Mathematicall RECREATIONS. OR, A Collection of many Problemes, extracted out of the Ancient and Modern Philoso­phers, as Secrets and Experiments in Arithmetick, Geometry, Cosmographie, Horologiographie, Astronomie, Navigation, Musick, Opticks, Architecture, Stati [...]k, Mechanicks, Chemistry, Water-works, Fire-works, &c. Not vulgarly mani­fest till now.

Written first in Greeke and Latin, lately compi'ld in French, by Henry Van Etten, and now in English, with the Examinations and Augmentations of divers Modern Mathematicians

Whereunto is added the Description and Use of the Generall Horologicall Ring: And The Double Horizontall Diall. Invented and written by WILLIAM OUGHTRED.

LONDON: Printed for William Leake, at the Signe of the Crown in Fleetstreet, between the two Temple Gates, MDCLIII.

On the Frontispice and Booke.

ALL Recreations do delight the minde,
But these are best being of a learned kinde:
Here Art and Nature strive to give content,
In shewing many a rare experiment,
Which you may read, & on their Schemes here look
Both in the Frontispice, and in the Book.
Upon whose table new conceits are set,
Like dainty dishes, thereby for to whet
And winne your judgement, with your appetite
To taste them, and therein to taka delight.
The Senses objects are but dull at best,
But Art doth give the Intellect a feast.
Come hither then, and here I will describe,
What this same Table doth for you provide.
Here Questions of Arithmetick are wrought,
And hidden secrets unto light are brought,
The like it in Geometrie doth unfold,
And some too in Cosmographie are told:
It divers pretty Dyals doth descrie,
With strange experiments in Astronomie,
And Navigation, with each severall Picture,
In Musick, Opticks, and in Architecture:
In Statick, Machanicks, and Chymistrie,
In Water-works, and to ascend more hie,
In Fire-works, like to Joves Artillerie.
All this I know thou in this Book shalt finde,
And here's enough for to content thy minde.
For from good Authors, this our Author drew
These Recreations, which are strange, and true
So that this Book's a Centre, and tis fit,
That in this Centre; lines of praise should meet
MATHEMATICALL Recreations Or a Collection of sundrie excellent Problemes out of ancient & moderne Phylosophers Both vsefull and Recreatiue Printed for William Leake and are to be solde at the Crowne in fleet streete betweene the two Temple gates.

TO The thrice Noble and most generous Lo. the Lo. Lambert Verreyken, Lo. of Hinden, Wolverthem, &c.

My honourable Lo.

AMongst the rare and curious Propositions which I have learned out of the studies of the Mathematicks in the famous University of Pont a Mousson, I have taken singular pleasure in certaine Pro­blemes no lesse ingenious than recreative, which drew me unto the search of de­monstrations more difficult and serious; some of which I have amassed and caused to passe the Presse, and here dedicate them now unto your Honour; not that I account them worthy of your view, but in part to [Page] testifie my affectionate desires to serve you, and to satisfie the curious, who de­light themselves in these pleasant studies, knowing well that the Nobilitie, and Gentrie rather studie the Mathematicall Arts, to content and satisfie their affecti­ons, in the speculation of such admirable experiments as are extracted from them, than in hope of gaine to fill their Purses. All which studies, and others, with my whole indevours, I shall alwayes dedicate unto your Honour, with an ardent desire to be accounted ever,

Your most humble and obedient Nephew and Servant, H. VAN ETTEN.

By vvay of advertisement. Five or six things I have thought worthy to declare before I passe further.

FIrst, that I place not the spe­culative demonstrations with all these Problems, but con­tent my self to shew them as at the fingers end: which was my plot and intention, because those which understand the Mathematicks can conceive them easily; others for the most part will content themselves onely with the knowledge of them, without seeking the reason.

Secondly, to give a greater grace to the practice of these things, they ought to be concealed as much as they may, in the subtiltie of the way; for that which doth ravish the spirits is, an admirable effect, whose cause is unknowne: which if it were discovered, halfe the pleasure is l [...]st; therefore all the finenesse consists in the [Page] dexterity of the Act, concealing the meanes, and changing often the streame.

Thirdly, great care ought to be had that one deceive not himselfe, that would de­clare by way of Art to deceive another: this will make the matter contemptible to ignorant Persons, which will rather cast the fault upon the Science, than upon him that shewes it: when the cause is not in the Mathematicall principles, but in him that failes in the acting of it.

