...]; di­vide also 12 by the number you took, viz. 4, the Quotient is also 3. as was declared; therefore if the Quo [...]ient [...] be added unto the number thought, viz. [...], 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, seven, &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, vvhich makes 11, and so alvvayes adding the tvvo next together, aske him hovv much the first and last makes together, vvhich is 8. then take these summes, and place them in order, and adde all these together, vvhich vvere in the odde places: that is the first, third, and fifth, viz. 5, 9, [...], makes 22. In like manner adde all these numbets together, vvhich are in the even places, that is in the second and fourth places, viz. 7 and 1 [...] makes 18, substract this from the former 22, then there vvill remaine the double of the [Page 41] first number thought upon, viz. 4. which known, the rest is easily known: seeing you know the summe of the first and second; but if the multitude of numbers be even as these six numbers, viz. 2, [...], 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 be 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, substract the one from the other, there shall remaine the double of the second number thought upon, which known all the rest are knowne.

PROBLEM XXIV. 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 answer is very facill, for to be so he must be supposed to be in the centre of the earth: for as the heaven is above on every side, Coelum undique sursum, all that which looks to the heavens being distant from the centre is up­ward; and it is in this sense that Ma [...]olyeus in his Cosmographie, & first dialogue, reported of one that thought he was led by one of the Muses to hell, where he saw Lucifer sitting [Page 42] in the middle of the World, and in the Centre of the earth, as in a Throne: having his head and feet 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 Centre of the earth, and the other halfe on the other side: and that two at the Centre of the World at one instant being to ascend, the one towards us, and the other towards our Antipodes, they should in ascending go 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 Rod 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 Rod of ground posses­seth [Page 43] not only the exterior surface of the earth, but is master also of that which extends even to the Centre of the earth, and in this wise all he­ritages & possessions are as so many Pyramides, whose summets or points meet in the cen­tre of the earth, and the basis of them are no­thing else but each mans possession, field, or vi­sible quantity; and therefore if there were made or imagined so to be made, a descent to go to the bottome of the heritage, which would reach to the centre of the earth; it would be 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 either true North or true 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 he looketh South, because all the Meridians concurre in the Poles of the world, and if he be under the South-pole, he looks 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 think upon a number, and will him to multiply it by what number you think convenient: and to the pro [...]ct bid him adde what number you please, or [...] that se­cretly you consider, that it ma [...] be divided by that which multiplied, and [...] divide the sum by the number which he [...] by, and substract from this Quotient the number thought upon: In the same time divide apart the number which was add [...]d by that which multiplied, so then your Quotient shall be equall to his remainder, wherefore without asking him any thing, you shall tell him what did remaine, which will seem 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, yields 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 two, 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 only [...] recreate the spirits, but also [...] knowledge of admirable things, [...] measure be shewen in this [...] the meane time to pro­duce alwayes some of them: suppose that a man hold divers things in his hand, as Gold and [...]ilver▪ and in one hand he held the Gold, and in the other hand he held the Silver: to know subtilly, and by way of divination, or artificially in which hand the Gold or Silver is; attribu [...]e t [...] the Gold, or suppose it have a cer­taine price, and so likewise attribute to the Sil­ver another price, conditionally that the one be odd, and the other even: as for example, bid h [...]m that the Gold be valued at 4 Crownes, or Shillings, and the Silver at [...] 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, & 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 Gold is in the right hand; if odde, the Gold 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 John, two numbers, or pieces 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 privately to themselves: how artificially, according to the congruity, and excellency of numbers, to finde which of them did take 10. and which 9. without asking any qustion: and this seems 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 took, and do you privately double also your greatest num­ber; then bid John to triple the number which he hath, and do you the like upon your last number: adde your two products together, & mark 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 do, then immediately tell Peter he took 10. and John 9. because the aggregate of the double of 4. and the triple of 3. makes odde, and such [Page 47] would be the aggregate or summe of the double of Peters number and Johns number, if Peter had taken 10. if otherwise, then they might have taken halfe, and so John should have ta­ken 10. and Peter 9. as suppose Peter had taken 10. the double is 20. and the triple of 9. the o­ther [...]umber 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. & the triple of 10. is 30. which set together makes 48. the halfe of it may be taken: therefore in such case Peter the took lesse number: and John the greater, and this being don 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 be 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 Arch in the two points E. and F. and placed in C. crosse the Arch 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 Centre of the Cir­cumference of a Cir­cle, which will passe by the said three points A.B.C. or it may be inverted, ha­ving a Circle drawne; to finde the Centre of that Circle, make 3. points in the circumference, and then use the same way: so shall you have the Centre, a thing most facill to every practitioner in the prin­ciples of Geometrie.

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

M [...]ke a Circle upon past-board or other materiall, as the Circle A.C.D.E. of which A. is the Centre; then cut it into 4 quar­ters, and dispose them so, that A. at the centre of the Circle may alwayes 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 Diameter B.D. Now here is to be noted that the square is greater then the Circle by the vacuity in the middle, viz. M.

PROBLEM. XXXIII. With one and the same comp [...]sses, and at one and the same extent, or opening, how to describe many Circles concentricall, that is, greater or lesser one then another?

IT is not without cause that many admire how this Proposition is to be resolved; yea in the judgement of some it is thought impossible: who consider not the industrie of an ingenious Geometrician, 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 Centre of that Circle, a small pegge of wood be placed, to be raised up and put downe at pleasure by help of a small ho [...]e made in the Centre, then with the same opening of the Compasses, you may describe Circles Con­centricall, that is, one greater or lesser than a­nother; for the higher the Center is lifted up, the [Page 50] lesser the Circle will

be. Secondly, the compasse being at that extent upon a Gibus body, a Circle may be 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. Prop. of the first of Euclids, for the Diameter [...]. D. is lesse than the line AD.A.E. taken together, and the lines AD.AE. being equall to the Diameter BC. because of the same distance or extent of opening the compasses, it followes that the Diameter E.D. and all his Circles together is much lesse than the Diameter, and the Circle BC. which was to be performed.

PROBLEM XXXIV. 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 multiply that summe by 5. and unto it adde 10. and unto this product add the next number thought upon; multiply this same againe by 10. and adde unto it the next number, and so proceed: now if he declare the last summe; marke if he thought but upon one figure, for then subtract only 35. from it, and the first figure in the place of tennes is the number thought upon: if he thought upon two figures, then subtract also the said [...]5. from his last summe, and the two figures which remaine are the number thought upon: if he thought upo [...] three figures, then subtract 350. and then the first three figures are the numbers thought upon, &c. so if one thought upon these numbers 5.7.9.6. double the first, makes 1 [...]. to which adde 5. makes 15. this multiplied by 5. makes 75. to which adde 1 [...]. makes 85. to this adde the next number, viz. 7. makes 92. this multiplied by 10. makes 920. to which adde the next number, viz. 9. makes 929. which multiplied 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 works by Magick, or Witchcraft: But in effect it is nothing else but a nimble act of Arithmetick, 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 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 think 4. numbers: for example, if the fourth person had the Ring in his left hand, and upon the fifth finger, and third joynt, and I would divine and finde it out: thus I would proceed, as in the 34 Problem: 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 add 5. makes 13. this multiplied by 5. makes 65. put 10. to it, makes 75. unto this put [...]. for the num­ber belonging to the left hand, and so it makes 77. which multiplied by 10. makes 770. to this adde the number of the fingers upon which the Ring is, viz. 5. makes 775. this multiplied by 10. makes 7750. to which adde the number for the joynt upon which the Ring is, viz the third joynt, makes 7 [...]53. 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, substract 3514▪ and there will remaine 4.2.5.3. which figures in order declares the whol mystery of that which is to be known: 4. signifieth the fourth person, 2. the left hand, 5. the fifth finger, and 3. the third joynt of that finger.

PROBLEM. XXXVI. The Play of 34. or more Dice.

THat which is said of the two precedent Problemes may be applied to this of Dice (and many other particular things) to finde what number appeareth upon each Dice being cast by 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, one ha­ving [Page 54] thrown three Dice, and 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 multiply 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 [...]50. 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 neere full of water or other liquor; and setting one hand upon the foot of it, to hold it fast: turne slightly one of the fingers of your other hand upon the brimme, or edge of the Glasse; having before privately 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 sen­sibly appeare to tremble, with notable rarefa­ction and condensation: thirdly, the water will shake, seeme to boyle: fourthly, it will cast it selfe out of the Glasse, and leap 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 Rarefaction of the parts of the Glasse, occasioned by the motion and pressure of the finger.

PROBLEM. XXXVIII. Of a fine vessell which holds wine or water, being cast in [...]o 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 bottom of the vessell as at [...] ▪ are open; let the end [...] 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; [Page 57] then poure in water or wine, and as long as it mounts not above E, it is safe; but if you poure in the water so that it mount above it, fare­well all: for it will not cease untill it be all gone 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 go well; but if you fill it unto H. you will see fine sport, for then all the vessell will be 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 those which know the nature of a Cock 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 run 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 then the water, it will run out without ceasing: but if the mouth of the Pipe be higher then 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 keep 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 seem but one, but there is a little space between them. No [...] poure Wine or other liquor between the two [Page 59] edges by help of a Tunnell, into a little hole left to this end, so vvill there appeare tvvo fine delusions or fallacies; for though there be not a drop of Wine vvithin the hollovv of the Glasse, it vvill seem to those vvhich behold it that it is an ordinary Glasse full of Wine, and that especially to those vvhich are side-vvise of it, and if any one move it, it vvill much confirme it, because of the motion of the Wine; but that vvhich vvill give most delight, is that, if a­ny one shall take the Glasse, and putting it to his mouth shall think to drink the Wine, in­stead of vvhich he shall sup the Aire, and so vvill cause laughter to those that stand by, vvho being deceived, vvill hold the Glass to the light, & thereby considering that the raies or beames of the light are not reflected to the eye, as they vvould be if there vvere a liquid substance in the Glasse, hence they have an assured proofe to conclude, that the hollovv of the Glasse is totally empty.