Fourthly, in certaine Arithmeticall propositions they have onely their an­swers as I found them in sundry Authors, which any one being studious of Mathe­maticall learning, may finde their origi­nall, and also the way of their operation.

Fifthly, because the number of these Problemes, and their dependances are ma­ny, and intermixed, I thought it conve­nient to gather them into a Table: that so each one according to his fancie, might make best choise of that which might best please his palate, the matter being not of one nature, nor of like subtiltie: But who­soever will have patience to read on, shall finde the end better than the beginning.

To the Reader.

IT hath been observed by many, that sundry fine wits as well amongst the Ancient as Mo­derne, have sported and de­lighted themselves upon seve­rall things of small consequence, as upon the foot of a fly, upon a straw, upon a point, nay upon nothing; striving as it were to shew the greatnesse of their glory in the smalnesse of the subject: And have amongst most solid and artificiall conclusions, com­posed and produced sundry Inventions both Philosophicall and Mathematicall, to so­lace the minde, and recreate the spirits, which the succeeding ages have imbraced, and from them gleaned and extracted many admirable, and rare conclusions; judging that borrowed matter often-times yeelds praise to the industry of its author. Hence for thy use (Courteous Reader) I have with [Page] great search and labour collected also, and heaped up together in a body of these pleasant and fine experiments to stirre up and delight the affectionate, (out of the writings of Socrates, Plato, Aristotle, Demosthenes, Pythagoras, Democrates, Plinie, Hypar­chus, Euclides, Vitruvius, Diaphantus, Pergaeus, Archimedes, Papus Alexandri­nus, Vitellius, Ptolomaeus, Copernicus, Proclus, Mauralicus, Cardanus, Valal­pandus, Kepleirus, Gilbertus, Tychonius, Dureirus, Josephus, Clavius, Gallileus Maginus, Euphanus Tyberill, and others) knowing Art imitating Nature that glories alwayes in the variety of things, which she produceth to satisfie the minde of curious inquisitors. And though perhaps these labours to some humourous per­sons may seeme vaine, and ridiculous, for such it was not undertaken: But for those which intentively have desired and [...]ought after the knowledge of those things, it being an invitation and motive to the search of greater matters, and to imploy the minde in usefull knowledge, rather than to be busi­ed in vaine Pamphlets, Play-books, fruit­lesse Legends, and prodigious Histories that are invented out of fancie, which abuse many Noble spirits, dull their wits, & alien­ate [Page] their thoughts from laudable and ho­nourable Studies. In this Tractate thou maist therefore make choise of such Mathe­maticall Problemes and Conclusions as may delight thee, which kinde of learning doth ex­cellently adorne a man; seeing the usefulnesse thereof, and the manly accomplishments it doth produce, is profitable and delightfull for all sorts of people, who may furnish and adorne themselves with abundance of matter in that kinde, to help them by way of use, and discourse. And to this we have also ad­ded our Pyrotechnie, knowing that Beasts have for their object only the surface of the earth; but hoping that thy spirit which fol­loweth the motion of fire, will abandon the lower Elements, and cause thee to lift up thine eyes to soare in an higher Contempla­tion, having so glittering a Canopie to be­hold, and these pleasant and recreative fires ascending may cause thy affections also to ascend. The Whole whereof we send forth to thee, that desirest the scrutability of things; Nature having furnished us with matter, thy spirit may easily digest them, and put them finely in order, though now in disorder.

A Table of the particular heads of this Book, contracted ac­cording to the severall Arts specified in the Title-page.

Experiments of Arithmetick.
PAge 1, 2, 3, 16, 19, 22, 28, 33, 39, 40, 44, 45, 51, 52, 53, 59, 60, 69, 71, 77, 83, 85, 86, 89, 90, 91, 124, 134, 135, 136▪ 137, 138, 139, 140, 178 179, 181, 182, 183, 184, 185, 188, 208, 210, 213.
Experiments [...]n Geometrie.
Pag. 12, 15, 24, 26, 27, 30, 35, 37, 41, 42, 47, 48, 49, 62, 65, 72, 79, 82, 113, 117, 118, 119, 214, 215, 217, 218, 234, 235, 236, 239, 240.
Experiments in Cosmographie.
Pag. 14, 43, 75, 106, 107, 219, 220, 225, 227, 228, 229, 230, 232.
Experiments in Horologiographie.
Pag. 137, 166, 167, 168, 169, 171, 234.
Experiments in Astronomie.
Pag. 220, 221, 222, 223, 224.
Experiments in Navigation.
Pag. 105, 233, 234, 237, 238.
Experiments in Musick.
Pag. 78, 87, 126.
[Page] Experiments in Opticks.
Pag. 6, 66, 98, 99, 100, 102, 129, 131, 141, 142, 143, 144, 146, 149, 151, 152, 153, 155, 156, 157, 158, 160, 161, 162, 163, 164, 165.
Experiments in Architecture.
Pag. 16, 242, 243.
Experiments in Staticke.
Pag. 27, 30, 32, 71, 199, 200, 201, 283, 204, 205, 207.
Experiments in Machanicks.
Pag. 56, 58, 68, 88, 95, 108, 110, 128, 173, 174, 176, 246, 248, 258, 259.
Experiments in Chymistrie.
Pag. 198, 255, 256, 257, 260, 262, 263, 264.
Experiments in Water-workes.
Pag. 190, 191, 192, 193, 194, 196, 247, 249, 250, 252, 253.
Experiments in Fireworkes.
From page, 265. to the end.