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

BId him that holds the money that he put out of one hand into the other vvhat num­ber you think 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 Shi [...] ­lings 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 be in the left hand [...]4, 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 mark how many points a paire are upwards, which adde unto the former summe: then put one of these Dice a­way not changing the side, mark the points which are under the other Dice, and adde it to [Page 61] 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 do appeare upward upon the three Dice, which adde privately together, and unto it adde [...]1 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 he formerly made. As if he should throw three Dice, and there should appeare upward 5, 3, 2. the sum of them is 10. and set­ting one of them apart, (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 19 makes 24. and setting one of these two Dice apart 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 whi [...]h he had; and in the same manner you may practise with 4, 5, 6, or many Dice or o­ther bodies, observing only 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 [Page 62] underneath 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 kin [...] 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 Emperour 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 h [...]s 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 think that in this Probleme one must be guided only 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 know not other­wise. A Mathematician accounts it an easie proposition, & will infallibly chuse the chest of Gold, and leave the chest of Lead, without ei­ther breaking, or opening any of the chests, and not go by chance and fortune: for if he may be permitted to weigh those chests first in the Aire, then in the water: it is a thing cleare by [Page 63] the proportion of

Mettalls, & accord­ing to the princi­ples of Archimedes, that the Gold shal be lesse weighty by his eighteenth part, & the Lead by his 11 [...] part, where­fore there may be gathered in which is the Gold, and in which is the Lead.

But because that this experiment in water hath divers accidents, and therefore subject to a caution; and namely, because the matter of the chest, mettall 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 equal quantity: and seeing that Gold is the heaviest of all mettalls, it will occupie less 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 therefore presented two Globes or Chests of wood or other matter alike, & equall one to the other, in one of which in the middle there is another Globe or body of lead weighing 12. l. (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 broad paire of Compasses, and clip one of the Coffers or Globes somewhat from the middle, as at D. then fix in the Chest or Globe a small piece of Iron between the feet of the Compasses, as EK, at the end of which hang a vveight G, so that the other end may be coun­terpoysed, and hang in aequilibrio: and do the like to the other Chest or Globe. Novv if that the other Chest or Globe being clipped in like distance from the end, and hanging at the other end the same weight G. there be found no difference; then clip them nearer tovvards the middle, that so the points of the Compasse may be against some of the mettall vvhich 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, vvhich takes up a part of the Lead, (because it occu­pies a greater place than the Gold) therefore that shall help the vveight G. to vveigh, and so vvill not hang in aequilibrio, except G be placed neare to F. hence vve may conclude, that there is the Lead; and in the other Chest or Globe, there is the Gold.

PROBLEM. XLIII. Two Globes of diverse mettalls, (as one Gold, and the other Copper) yet of equall weight being put into a box, as BG, to finde in which end the Gold or Copper is.

THis is discovered by the changing of the places of the tvvo Bovvles or Globes, having the same counterpoyse H to be hung at the o­ther side, as in N. and if the Gold vvhich is the lesser Globe, vvere before the nearest to the handle D [...], having novv changed his place vvill be farthest from the handle DE, as in K. [Page 66] therefore the Centre of gravity of the two Globes taken together, shall be farther separate from the middle of the

handle (under which is the Centre of gra­vity of the Box) than it was before, and see­ing that the handle is alwayes in the middle of the Box, the vveight N. must be augment­ed▪ to keep it in equi­l [...] and by this way one may knovv, that if at the second time, the counterpoise be too light, it is a signe that the Gold is farthest off the handle, as at the first triall it vvas nearest.

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

THe Rainbovve is a thing admirable in the vvorld, vvhich ravisheth often the eyes and spirits of men in consideration of his rich inter­mingled colours vvhich are seen under the clouds, seeming as the glistering of the Starres, precious stones, and ornaments of the most beauteous flovvers: some part of it as the re­splendent stars, or as a Rose, or burning Cole of fire▪ in it one may see Dyes of sundry sorts, [Page 67] the Violet, the Blew, the Orange, the Saphir, the Jacinct, and the Emerald colours, as a lively plant placed in a green soile: and as a most rich treasure of nature, it is a high work of the Sun who casteth his raies or beames as a curious Painter drawes strokes with his pensill, and placeth his colours in an exquisite situation; and Solomon saith, Eccles. 43. it is a chiefe and principall work 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 seen 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, make a kinde of coloured Rainbowe: or may we not see in houses or Gardens of plea­sure artificiall fountaines, which poure forth their droppie streames of water, that being be­tween the Sunne and the fountaine, there will be presented as a continuall Rainbowe? But not to go farther, I will shew you how you may do it at your doore, by a fine and facill expe­riment.

Take water in your mouth, and turne your back to the Sunne, and your face against some obscure place, then blow out the water which is in your mouth, that it may be sprinkled in small drops and vapours: you shall see those [Page 68] atomes vapours in the beames of the Sunne to turne into a faire Rainebowe, 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 raies that passe through strike upon a shadowed place, you will have pleasure to see the fine forme of a Rainebovve by this reflection. Or take a Trigo­nall Glasse or Crystall Glasse of diverse An­gles, and look through it, or let the beames of the Sunne passe through it; or vvith a candle let the appearances be received upon a shadovv­ed place: you vvill have the same contentment.

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

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

Of like fineness or subtiltie may it be that a bowle of Iron falling from a high place upon a plaine pavement of thin Glasse, it were impos­sible [Page 69] any wise to break 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 parts which are 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 be concluded as speaking naturally, that such a bovvle falling upon such a glasse vvill not break it. But this matter is meere Metaphysicall, and all the vvorkmen in the vvorld cannot ever vvith all their industrie make a bovvle 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 di­divided by 7, there will re­maine nothing.

IN many Authors of Arithmetick this Pro­bleme is thus proposed: A vvoman carrying Egges to Market in a basket, met an unruly fel­lovv who broke them: who vvas 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 vvill still remaine a unite▪ multiply these numbers together, makes 720, to which adde 1; so have you the number, viz. 721. in like manner 301 vvill be measured by 2, 3, 4, 5, 6; so that 1 remaines: but being measured by 7, nothing vvill remaine; to vvhich continually adde 220, and you have other numbers vvhich vvill do the same: hence it is doubtfull vvhat number she had, therefore not to faile, it must be knovvn vvhether they did exceed 400, 800, &c. in vvhich it may be conjectured that it could not exceed 4 or 5 hundred, seeing a man or vvoman could not carry 7 or 8 hundred Egges, therefore the number vvas the former [...]01. vvhich she had in her Basket: vvhich be­ing counted by 2 and 2, there vvill remaine 1, by 3 and 3, &c. but counted by 7 and 7, there vvill remaine nothing.

PROBLEM. 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 & 6, there rested 5. but counting them by 7 and 7, there remained nothing: how many crownes might he 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, vvhich multiplied by 7, and then di­vided by 2, 3, 4, 5, 6; there may alvvayes remaine a number lesse by 1 than the Divisor: Novv the first number vvhich arrives in this nature is 119, unto vvhich if 420 be added, makes 539, vvhich also vvill do 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 between 1 pound and 40 pound, and so unto 121, & 364 pound.

TO vveigh things betvveen 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 [...] 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 between 1 and 40 pound may be 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 between 1 pound, and 121 pound: and taking those 6 weights▪ as 1, 3, 9, 2 [...], 81, 243, you may weigh even from 1 pound unto 364 pound: this de­pends upon the property of continued propor­tionals, the latter of which containing twice all the former.

PROBLEM. XLIX. Of a deceitfull ballance which being c [...]ty seemes i [...] be just, because it hangs in aequilibrio: not­ [...]ithstanding putting 12 pound in one ballance, and 11 in the other, it will remaine in aequilibrio.

ARistotle maketh mention of this ballance in his mechanick 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 then another, by the same proportion, that one weight is heavier then 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 be done: then after­wards put into the ballance two unequal 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 seem most sincere and just; though it be 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 vveights themselves: for in one and the same counterpoise, by hovv much it is farther from the Centre of the handle, by so much it seems heavier, therefore if there be a diversitie of distance that the ballances hang from the handle, there must necessarily be an ineqality of weight in these ballances to make them hang in aequilibrio, and to discover if there be deceit, change the weight into the other ballance, for as soone as the greater vveight is placed in the ballance that hangs on the longer parts of the beame: it vvill vveigh dovvne the other instantly.

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

TAke a stravv that is not bruised, bovv it that it make an Angle, and put it into the bottle so that the greatest end be in the neck, then the Reed being put in

the bovved part vvil cast side-vvise, and make an Angle as in the figure may be seen: then may you take the end vvhich is out of the bottle in your hand, and heave up the bottle, [Page 75] and it is 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 come out of the bottle.

PROBLEM. LI. How in the middle of a wood or desert, without the sight of the Sunne, Starres, Shadow 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 help 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 mark the many circles that are about the sap or pith of the tree, which seem nearer together in some part than in o­ther, which is by reason of the Suns motion a­bout the tree: for that the humiditie of the parts of the tree towards the South by the heat of the Sun is rarified, and caused to extend: and the S [...]n not giving such heat towards the North-part of the tree, the sap is lesser rarefied, but condensed; by which the circles are nearer together on the North-part, than on the South-part: therefore if a line be drawne from the [Page 76] widest to the narrowest

part of the circles, it shall shew the North & South of the world. Another Experiment may be thus: Take a small needle, such as women work with: place it gently downe flatwise upon still wa­ter, and it will not sink, (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 be great and will not swim, thrust it through a small piece of Cork, or some such like thing, and then it will do the same: for such is the property of Iron when it is placed in aequilibrio, it strives to finde out the Poles of the world 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 ma [...] be divided by 4, and 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.

PROBLEM. LIII. How to make a consort of musick of many parts with one voyce, or one instrument only?

THis Probleme is resolved, so that a finger 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 answer, he may be­gin sol, and pronounce it in the same tune that [...]he Echo answereth, by which meanes you [...]ave a fifth, agreeable consort of musick: then [Page 79] in the same time that the Echo followeth, to sound the second note sol, he may sound forth another sol higher or lower to make an eight, the most perfect consort of musick, and so of others, if he will continue his voice 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 are many more parts in the musick of a Quire, then in effect truly there are because of the resounding and multiplying of the voic [...], and redoubling of the Quire.

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

THere are many fine wayes in Geometricall practices, to make an Ovall figure or one neare unto it, by severall centres: any of which I will not touch upon, but shew how it may be done promptly upon one centre 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 deep 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 be circular but ovall-wise: and a paire of Compasses may be so accommodated, that it may be done also upon a plaine thus. As let the length of the Ovall be H.K, fasten 2 pinnes or nailes 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 be­tween his legges: and placing one foot of this Compasse in the Cen­tre of the Ovall, and guiding the threed by the other foot of the Compasses, and so carry­ing it about: the spring will help to describe and draw the Ovall forme. But in stead of the Compasses it may be done with ones hand only, as in the figure may appeare.

PROBLEM. LV. Of a pu [...]se difficult to be opened.

IT is made to shut and open with Rings: first at each side there is a strap or string, as AB. [Page 81] and CD, at the end of which are 2 rings, B & D, and the string CD 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 CD, then over the purse, there is a piece of Leather EFGH, which covers the opening of the purse, and there is another piece of Leather AE, which passeth

through many rings: which hath a slit to­wards the end I, so great that the string BC 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 BC slide through the side at I, therefore bring down 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 seem 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 DC, 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 Centre of it?

IT is said that upon a time past, two Mathe­maticians met, and they would make tryall of their industry: the one made instantly a Perfect circle without Compasses, and the other immediately pointed out the Centre thereof with the point of a needle; now which is the chiefest action? it seems the first, for to draw the most noblest figure upon a plaine Table without other help than the hand, and the minde, is full of admiration; to finde the Cen­tre is but to finde out only one point, but to draw a round, there must be almost infinite points, equidistant from the Centre or middle; that in conclusion it is both the Circle and the Centre together. But contrarily it may seem that to finde the Centre 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 keeps alwayes the same distance, and is guided by a halfe di­stance to finish the rest; but he that must finde the Centre, must in the same time take heed to the parts about it, and choose one only point which is equall distant from an infinite of other [Page 83] points which are in the circumference; which is very difficult. Aristotle confirmes this amongst his morals, and seems 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 Centre of the end of a right mediocritie of a vertuous action; for to do well it must touch the middle point which is but one, and there must be a true point which respects the end, and that's but one only. Now to judge which is the most difficult, as before is said, either to draw the round or to finde the Centre, the round seems to be harder than to finde the Centre, because that in finding of it, it is done at once, and hath an equall distance from the whole; But, as before, to draw a round there is a visible point imagined, about which the circle is to be drawne. I esteeme that it is as difficult therefore, if not more, to make the cir­cle without a Centre, as to finde the middle or Centre of that circle.

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

THis is to be exercised upon a full Pack 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 shew the number of points that his three Cards which he took at the first did conteine. 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, & 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 be subtracted from the Cards, which re­maines to make up the game: if there remaine nothing after the Subtraction, then the num­ber of Cards remaining doth justly shew the number of points which were in the Cards cho­sen. If the Subtraction 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; [Page 85] so the first wants 5, the second wants 2, the third wants 7, and 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 be 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 be 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 privately multiply 15 by 5, makes 75, to which adde 5 makes 80, from this take 52 the number of Cards, rest 28, to vvhich add the remainder of Cards, make 34. the summe with 8, 6, 10, 3, 7.

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

TAke 15 Cards, and place them in 3 heaps in rank-wise, 5 in a heap: now suppose any one had thought one of these Cards in any one of the heaps, it is easie to finde vvhich of the Cards it is, and it is done thus; ask him in vvhich of the heaps it is, vvhich place in the middle of the other tvvo; then throvv dovvne the Cards by 1 and 1 into three severall heaps in rank-vvise, untill all be cast dovvne, then aske him [Page 86] in which of the rankes his Card is, which heap place in the middle of the other two heaps al­wayes, and this do foure times at least, so in put­ting the Cards altogether, look upon the Cards, or let their back be 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 these 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, and put these 4 away, then take 4 other Cards, and shew them in like manner to the second person, and bid him think any one of these Cards, and so do 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, and lastly, upon them these of the fourth person, then shew unto eaeh of these parties each of these ranks, and aske him if his Card be in it which he thought, for infallibly that vvhich the first partie thought upon vvill be in the first rank, and at the bottome, the Card of the second person vvill be in the second ranke, [Page 87] the Card of the third thought upon will be in the third rank, and the fourth mans Card will be in the fourth rank, and so of others, if there be more persons use the same method. This may be practised by other things, ranking them by certaine numbers: allotted to pieces of mo­ney, or such like things.

PROBLEM. LIX. How to make an instrument to help hearing, as Galileus made to help the sight?

THink not that the Mathematickes (which hath furnished us with such admirable helps for seeing) is wanting for that of hearing, its well knowne that long trunks or pipes make one heare well farre off, and experience shewes us that in certaine places of the Or­cades in a hollow vault, that a man speaking but softly at one corner thereof, may be audi­bly understood at the other end: notwithstand­ing those which are between the parties can­not heare him speak at all: And it is a generall principle, that pipes do greatly help to strengthen the activitie of naturall causes: we see that [...] 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 vvater be­ing contracted into a pipe, causeth a violence in its passage. The Glasses of Galeileus 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 those which walk in the adjacent Gardens, which is by certaine vessels and pipes that answer from the Garden to the Hall. Vitruvius makes men­tion also of such vessels and pipes, to strengthen the voice and action of Comedians: and in these times amongst many noble personages▪ the new kinde of trunkes are used to help 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 voice, that so by the pipe applied to the eare it may be more uniform and lesse in danger to dissipate the voice, and so consequently more fortified.

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

IT must be observed that the vessell in which the oile 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 solid matter: lastly, this second circle hath two pinnes, which may hang within some box to containe the whole lamp, in such manner, that there be 6 pinnes in different po­sition: Now by the aid of these pegges or pinnes, the lamp that is in the middle will be alwayes well situated according to his Centre of gravity, though it

be turned any way: though if you endea­vour to turne it up­side downe, it will lie levell▪ which is plea­sant and admirable to behold to those which know not the cause: And it is fa­cil from his to make a place to rest quiet in, though there be great a­gitation in the outvvard 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 think one of them, but let him note vvhether it be the first, second, third, &c. then vvith promptness learn vvhat number of Cards [Page 90] you had in your hands, and take the other part of the Cards, and place them 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 AB.C. carried apples to a marke 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 answer 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 she 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] cheap, but would sell them for 3 pence the peece, which she did: and so her apples came to 54 pence, and B having left but 13 apples sold them at the same rate, which came to 39 pence: 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 a 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 [...], of 9 and 5, of 10 and 4, of 11 and 3, of 12 and 2, of 13 and [...]. 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 multiplied 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, but in broken numbers there are infinite, whose summe and product will be equall one to ano­ther. As Clavius shewes upon the 36 Pro. of the 9t ^h book 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 multiplied by themselves, do 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▪ 50 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 [...]7 it makes 3 times 9, so vvhether the nines be rejected of 27, or of the summe of 2 and 7, it is all one, so if the nines vvere taken avvay of 240. it is all one, if the nines vvere taken avvay of 2, 4, and 0; for there vvould remaine 6 in either; and so of others.

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

Seventhly, the numbers 220 and 284 being unequall, notwithstanding the parts of the one number do alwayes equalize the other num­ber: so the aliquot parts of 220 are 110, 54, 44, 22, 20, 11, 10, 5, 4, 2, 1, which together makes 284. the aliquot 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 book of Euclide, was ground­ed, that the square of the Hypothenusal in any such Triangle, is equal to the square of the other two sides: that

is 5, the Hypothenusal multiplied in 5 makes 25, and 4 multiplied 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, be doubled, viz. 6, 8, 10: the square of 10 is equall to the quare 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 be applied thus, for the help of such which plot out Gardens, Houses, encamp Horse or Foot, &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 plat, a square building plat, or other long square. As suppose there is a figure [...]DGF. to be plotted, ED of 60 yards broad, and DG 100 yards long. First measure out ED 60 yards, 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 Triangle AB be in the line ED, which suppose at A make the cord AB fast in E and A, then put the other two cords of the Triangle untill they meet, 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 [Page 95] pegge; then at F, apply your Triangular cord, as you did at [...], and so may you draw the line FG as long as ED, viz. 60 yards. Lastly, it is easie to draw the line GD, and so the rectangu­led figure or long square shall be plotted, whose breadth is 60 yards, & length 100 yards, as was required: and to examine this, measure EG, then if FD be as long, the figure is true: other­wise it is defective, and may easily be amended.

I [...] one be taken from any square number which is odde, the square o [...] halfe of it being added to the first square, will make a square number.

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

If odde numbers be continually added from the unitie successively, there will be made all square numbers, and if cubick numbers be ad­ded successively from the unitie, there will be likewise made square numbers.

PROBLEM. LXIV. Of an excellent lamp, which serves or furnisheth it selfe with oile, and burnes a long time.

I Speak not here of a common lamp which Ca [...]danus writes upon in his book de subtilita­ [...], for that's a little vessell in columne-wise, [Page 96] which is full of Oile, and because there is but one little hole at the bottome neare the weeke or match; the oile runnes not, for feare that there be emptinesse above: when the match is kindled it begins to heat the lamp, and rare­fying the oile it issueth by this occasion: and so sends his more airie parts above to avoid va­cuitie.

But that which I

here deliver, is more ingenious, the princi­pall peece of which is a vessel, as CD. 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, near 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 oile running out will stop the hole at E, or throwing in oile into the vessell underneath, untill E be stop­ped; then the oile at C will not runne▪ because no aire can come into the pipe DE. Now as the oile burneth and consumeth in the vessell AB, the hole at E, will begin to be open, then immediately will C begin to runne to fill up AB. and E being stopped with the oile, the oile at C ceaseth to run.