A Table of the Contents, and chiefe points con­teined in this Book. PROBLEM. II.

  • HOw visible objects that are without, and things that passe by, are most lively re­presented to those that are within. Page 6
  • Prob. 1 Of finding of numbers conceived in the minde. 1, 2, 3
  • Prob. 5 Of a Geographicall Garden-plot fit for a Prince or some great personage. 14
  • Prob. 37 Any liquid substance, as water or wine, placed in a Glasse, may be made to boile by the motion of the finger, and yet not touch­ing it. 54
  • Prob. 3 How to weigh the blow of ones fist, of a Mallet, a Hatchet or such like. 9.
  • Prob. 30 Two severall numbers being ta­ken by two sundry persons, how subtilly to discover which of those numbers each of them took. 46
  • Prob. 4 That a staffe may be broken▪ placed upon two Glasses, without hurting of the Glasses. 12
  • Prob. 7 How to dispose Lots that the 5, 6, 9, &c. of any number of persons may escape. 16
  • [Page] Prob. 13 How the weight of smoke of a combustible body, which is exhaled, may be weighed. 27
  • Prob. 12 Of three knives which may be so disposed to hang in the aire, and move upon the Point of a needle. 27
  • Prob. 17 Of a deceitfull bowle, to bowle withall. 32
  • Prob. 16 A ponderous or heavy body may be supported in the aire without any one touching it. 30
  • Prob. 18 How a Peare, or Apple, may be parted into any parts, without breaking the rinde thereof. 33
  • Prob. 15 Of a fine kinde of dore which o­pens and shuts on both sides. 30
  • Prob. 9 How the halfe of a Vessell which containes 8 measures may be taken, being but onely two other measures, the one being 3, and the other 8 measures. 22
  • Prob. 8 Three persons having taken each of them severall things, to finde which each of them hath taken. 19
  • Prob. 6 How to dispose three staves which may support each other in the aire. 15
  • Prob. 14 Many things being disposed Circular (or otherwise) to finde which of them any one thinks upon. 28
  • Prob. 19 To finde a number thought upon [Page] without asking questions. 33
  • Prob. 11 How a Milstone or other pondero­sity may hang upon the point of a Needle with­out bowing, or any wise breaking of it. 26
  • Prob. 20 and 21 How a body that is uni­forme and inflexible may passe through a hole which is round, square and Triangular; or round, square and ovall-wise, and ex­actly fill those severall holes. 35, 37
  • Prob. 10 How a stick may stand upon ones finger, or a Pike in the middle of a Court without falling. 24
  • Prob. 22 To finde a number thought upon after another manner than those which are formerly delivered. 39
  • Prob. 23 To finde out many numbers that sundry persons or any one hath thought upon. 40
  • Prob. 24 How is it that a man in one & the same time may have his head upward, and his feet upward, being in one and the same place? 4 [...]
  • Prob. 25 Of a Ladder by which two men ascending at one time, the more they ascend, the more they shal be asunder, notwith stan­ding the one be as high as the other. 42
  • Prob. 26 How is it that a man having but a Rod or Pole of land, doth brag that he [Page] may in a right line passe from place to place 3000 miles. 42
  • Prob. 27 How is it that a man standing upright, and looking which way he will, he looketh true North or South. 43
  • Prob. 28 To tell any one what number re­maines after certaine operations being ended, without asking any question. 44
  • Prob. 29 Of the play with two severall things. 45
  • Prob. 31 How to describe a circle that shall touch 3 points placed howsoever upon a plaine, if they be not in a right line. 47
  • Prob. 32 How to change a circle into a square forme. 48
  • Prob. 33 With one and the same compasses, and at one and the same extent or opening, how to describe many circles concentricall, that is, greater or lesser one than another. 49
  • Prob. 34 Any number under 10. be­ing thought upon, to finde what numbers they were. 51
  • Prob, 35 Of the play with the Ring. 52
  • Prob. 36 The play of 3, 4, or more Dice. 53