[Page 197]It is certaine that such a lampe the Atheniaus used, which lasted a whole yeare without being touched: which was placed before the statue of Minerva, for they might put a certaine quan­titie of oile in the lamp CD, and a match to burne without being consumed: such as the naturalists write of, by which the lamp will furnish it selfe, and so continue in burning: and here may be noted that the oile may be poured in, at the top of th [...] vessell at a little hole, and 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 with 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 be de­monstrated, that if the hand of him that playes were so well assured by experience, as reason in­duceth one thereto; one might at one blow strike downe all the Keyles, of 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 strike it, that it should strike down the Keyles 2 and 5, and these might in their violence strike [Page 98]

downe the Keyles 3, 6, and 9, and the bowle being in mo­tion may strike down the Keyle 4, and 7; which 4 Keyle may strike the Keyle 8, & so all the 9 Keyles may be striken down at once.

PROBLEM. LXIV. Of Spectacles of pleasure.

SImple Spectacles of blew, yellow, red or green colour, are proper to recreate the sight, and will present the objects died in like colour that the Glasses are, only those of the greene do 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.

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

WHo 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 black 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 vvere 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 pene­trateth insensibly 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 naile, the point of a knife, or other pieces of Iron? Is it not a pleasure to see how the magnes will turne file dust, or move needles, or nailes be­ing upon a Table, or upon a piece of paper? for as soone as the magnes turnes or moves over, it moves also: who is it that would not be 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 run up and downe a Turret: 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 minds than a great massie substance of Iron to hang in the aire in the middest of a building without any thing in the world touching it, only but the aire? As some histories assure us, that by the aid of a Magnes or Adamant, placed at the roof of one of the Turkish Synagogues in Meca: the sepulchre of that infamous Mah [...]met rests sus­pended in the aire; and Plinie in his naturall Historie writes that the Architect or Democrates did begin to vault the Temple of A [...]sin [...]e in A­lexandria, with store of magnes to produce the like deceit, to hang the sepulchre of that God­desse likewise in the aire.

I should passe the bounds of my counter­poise, if I should divulge all the secrets of this [Page 105] stone, and should expose my selfe to the laugh­ter of the world: if I should brag to shew others the cause how this appeareth, than in its owne naturall sympathy, for why is it that a magnes with one end will cast the Iron away, & attract it with the other? from whence commeth it that all the magnes is not proper to give a true touch to the needle, but only in the two Poles of the stone: which is known by hanging the stone by a threed in the aire untill it be quiet, or placed upon a peece of Cork in a dish of water, or upon some thinne 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 only 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?

What's the cause that fire and Garlick takes away the propertie of the magnes? There are many great hidden mysteries 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 rrue cause thereof.

Some say, that by help of the Magnes per­sons which are absent may know each others [Page 106] minde, as if one being here at London, and a­nother at Prague 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 needle which is at Prague shall move, 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 moving 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 bowels to blow the fire.

THese are concave vessels of Brass or Copper or other material, 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 be hot there is no effect seen; but as­soone as the heat doth penetrate it, the water begins to rarefie, & issueth forth with a hidious and marvelous force; it is pleasure to see how it blowes the fire with great noise.

Vitruvius in his first book of Architecture, Cap. 8: approves from these Engines, that winde is no other thing than a quantity of vapours and ex­halations agitated with the aire by rare­faction and condensation, and we may draw a consequence from it, to shew that a little wa­ter [Page 109] may ingender a very great quantitie of va­pours and aire: for a Glasse of water throwne into an Aeolipile will keep blowing neare a vvhole houre, sending forth his vapours a thousand times greater than it is extended.

Novv touching the forme of these vessels, they are not made of one like fashion: some makes them like a bovvle, some like a head painted representing the vvinde, some make them like a Peare: as though one vvould put it to rost at the fire, vvhen one vvould have it to blovv, for the taile of it is hollovv, in forme of a funnell, having at the top a very little hole no greater than the head of a pinne.

Some do accustome to put vvithin the Aeo­lipile a crooked funnell of many foldings, to the end that the vvinde that impetuously rolles▪ to and fro vvithin, may imitate the noise of thun­der. Others content themselves vvith a simple funnell placed right upvvard, somevvhat vvi­der at the top than elsevvhere like a Cone, vvhose basis is the mouth of the funnell: and there may be placed a bovvle of Iron or Brasse, vvhich by the vapours that are cast out vvill cause it to leap up, and dance over the mouth of the Aeolipile.

Lastly, some apply near to the hole smal Wind­mils, or such like, vvhich easily turne by reason of the vapours; or by help of tvvo or more bovved funnels, a bowle may be made to turne [...] these Aeolipiles are of excellent use for the melt­ing of mettalls and such like.

[Page 110]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.

Heat the Aeolipiles being empty, and the aire which is within it will become extreamely rarefied; then being thus hot throw it into wa­ter, and the aire will begin to be 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, yellow, green, or such like: such as is not thick. This being done the use may be thus.

First, I say, that as the aire inclosed in the Thermometer is rarefied or condensed, the water will evidently ascend or descend in the Cylin­der: which you may try easily by carrying the Thermometer 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 [Page 111] ball of the Thermometer: the Glasse being so thinne, and the aire so capable of rarefaction, that at the very instant you may see the water descend: and your hand being taken away, it will softly ascend to his formes place againe. This is yet more sensible when one heats the ball at the top 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 rarefie 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 heat and cold, which are in the aire each houre of the day; forasmuch as the exterior aire is either hot or cold, the aire which is inclosed in the Thermometer doth likewise either rarefie 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 be full, but [Page 112] in Summer, it will descend so low that scarce there will be perceaved in it any water at all.

Those 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 Physicians, dividing each of these 8 into 8 others: to have in all 64 divisions, & by this vvay they may not only distinguish upon vvhat degree the vvater ascendeth in the morning, at midday, & at any other houre: but also one may knovv hovv much one day is hotter or colder than another: by marking hovv many degrees the vvater ascend­eth or descendeth, one may compare the hot­test and coldest dayes in a vvhole year together vvith these of another year: againe one may knovv hovv much hotter one roome is than an­other, by vvhich also one might keep a cham­ber, a furnace, a stove, &c. alvvayes in an equalitie of heat, by making the vvater of the Thermo­meter rest alvvayes upon one & the same degree: in brief, one may judge in some measure the burning of Fevers, and neare unto what exten­sion the aire can be rarefied by the greatest heat.

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

PROBLEM. LXX. Of the proportion of humane 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, & according to the Max­ime of Philosophers, that which is most perfect and the first in rank, measureth all the rest.

Secondly, because in effect the ordinary mea­sure of a foot, the inch, the cubit, the pace, have taken their names and greatnesse from humane bodies.

Thirdly, because the symmetrie 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 pieces 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 breadth 50 Cubits, in height or depth 30 cubits, so that the length containes the breadth 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 Solo­mon (that chieftaine of works) was modula­ted all of good Architecture, and curiously to be observed in many pieces to keep the same proportion as the body to his parts: so that by the greatnesse of the work 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 foot; the top, and roofe as the head; the rest as the body: those which have beene somewhat more curious, have noted that as in humane 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 equality in the same Architecture.

In like manner, some have been 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 vvrithing of haire, the channells of columnes, to the fouldings of vvomens Robes, &c.

Novv building being a vvork of the best Ar­tist, there is much reason vvhy man ought to make his imitation from the chiefe vvork of na­ture; vvhich is man.

Hence it is that Vitru [...]ius in his third book, [Page 115] and all the best Architectes, treate of the proportion of man; amongst others Albert Durens hath made a whole book of the mea­sures of mans body, from the foot to the head, let them read it who wil, they may have a prefect knowledge thereof: 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. Andrews Crosse, placing one foot 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 neck, make 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 thicknesse of the body taken from the belly to the back; 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 between 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 work; and by the same is made most evident, that which is related of the images and statues of Greece, that upon a day diverse workmen 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 help of proportion, the body of Hercules was measured by the knowledge of his foot onely, a Lion by his claw, the Giant by his thumb, and a man by any part of his body. For so it was that Pythagoras having measured the length of Hercules foot, by the steps which were left upon the ground, found out all his height: and so it was that Phidias having onely the claw of a Lion, did figure and draw out all the beast ac­cording to his true type or forme, so the exqui­site Painter Timantes, having painted a Pygmey or Dwarfe, which he measured with a fadome made with the inch of a Giant, it was suffici­ent [Page 117] to know the greatnesse of that Giant-

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

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

THe Mathematickes often findeth place in sundry Games to aid 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 Ball or Bowle by Geometrical lines and angles.

But the exercise, experience, and dexteritie [...] the player seems more in this action than an [...] 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 trapstick striketh the Ball, the moving of the Ball is made in a right line, which is drawne from the Centre of the Bowle by the point of contingencie.

Secondly, in all kinde of such motion; when a Ball 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 canclude, first, in what part of the wood or wall, one may make the Bowle or Ball go 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 go and meet with the third, keep­ing the reflection or Angle of incidence equal.

Thirly, how one may touch a Bowle to send it to what part one pleaseth: such and many other practices may be done. At the exercises at Keyls there must be taken heed that the motion slack or diminish by little and little, and [Page 124] may be 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 applied, as before in part is shewed, and here I will say something 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 were appointed 9 men to defend each front thereof, some at the gates, & the other at each corner or Angle, so that each Angle served to assist two faces of the square, if need 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 be entertained into service, who told them he 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] keep still the order

of 9, for each face of the square to defend the Angles & Gates, to which the Gover­nours agreed, and these Souldiers being there some few weeks liked not their service but indeavoured to remove themselves, and so laboured with some of the the rest; that each of these foure Souldiers took away his cumrade with him, and so departed; yet left to defend each side of the passage, and how may this be?

It's 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 Cum­rade, and placing two of these men which are at each Gate to each Angle, there will be 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? & this is according to the figures, F.G.H

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

THis is a miracle in musick, yet easie to be experimented. Take a Viole or other Instru­ment, and choose two strings, so that there be one between them; make these two strings, a­gree in one and the same tune: then move the Viole-bowe 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 sensibly shake without any one touch­ing it; and it is more admirable that the string which is between 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 sensibly shake.

Now it may be demanded, how comes this shaking, is it in the occult sympathie, or is it in the strings being wound up to like notes or tunes, that so easily the other may receive the impression of the aire, which is agitated or moved by the shaking or the trembling of the other? & whence is it that the Viole-bowe mo­ved upon the first string, doth instantly in the same time move the third string, and not the second? if the cause be not either in the first or second? I leave to others to descant on.

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

THe vessell is thus made, it must be divided into three Cells for to conteine the three li­quors, which admit to be Sack, Claret, and White-wine: Now in the bung-hole there is an Engine with three pipes, each extending to his proper Cell, into which there is put a broach or funnell pierced in three places, in such sort, that placing one of the holes right against the pipe which answereth unto him, the other tvvo pipes are stopped; then vvhen it is full, turne the fun­nel, and then the former hole vvill be stopped, and another open, to cast in other vvine vvith­out mixing it vvith the other.

Novv to dravv out also vvithout mixture, at the bottome of the vessell there must be placed a pipe or broach, vvhich may have three pipes; and a cock piersed vvith three holes so artifici­ally done, that turning the cock, the whole vvhich ansvvereth to such of the pipes that is placed at the bottom, may issue forth such vvine as belongeth to that pipe, & turning the Cock to another pipe, the former hole vvil be stopped; [Page 129] and so there will issue

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

PROBLEM. LXXV. Of burning-Glasses.

IN this insuing discourse I will shew the in­vention of Prom [...]theus, how to steale fire from Heaven, and bring it down 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-brands to wake them burne, melt Lead, [...]inne, Gold, and Silver, in a little time▪ with as great ease as though it had been put into a Cruzet over a great fire.

Have you not read of Archimedes of Syracu­ [...], who when he could not come to the Ships of Ma [...]cellus ▪ which besieged that place, to hin­der and impeach their aproach, he flung huge stones by his Ingines to sink them into the Sea, and transformed himselfe into Jupiter, thunder­ing downe from the highest Towers of the [Page 130] Town, his thunder-bolts of lightning into the Ships causing a terrible burning, in des­pite

of Neptune and his watery region: Z [...] ­naras witnesseth that Proclus a brave Ma­thematician, burned in the same manner the Ships of Vitali­an, which were come to besiege Constanti­nople; and daily 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 cloaths.

But this is nothing to the burning of those Glasses which are hollow, namely those which are of steele well polished, according to a pa­r [...]bolicall or ovall section. A sphericall Glasse, or that which is according to the segment of a Sphere, burnes very effectually about the fourth part of the Diameter; notwithstanding the Pa­rabolie and Ecliptick sections have a great ef­fect: by which Glasses there are also diverse fi­gures represented forth to the eye.

The cause of this burning is the uniting of the beames of the Sunne, which heat migh­tily in the point of concourse or inflammation, [Page 131] which is either by transmissi [...]n 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 promise to make Glasses which shall burne a great distance off, but yet not seen vulgarly produced, of which if they were made, the Parabolie makes the greatest eff [...]ct, and is g [...]nerally held to be the invention of Archimedes or Pro [...]us.

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 shadow, or neare such a place where the Sunne shines not, which is by help of a flat Glasse, by which may be made a percussion of the beames of the Sun into the concave Glasse, adding unto it that it serves to good use to put fi [...]e to a Mine, provided that the combustible matter be well applyed before the concave Glasse; in which he saies true: but because all the effect of the practice depends upon the placing of the Glasse and the Powder which he speaks not of: I will deliver here a rule more generall.

How one may place a Burning-glasse with his combust [...]ble matter in such sort, that at a con­venient houre of the day, the Sun shining, it shall take fire and burne: Now it is certaine [Page 132] that the point of inflammation or burning, is changed as the Sun changeth place, and no more nor lesse, than the shadow turnes about the style of a Dyall; therefore have regard to the Suns motion, and [...]is height and place: a Bowle of Crystall in the same place that the top of the style 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 Suns arch for that day: now the Sunne comming to the houre of 12, to [...], 2, [...], &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 m [...]ny ple [...]sant Questions by way of Arithmetick [...].

J Will not in [...]ert i [...] this Probleme that which is drawne from the [...]reek Epigrams, but proposing the Question immediately will give the an [...]wer also, without [...]aying to shew the manner how they are answered; in this J will [...] be tied to the [...]reek tearms, w [...]ch J ac­count no [...] proper to this place, nei [...]er to my purpose: [...]et t [...]o [...]e [...]ead that will Di [...]phanta S [...]biliu [...] upon Eu [...]li [...] and others, and they may be satisf [...]ed

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 vvhich reflect: therefore I vvill novv produce some experiments upon them, not that vvill dive into their depth (that vvere to lay open a mysterious thing) but that vvhich may delight and recreate the spirits: Let us suppose therefore these principles, upon which is built the demonstration of the appa­r [...]nces which are made [...]n all sort of Glasses.

First, that the rayes or beames, vvhich reflect upon a Glasse, make the Angle of incident equall to the Angle of Reflection, by the first Theo. of the Catoptick of Euc.

Secondly, that in all plain Glasses, the Images are seen in the perpendicular line to the Glasse, [Page 142] as far within the glass as the object is without it.

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

And here you are to understand, that there is not meant only those 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, Mettal, 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 be placed three Glasses in the two Chambers, of which one of them shall be 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 FE, shall fall upon the Glasse BA, and reflect upon the superficies of the Glasse BC, in the point G; so that if the

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

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

AS let the eye be at O ▪ and the mark C, place a plaine Glasse perpendicular as AB. so the marke C shall be seen in Catheti CA, viz. [Page 163] in D, and the line of

reflexion is D, now let the Musket FE, upon a rest▪ be mo­ved to and fro untill it be seen in the line OD, which admit to be HG, so giving fire to the Musket, it shall undoubtedly strike the mark.

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

TAke two Glasses, and place them at right Angles one unto the other, as admit AB, and CB, of which admit CB, Ho [...]izontall, and let the eye be at H, and the object or image to

be DE; so D will be reflected at F, so to N, so to HE: then at G, so to [...] and then to H, and by a double reflection ED will seeme in QR, the highest point D in R, and the point L in Q inversed as was said, taking D for the head, and E for the feet; so it will be a man inversed, which will seem to be flying in the aire, if the Jmage had wings unto it, and had secretly [...] 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 seen by that motion.

PROBLEM. LXXXI. How to make a company of representative Souldiers seeme to be a Regiment, or how few in number may be multiplyed to seem 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 GH. and FI, one placed right against another per­pendicular to the Horizon, upon a plaine levell Table: betvveene

vvhich Glasses let there be ranged in Battalia-vvise upon the same Table a number of small men according to the square G, H, I, F, or in any other forme or posture: hen may you evidently see hovv the said battel vvill be multiplyed and seem farre bigger in the appear­ance than it is in effect.

PROBLEM. LXXXII. Of fine and pleasant Dyal [...].

COuld you choose a more ridiculous one than the natural Dyall written amongst the Greek Epigrams, upon which some sound Poet made verses; shewing that a man carrieth a­bout him alwayes a Dyall in his face by meanes of the Nose and Teeth? and is not this a jolly Dyall? for he need not but open the mouth, the lines shall be all the teeth, and the nose shall serve for the style.

PROBLEM. LXXXIII. Of Cannons or great Artillery. Souldiers, and o­thers would willingly see [...] Problems, which containe: three or foure subtile questions:

PROBLEM. LXXXIIII. Of predigious progression and multiplication, of Creatures, Plants, Fruits, 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 needs 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 Arithmetick, and first of the number of sands.

IT may be said incontinent, that to undertake this were impossible, either to number the Sands of Lybia, or the Sands of the Sea; and it vvas this that the Poets sung, and that vvhich the vulgar beleeves; nay, that vvhich long a­go certaine Philosophers to Gelon King of Sici­ly [Page 209] reported, that the graines of sand vvere in­numerable: But I ansvvere vvith Archimedes, that not only one may number those vvhich are at the border and about the Sea; but those vvhich are able to fill the vvhole vvorld, if there vvere nothing else but sand; and the graines of sands admitted to be so small, that 10 may make but one graine of Poppy: for at the end of the account there need not to expresse them, but this number 30840979456, and 35 Ciphers at the end of it. Clavius and Archimedes make it somevvhat 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 are requisite to fill another vvorld, in comparison that our visible vvorld vvere but as one graine of sand, an a­tome or a point; for there is nothing to do but to multiply the number by it selfe, vvhich vvill amount to ninety places, vvhereof tvventie are these, 95143798134910955936, and 70 Ci­phers at the end of it: vvhich 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 bovvle of a graine of Poppy, vvith a bovvle of an inch or of a foot, & that to be compared vvith that of the earth, and then that of the earth vvith that o the firmament; and so of the rest.

PROBLEM. LXXXVII. Witty suits or debates between Caius and Sempro­nius, upon the forme of f [...]gures, which Geo­metricians call Isoperimeter, or equall in circuit or compasse.

MArvell [...]ot at it if I make the Mathema­ticks take place at the Ba [...]e, and if I set [Page 215] forth here B [...]rtoleus, who witnesseth of him­selfe, that being then an ancient Doctor in the Law, he himselfe took upon him to learne the elements and principles of Geometry, by which he might set forth certaine Lawes touching the divisions of Fields, Waters, Islands, and other incident places: now this shall be to shew in

passing by, that these sciences are profi­table and behovefull for Judges, Coun­sellors, or such, to ex­plaine many things which fall out in Lawes, to avoid am­biguities, contentions, and suits often.

PROBLEM. LXXXVIII. Containing sundry Questions in matter of Cosmography.

FIrst, it may be demanded, vvhere is the mid­dle of the vvorld? I speak not here Mathe­matically, but as the vulgar people, vvho ask, vvhere is the middle of the vvorld? in this sence to speak absolutely there is no point vvhich may be said to be the middle of the surface, for the middle of a Globe is every vvhere: not­vvithstanding the Holy Scriptures speake re­spectively, and make mention of the middle of the earth, and the interpreters apply it to the Citie of Jerusalem placed in the middle of Pa­lestina, and the habitable vvorld, that in effect taking a mappe of the vvorld, and placing one foot of the Compasses upon Jerusalem, and ex­tending the other foot to the extremity of Eu­rope, Asia, and Afric [...], you shall see that the Citie of Jerusalem is as a Centre to that Circle.

PROBLEM. XCII. To finde the Bissextile yeare, the Dominicall letter, and the letters of the moneth.

LEt 123, or 124, or 125, or 26, or 27, (which is the remainder of 1500, or 1600) be divi­ded by 4, which is the number of the Leape-yeare, and that which remaines of the division shewes the leap-yeare, as if one remaine, it shewes that it is the first yeare since the Bis­sextile or Leap-year, if two, it is the second year &c. and if nothing remaine, then it is the Bis­sextile or Leap-yeare, and the Quotient shews you how many Bissextiles or Leap-yeares there are conteined in so many yeares.

PROBLEM. XCIII. To finde the New and Full Moone in each Moneth.

ADde to t [...]e Epact for the yeare, the Moneth from March, then subtract that surplus from 30, and the rest is the day of the Moneth that it vvill be New Moone, and adding unto it 14, you shall have that Full Moone.

PROBLEM. XCIV. To finde the Latitude of [...] Countrey.

THose that dwell between the North-Pole and the Tropicke of Cancer, have their Spring and Summer between the 10 of March, and the 13 of September: and therefore in any day between that time, get the sunnes distance by instrumentall observation from the zenith at noone, and adde the declination of the sun for that day to it: so the Aggragate sheweth such is the Latitude, or Poles height of that Coun­trey. Now the declination of the sunne for any day is found out by Tables calculated to that end: or Mechanically by the Globe, or by In­strument it may be indifferently had: and here note that if the day be between the 13 of Sep­tember and the 10 of March, then the sunnes declination for that day must be taken out of the distance of the sunne from the zenith at noone: so shall you have the Latitude, as be­fore.

PRBOLEM XCV. Of the Climates of countreys, and to finde in what C [...]imate any countrey is under.

CLimates as they are taken Geographically signifie nothing else but when the l [...]ngt [...] [Page 226] of the longest day of any place, is half an houre longer, or shorter than it is in another place (and so of the sh [...]rtest day) and this account to begin from the Equinoctia [...]l Circle, seeing all Countreys under it have the shortest and lon­gest day that can be but 12 houres; But all o­ther Countreys that are from the Equinoctiall Circle either towards the North or South of it unto the Poles themselves, are said to be in some one Climate or other, from the Equinoctiall to either of the Poles Circles, (which are in the Latitude of 66 degr. 30 m.) between each of which Polar Circles and the Equinoctial Circle there is accounted 24 Climates, which differ one from another by halfe an hours time: then from each Polar Circle, to each Pole there are rec­koned 6. other Climates which differ one from another by a moneths time: so the whole earth is divided into 60 Climates, 30 being allotted to the Northerne Hemisphere, and 30 [...] to the Southerne Hemispheare. And here note, that though these Climats which are betweene the Equinoctiall and the Polar Circles are equall one unto the other in respect of time, to wit, by halfe an houre; yet the Latitude, breadth, or internall, conteined between Climate and Climate, is not equall: and by how much any Climate is farther from the Equinoctiall than another Climate, by so much the lesser is the intervall between that Climate and the next: so those that are nearest the Equinoctial are larg­est, [Page 227] and those which are farthest off most con­tracted: and to finde what Climate any Coun­trey is under: subtract the length of an Equi­noctiall day, to wit, 12 houres from the length of the longest day of that Countrey; the re­mainder being doubled shews the Climate: So at London the longest day is neare 16 houres and a halfe; 12 taken from it there remaines 4 houres and a halfe, which doubled makes 9 halfe houres, that is, 9 Climates; so London is in the 9 climate.

PROBLEM. XCVI. Of Longitude and Latitude of the Earth and of the Starres.

LOngitude of a Countrey, or place, is an arcke of the Aequator conteined between the Me­ridian of the Azores, and the Meridian of the place, and the greatest Longitude that can be is 360 degrees.

PROBLEM. XCVII. To make a Triangle that shall have three right Angles.

OPen the C [...]passes at p [...]easure: and upon A, describe an Arke BC. then at the same o­pening, place one of the feet in B, and describe [Page 235] the Ark AC. Lastly, place one of the feet of the Compasses in C. and describe the Arke AB· so shall you have the

sphericall Aequilate­rall Triangle ABC, right angled at A, at B, and at C. that is, each angle compre­hended 9 [...]. degrees: which can never be in any plaine Tri­angle, whether it be Equilaterall, Isocelse, scaleve, Orthogonall, or Opi­gonall.

PROBLEM. XCVIII. To divide a line in as many equall parts as one will, without compasses, or without seeing of it.

THis Proposition hath a fallacie in it, & can­not be practised but upon a Maincordion: for the Mathematicall line which proceeds from the flux of a point, cannot be divided in that wise: One may have therefore an Instru­ment which is called Maincordion, because there is but one cord: and if you desire to di­vide your line into 3 parts, run your finger up­on the frets untill you sound a third in musick: if you would have the fourth part of the line, [Page 236] then finde the fourth sound, a fifth, &c. so shall you have the answer.

PROBLEM. XCIX. To draw a line which shall incline to another line, yet never meet: against the Axiome of Parallels.

THis is done by help of a Conoeide line, pro­duced by a right line upon one & the same plaine, held in great account amongst the An­cients, and it is drawne after this manner.

Draw a right line infinitely, and upon some end of it, as at I, draw a perpendicular line I

A. augment it to H. then from A. draw lines at pleasure to intersect the line I. M. in each of which lines from the right line, IM. transferre IH. viz. KB.LC.OD.PE.QF.MG. then from those points draw the line H.B.C.D.E.F.G. which will not meet with the line IM. and yet incline nearer and nearer unto it.

PROBLEM. C. To observe the variation of the compasses, or needle in any places.

FIrst describe a Circle upon a plaine, so that the Sun may shine on it both before noone and afternoone: in the centre of which Circle place a Gn [...]on or wire perpendicular as AB, and an houre before noone marke the extremitie of the shadow of AB, which suppose it be at C. describe a Circle at that semidiamiter CDF. then after noone mark when the top of the shadow of AB. toucheth the Circle, which admit in D; divide the distance CD into two equall parts, which suppose at E. draw the line EAF. which is the Meridian line, or line of North & South: now

if the Arke of the Circle CD. be divi­ded into degrees. place a Needle GH, upon a plaine set up in the Centre, and marke how many de­grees the point of the Needle G, is from E. so much doth the Needle vary from the North in that place.

PROBLEM. CI. How to finde at any time which way the wind is in ones Chamber, without going abroad?

VPon the Plancking or floore of a Chamber, Parlor, or Hall, that you intend to have this device, let there come downe from the top of the house a hollow post, in which place an Iron rod that it ascend above the house 10, or 6

foot with a vane or a scouchen at it to shew the winds with­out: and at the lower end of this rod of Iron, place a Dart which may by the moving of the vane with the wind with­out, turne this Dart which is within: about which upon the plaister must be described a Circle divided into the 32 points of the Mari­ners Compasse pointed and distinguished to that end, then may it be marked by placi [...] to Compasse by it; for having noted the North point, the East, &c▪ it is easie to note all the rest of the points: and so at any time comming into this Roome, you have nothing to do but to look up to the Dart, which will point you out what way the winde bloweth at that in­stant.

PROBLEM. CII. How to draw a parallel sphericall line with great ease?

FIrst draw an obscure line GF. in the middle of it make two points AB, (which serves for Centres then place one foot of the Compasses in B, and extend the other foot to A, and de­scribe the semicircle AC. then place one foot of the Compasses in A, and extend the other foot to C, and describe the semicircle CD. Now place the Compasses in B, and extend the other foot unto D, and describe the semicircle DF, and so ad infinitum; which being done neatly, that there be

no right line seene nor where the Com­passes were placed, will seeme very strange how possibly it could be drawne with such exactnes, to such which are ig­norant of that way.

PROBLEM. CIII. To measure an in accessible distance, as the breadth of a River with the help of ones hat onely.

THe way of this is easie: for having ones hat upon his head, come neare to the bank of the River, and holding your head upright (which may be by putting a small stick to some one of your buttons to prop up the chin) pluck downe the brim or edge of your hat untill you may but see the other side of the water, then turne about the body in the same posture that it was before towards some plaine, and marke where the sight by the brimme of the hat glaunceth on the ground▪ for the distance from that place to your standing, is the breadth of the River required.

PROBLEM. CIIII. How to measure a height with two strawes or two small stickes.

TAke two strawes or two stickes which are one as long as another, and place them at right Angles one to the other, as AB. and AC. then holding AB. parallel to the ground, place the end A to the eye at A. and looking to the other top BC. at C. by going backward or for­ward [Page 241] untill you may

see the top of the Tower or tree, which suppose at E. So the distance from your standing to the Tow­er or Tree, is equall to the height thereof above the levell of the eye: to which if you adde your ovvne height you have the whole height.

PROBLEM. CV. How to make statues, letters, bowles, or other things which are placed in the side of a high build­ing, to be seen below of an equall bignesse.

LEt BC. be a Pillar 7 yards high, and let it be required that three yards above the le­vell of the eye A, viz. at B. be placed a Globe, and 9 yards above B. be placed another, & 22. yards above that be placed another Globe: how much shall the Diameter of these Globes be, that at the eye, at A,

they may all appeare to be of one and the same Magnitude: It is thus done, first draw a line as AK. and upon K. erect a perpendicular KX. divide this line into 27 parts▪ and accord­ing to AK. describe an Arke KY. then from K▪ in the perpendicular KX, account [...] ▪ par [...]s, viz at L. which shall re­present the former three yardes, and draw the line LA. from L, in the said perpendicular reckon the diameter of the lesser Globe of what Magnitude it is intended to be: suppose SL, and draw the line SA. cutting the Arke VK. in N. then from K. in the perpendicular ac­count 9 yards, which admit at T. draw TA, cut­ting YK. in O transferre the Arke MN, from [Page 243] A to P. and draw AP. which will cut the per­pendicular in V. so a line drawne from the mid­dle of VF. unto the visuall lines AI, and AV, shall be the diameter of the next Globe: Lastly, account from K. in the perpendicular XK. 22 parts, and draw the line WA. cutting YK in Q. then take the Arke MN, and transferre it from Q to R and draw AR▪ which will cut the per­pendicular in X so the line which passeth by the meddle of XW. perpendicular to the visuall line AW, and AX. be the Diameter of the third Globe, to wit 5, 6. which measures transferred in the Pillar BC. which sheweth the true Mag­nitude of the Globes 1, 2, 3. from this an Archi­tect or doth proportion his Images, & the fould­ing of the Robes which are most deformed at the eye below in the making, yet most perfect when it is set in his true height above the eye.

PROBLEM. CVI. How to disg [...]is [...] or disfigure an Image, as a head, an arme, a whole body, &c. so that it hath no pro­portion the eares to become long: the nose as that of a swan, the mouth as a coaches en­trance, &c yet the eye placed at a certaine point will be seen in a direct & exact proportion.

I Will not strive to set a Geometricall figure here, for feare it may seeme too difficult to un­derstand, [Page 244] but I will indeavour by discourse how Mechanically with a Candle you may perceive it sensible: first there must be made a figure upon Paper, such as you please, according to his just proportion, and paint it as a Picture (which painters know well enough to do) af­terwards put a Candle upon the Table, and in­terpose this figure obliquely, between the said Candle and the Bookes of Paper, where you desire to have the figure disguised in such sort that the height passe athwart the hole of the Picture: then will it carry all the forme of the Picture upon the Paper, but with deformity; follow these tracts and marke out the light with a Coles black head or Ink: and you have your desire.

To finde now the point where the eye must see it in his naturall forme: it is accustomed ac­cording to the order of Perspective, to place this point in the line drawne in height, equall to the largenesse of the narrowest side of the deformed square, and it is by this way that it is performed.

PROBLEM. CVII. How a Cannon after that it hath shot, may be covered from the battery of the enemy.

LEt the mouth of a Cannon be I, the Cannon M. his charge NO, the wheele L, the axle­tree PB. upon which the Cannon is placed, at [Page 245] which end towards B, is placed a pillar AE· supported with props D, C, E, F, G ▪ about which the Axeltree

turneth: now the Cannon being to shoot, it retires to H, which cannot be di­rectly because of the Axletree, but it make a segment of a circle▪ and hides himselfe behind the wal QR, and so preserves it selfe from the Enemies bat­tery, by which meanes one may avoid many inconveniences which might arise: and more­over, one man may more easily replace it againe for another shot by help of poles tyed to the wall, or other help which may multiply the strength.

PROBLEM. CVIII. How to make a Lever, by which one man may alone place a Cannon upon his carriage, or raise what other weight he would.

FIrst place two thick boards upright, as the figure sheweth, pierced with holes, alike opposite one unto another as CD, and EF: & let L, and M, be the two barres of Iron which passeth through the holes GH, and F, K, the [Page 246]

two supports, or props, AB. the Can­non, OP, the Lever, RS, the two notches in the Lever, and Q, the hooke where the burthen or Cannon is tyed to. The rest of the operation is [...]cill, that the young­est schollers or lear­ners cannot faile to performe it: to teach Mi­nerva were in vaine, and it were to Mathema­ticians injury in the succeeding Ages.

PROBLEM. CIX. How to make a Clock with one onely wheele.

MAke the body of an ordinary Dy­all, and divide the houre in the Circle into 12. parts: make a great wheele in height above the Axletree, to the which you shall place the cord of your counterpoize▪ so that it may descend, that in 1 [...] [Page 247] houres of time your Index or Needle may make one revolution, which may be knowne by a watch which you may have by you: then put a balance which may stop the course of the Wheele, and give it a regular motion, and you shall see an effect as just from this as from a Clock with many wheeles.

PROBLEM. CX. How by help of two wheeles to make a Childe to draw up alone a hogshead of water at a time: and being drawne up shall cast out it selfe into another vessell as one would have it.

LEt R be the Pit from whence water is to be drawne, P the hook to throw out the water when it is brought up (this hook must be moveable) let AB be the Axis of the wheele SF, which wheele hath divers forkes of Iron made at G, equally fastened at the wheele; let I, be a Card, which is drawne by K, to make the wheele S, to turne, vvhich vvheele S, beares proportion to the vvheele T, as 8 to [...]. let N be a Chaine of Iron to vvhich is tyed the vessel O, and the other vvhich is in the Pit: E [...] is a piece of vvood vvhich hath a mortes in 1, and [...], by vvhich the Cord I, passeth, tyed at the vvall, as KH, and the other piece of timber of the little vvheele as M, mortified in likevvise for the [Page 248] chaine to passe through: draw the Cord I, by K, and the wheele will turne, & so consequent­ly the wheele T, which will cause the vessell O to raise; which being

empty, draw the cord againe by Y, and the other vessell which is in the pit [...]ill come out by the same rea­son. This is an in­vention which will save labour if practi­sed; but here is to be noted that the pit must be large enough, to the end that it conteine two great vessels to passe up and downe one by another▪

PROBLEM. CXI. To make a Ladder of Cords, which may be carryed in ones pocket: by which one may easily mount up a Wall, or Tree alone.

TAke two Pullies A, & D, unto that of A, let there be fastened a Cramp of Iron as B; and at D, let there be fastened a staffe of a foot and a halfe long as F, then the Pully A: place a hand of Iron, as E, to vvhich tie a cord of an halfe inch thick (vvhich may be of silk because it is for the pocket:) then strive to make fast the [Page 249] Pully A, by the help of the Crampe of Iron B, to the place that you intend to scale; and the staffe F, being tyed at the Pully D, put it be­tvveen your legges as though you vvould sit upon it: then hold­ing

the Cord C in your hand, you may guide your selfe to the place required▪ vvhich may be made more facill by the multiplying of Pul­lies. This secret is most excellent in Warre, and for lo­vers, its supportablenesse avoids suspition.

PROBLEM. CXII. How to make a Pumpe whose strength is marve­lous by reason of the great weight of water that it is able to bring up at once, and so by continuance.

LEt [...], be the height of the case about two or three foot high, and broader ac­cording to discretion: the rest of the Case or concavity let be O: let the sucker of the Pumpe vvhich is made, be just for the Case or Pumpes head [...], & may be made of vvood or brasse of 4 inches thick, having a hole at E, vvhich de­scending [Page 250]

raiseth up the cover P, by which issueth forth the water, & ascend­ing or raising up it shuts it or makes it close: RS, is the handle of the sucker tyed to the handle TX, which works in the post VZ. Let A, B, C, D, be a piece of Brasse, G the piece which enters into the hole to F, to keep out the Aire. H, I, K, L, the piece tyed at the funnell or pipe: in which playes the Iron rod or axis G, so that it passe through the other piece MN, which is tyed with the end of the pipe of Brasse.

Note, that the lower end of the Cisterne ought to be rested upon a Gridiron or Iron Grate▪ which may be tyed in the pit, by which means lifting up and putting downe the handle, you may draw ten times more water than o­therwise you could.

PROBLEM. CXIII. How by meanes of a Cisterne, to make water of a Pit continually to ascend without strength, or the assistance of any other Pumpe.

LEt IL, be the Pit where one would cause water to ascend continually to [...]ach office [Page 251] of a house or the places which are separated from it: let there be made a receive [...] as A, well closed up with lead or other matter that aire enter not in, to which fasten a pipe of lead as at E, which may have vent at pleasure, then let there be made a Cisterne as B, which may be communicative to A, by helpe of the pipe G, from vvhich Cistern

B, may issue the vva­ter of pipe D, vvhich may descend to H, vvhich is a little be­lovv the levell of the vvater of the pit as much as is GH. to the end of vvhich shall be soldered close a Cock vvhich shall cast out the vvater by KH. Novv to make use of it, let B be filled full of vvater, and vvhen you vvould have it run turne the Cock, for then the vvater in B, vvill descend by K. and for feare that there should be vacuity, nature vvhich abhors it, vvill labour to furnish and supply that emptinesse out of the spring F, and that the Pit dry not, the Pipe ought to be small of an indifferent capacity according to the greatnesse or smalnesse of the spring.

PROBLEM. CXIIII. How out of a fountaine to cast the water very high: different from a Probleme formerly delivered.

LEt the fountaine be BD, of a round forme (seeing it is the most capable and most per­fect figure) place into it two pipes conjoyned as EA, and HC, so that no Aire may enter in at the place of joyning: let each of the Pipes

have a cock G, & L: the cocke at G, being closed, open that at I ▪ & so with a squirt force the water through the hole at H, then close the Cocke at A, & draw out the squirt, and open the cock at G. the Aire being before rarified will extend his dimensions, and force the water with such vio­lence, that it will amount above the height of one or two Pipes: and so much the more by how much the Machine is great: this violence will last but a little while if the Pipe have too great an opening, for as the Aire approacheth to his naturall place, so the force will diminish.

PROBLEM. CXV. How to empty the water of a Cisterne by a Pipe which shall have a motion of it selfe.

LEt AB, be the vessell; CDE, the Pipe: HG, a little vessell under the greater, in which one end of the Pipe is, viz. C, and let the o­ther end of the Pipe E. passing through the bottome of the ves­sell

at F, then as the vessell filleth so will the Pipe, and when the vessell, shall be full as farre as PO, the Pipe will begin to runne at E, of his owne accord, and ne­ver cease untill the vessell be wholly empty.

PROBLEM CXVI. How to squirt or spout out a great height, so that one pot of water shall last a long time.

LEt there be prepared two vessels of Brasse, Lead, or of other matter of equal substance as are the two vessels AB, and BD, & let them be joyned together by the two Pillars MN, & EF: then let there be a pipe HG. which may passe through the cover of the vessell CD, and passe through AB, into G, making a little bunch or rising in the cover of the vessell AB, so that the pipe touch it not at the bottome: [Page 254] then let there be soldered fast another Pipe IL, which may be separated from the bottome of the vessell, and may have his bunchie swelling as the former without touching the bottome: as is represented in L, and passing through the bottome of AB, may be continued unto I, that

is to say, to make an opening to the cover of the vessell AB, & let it have a little mouth as a Trumpet: to that end to receive the water. Then there must further be ad­ded a very smal Pipe which may passe through the bottome of the vessell AB, as let it be OP, and let there be a bunch; or swelling over it as at P, so that it touch not also the bot­tome: let there be further made to this lesser vessell an edge in forme of a Basin to receive the water, which being done poure water into the Pipe IL, untill the vessell CD, be full, then turne the whole Machine upside downe that the vessell CD, may be uppermost, and AB, undermost; so by helpe of the pipe GH, the water of the vessell CD, will runne into the vessel AB, to have passage by the pipe PO. This motion is pleasant at a feast in filling the said vessel with wine, which will spout it out as though it were from a boyling fountaine, in the forme of a threed very pleasant to behold.

PROBLEM. CXVIII. How to practise excellently the reanimation of sim­ples, in case the plants may not be transported to be replanted by reason of distance of places.

TAke what simple you please, burne it and take the ashes of it, and let it be calcina­ted two houres between two Creusets wel luted, and extract the salt: that is, to put water into it in moving of it; then let it settle: and do it two or three times, afterwards evaporate it, that is, let the water be boyled in some vessel, untill it be all consumed: then there will remaine a salt at the bottome, which you shall afterwards sowe in good Ground wel prepared: such as the Theatre of husbandry sheweth, and you shall have your desire.

PROBLEM. CVIII. How to make an infalliable perpetuall motion.

M [...]xe 5. or 6. ounces of ☿ with is equall weight of ♃, grinde it together with 10. or 12 ounces of sublimate dissolved in a celler upon a Marble the space of foure dayes, and it will become like Oile, Olive, which distill with fire of chaffe or driving fire, and it will [Page 256] sublime dry substance, then put water upon the earth (in forme of Lye) which will be at the bottom of the Limbeck, and dissolve that which you can; filter it, then distill it, and there will be produced very subtill Antomes, which put into a bottle close stopped, and keep it dry, and you shall have your desire, with astonishment to all the world, and especially to those which have travelled herein without fruit.

PROBLEM. CXIX. Of the admirable invention of making the Philo­sophers Tree, which one may see with his eye to grow by little and little.

TAke two ounces of Aqua fortis, and dis­solve in it halfe an ounce of fine silver re­fined in a Cappell: then take an ounce of Aqua fortis, and two drams of Quick-silver: which put in it, and mixe these two dissolved things together, then cast it into a Viall of halfe a pound of water, which may be well stopped; for then every day you may see it grow both in the Tree and in the branch. This liquid serves to black haire which is red, or white, without fading untill they fall, but here is to be noted that great care ought to be had in anointing the haire, for feare of touching the flesh: for this composition is very Corrosive or searching, that as soone as it toucheth the flesh it raiseth blisters, and bladders very painfull.

PROBLEM. CXX. How to make the representation of the great world?

DRaw salt Niter out of salt Earth▪ which is found along the Rivers side, and at the foot of Mountaines, where especially are Mi­nerals of Gold and Silver: mix that Niter well cleansed with ♃, then calcinate it hermetically▪ then put it in a Limbeck and let the receiver be of Glasse, well luted, and alwayes in which let there be placed leaves of Gold at the bottome, then put fire under

the Limbeck untill vapours arise which will cleave unto the Gold; augment your fire untill there a­scend no more, then take away your receiver, and close it hermetically, and make a Lampe fire under it untill you may see presented in it that which nature affords us: as Flowers, Trees, Fruits, Fountaines, Sunne, Moone, Starres, &c. Behold here the forme of the Limbeck, and the recei­ver: A represents the Limbeck, B stands for the receiver.

PROBLEM. CXXI. How to make a Cone, or a Pyramidall body move upon a Table without springs or other Artifi­ciall meanes: so that it shall move by the edge of the Table without falling?

THis proposition is not so thornie and sub­tile as it seemes to be, for putting under a Cone of paper a Beetle or such like creature,

you shall have plea­sure with astonish­ment & admiration to those which are ignorant in the cause: for this animall will strive alwayes to free herself from the cap­tivity in which she is in by the imprison­ment of the Cone: for comming neere the edge of the Table she will returne to the other side for feare of falling.

PROBLEM CXXII. To cleave an Anvill with the blow of a Pistoll.

THis is proper to a Warrier, and to performe it, let the Anvill be heated red hot as one [Page 259] can possible, in such sort that all the solidity of the body be softned by the fire: then charge the Pistoll with a bullet of silver, and so have you infallibly the experiment.

PROBLEM. CXXIII. How to r [...]st a Capon carried in a Budget at a Saddle-bowe, in the space of riding 5 or 6 miles?

HAving made it ready and larded it, stuffe [...]t with Butter; then heat a piece of steele which may be formed round according to the length of the Capon, and big enough to fill the Belly of it, and then stop it with Butter; then wrap it up well and inclose it in a Box in the Budget, and you shall have your desire: it is said that Count Mansfield served himse [...]fe with no others, but such as were made ready in this kinde, for that it loseth none of its substance, and it is dressed very equally.

PROBLEM. CXXIV. How to make a Candle burne and continue three times as long as otherwise it would?

VNto the end of a Candle half [...]burned stick a farthing lesse or more, to make it hang [Page 260] perpendicular in a vessel of water, so that it swimme above the water; then light it, and it

will susteine it self & float in this manner; and being placed in­to a fountaine, pond, or lake that runnes slowly, where many people assemble, it will cause an extreme feare to those which come therein in the night, knowing not what it is.

PROBLEM. CXXV. How out of a quantitie of wine to extract that which is most windy, and evill, that it hurt not a sick Person?

TAke two vials in such sort that they be of like great­nesse both in th [...] belly and the neck; fill one of them of wine, and the other of water: let the mouth of that which hath the water be placed into the mouth of that which hath the wine, so the water shall [Page 261] be uppermost, now because the water is heavier than the wine, it will descend into the other Viall, and the wine which is lowest, because it is highest will ascend above to supply the place of the water, and so there will be a mutuall in­terchange of liquids, and by this penetration the wine wil lose her vapors in passing through the water.

PROBLEM CXXVI. How to make two Marmouzets, one of which shall light a Candle, and the other put it out?

Upon the side of a wall make the figure of a Marmouzet or other animall or forme, and right against it on the other wall make another; in the mouth of each put a pipe or quill so artificially that it be not perceived; in one of which place salt peter very fine, and dry and pulverised; and at the end set a little match of paper, in the other place sulphur beaten smal, then holding a Candle lighted in your hand, say to one of these Images by way of command, Blow out the Candle; then lighting the paper with the candle, the salt-peter wil blow out the Candle immediatly, and going to the other I­mage (before the match of the Candle be out) touch the sulphur with it and say, Light the Candle, & it will immediatly be lighted, which will cause an admiration to those which see the action, if it be wel done vvith a secret dexterity.

PROBLEM. XXVII. How to keepe wine fresh as if it were in a celler though it were in the heat of Summer, and without Ice or snow, yea though it were carried at a saddles bow, and ex­posed to the Sun all the day?

SEt your wine in a viall of Glasse; and place it in a Box made of wood, Leather, or such like: about which vial place Salt-peeter, and it will preserve it and keep it very fresh: this ex­periment is not a little commodious for those which are not neare fresh waters, and whose dwellings are much exposed to the Sunne.

PUOBLEM. CXXVIII. To make a Cement which indureth or lasteth as marble, which resisteth aire and wa­ter without ever disjoyning or uncementing?

TAke a quantity of strong and gluing Mor­ter vvell beaten, mixe vvith this as much nevv slaked Lime, and upon it cast Oile of Olive or Linseed-Oile, and it vvill become hard as Marble being applyed in time.

PROBLEM. CXXIX. How to melt metall very quickly, yea in a shell upon a little fire.

MAke a bed upon a bed of metall with pou­der of Sulphur, of Salt-peeter, and saw-dust alike; then put fire to the said pouder with a burning Charcole, and you shall see that the metall will dissolve incontinent and be in a Masse. This secret is most excellent, and hath been practised by the reverend father Mercen [...] of the order of the Minims.

PROBLEM. CXXX. How to make Iron or steele exceeding hard?

QVench your Blade or other Instrument seven times in the blood of a male Hog mixt with Goose-grease, and at each time dry it at the fire before you wet it: and it will become exceeding hard, and not brittle, which is not ordinary according to other temperings and quenchings of Iron: an experiment of small cost, often proved, and of great consequence for Armorie in warlike negotiations.

PRBOLEM CXXXI. To preserve fire as long as you will, imitating the inextinguible fire of Vestales.

AFter that you have extracted the burning spirit of the salt of ♃, by the degrees of fire, as is required according to the Art of Chymistrie, the fire being kindled of it selfe, break the Limbeck, and the Irons which are found at the bottome will flame and appeare as burning Coles as soone as they feele the aire; which if you promptly inclose in a viall of Glasse, and that you stop it exactly with some good Lute: or to be more assured it may be closed up with Hermes wax for feare that the Aire get not in. Then will it keep more than a thousand yeares (as a man may say) yea at the bottome of the Sea; and opening it at the end of the time, as soone as it feeles the Aire [...] takes fi [...]e▪ with which you may light a Match. This secret merits to be travelled after and put in practice, for that it is not common, and full of astonishment, seeing that all kinde of fire lasteth but as long as his matter lasteth, and that there is no matter to be found that will so long in [...]e.