  • Prob. 38 Of a fine Vessell which holds Wine or Water being cast into it at a certain height, but being filled higher it will runne all out of its owne accord. 56
  • [Page] Prob. 39 Of a Glasse very pleasant. 58
  • Prob. 40. If any one should hold in each hand as many pieces of money as in the other, how to finde how much there is. 59
  • Prob. 41 Many Dice being cast, how ar­tificially to discover the number of the points that may arise. 60
  • Prob. 42 Two metals as Gold and Silver or of other kinde, weighing alike, being pri­vately placed into two like boxes, to finde in which of them the Gold or Silver is. 62
  • Prob. 43 Two Globes of divers metals (as one Gold the other Copper) yet of equall weight, being put in a Box as B.G. to finde in which end the Gold or Copper is. 65
  • Prob. 44 How to represent divers sorts of Rainbowes here below. 66
  • Prob. 45 How that if all the powder in the World were inclosed in a bowle of paper or glasse, and being fired on all parts, it could not break that bowle. 68
  • Prob. 46 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 will still remaine one, but being divided by 7 will remaine nothing. 69
  • Prob. 47 One had a certaine number of Crownes, and counting them by 2 and 2, [Page] there rested 1. counting them by 3, and 3, there rested 2. counting them by 4, and 4, there re­sted 3. counting them by 5, and 5, there rested 4. counting them by 6, and 6, there rested 5. but counting them by 7 and 7, there rested nothing, how many Crownes might he have? 71
  • Prob. 48 How many sorts of weights in the least manner must there be to weigh all sorts of things betweene one pound and 121 pound, and so unto 364 pound? 71
  • Prob. 49 Of a deceitfull balance which being empty seems to be just, because it hangs in Aequilibrio, notwithstanding putting 12 pound in one ballance, and 11 in the other, it will remaine in Aequilibrio. 72
  • Prob. 50 To heave or lift up a bottle with a straw. 74
  • Prob. 51 How in the middle of a wood or desert, without the sight of the Sun, starres, shadow, or compasse, to finde out the North, or South, or the 4 Cardinal points of the World, East, West, &c. 75
  • Prob. 52 Three persons having taken Counters, Cards, or other things, to finde how much each one hath taken. 7 [...]
  • Prob. 53 How to make a consort of Musick of many parts with one voice or one instru­ment onely. 78
  • Prob. 54 To make or describe an oval form, [Page] or that which is neare resembled unto it at one turning, with a paire of common Compasses. 79
  • Prob. 55 Of a purse difficult to be opened. 80
  • Prob. 56 Whether is it more hard and ad­mirable without Compasses to make a per­fect circle, or being made to finde out the Centre of it? 82
  • Prob. 56 Any one having taken 3 Cards, to finde how many points they containe. 83
  • Prob. 57 Many Cards placed in divers ranks, to finde which of those Cards any one hath thought. 85
  • Prob. 58 Many Cards being offered to sundry persons to finde which of those Cards any one thinketh upon. 86
  • Prob. 59 How to make an instrument that helps to heare, as Gallileus made to help to see. 87
  • Prob. 60 Of a fine Lamp which goeth not out, though one carries it in ones pocket, or being rolled on the ground will still burne. 88
  • Prob. 61 Any one having thought a Card amongst many Cards, how artificially to dis­cover it out. 89
  • [Page] Prob. 62 Three Women A, B, C. carried Apples to a Market to sell: A had 20. B had 30. C 40. they sold as many for a penny one as the other, and brought home one as much money as another, how could this be? 90
  • Prob. 63 Of the properties of some num­bers. 91
  • Prob. 64 Of an excellent Lamp which serves or furnisheth it selfe with Oile, and burnes a long time. 95
  • Prob. 65 Of the play at Keyles or Nine-pins. 97
  • Prob. 66 Of Spectacles of pleasure
  • Of Spectacles which give severall colours to the visage. 98
  • Of Spectacles which make a Towne seeme to be a City, one armed man as a Company, and a piece of Gold as many pieces. 99
  • How out of a Chamber to see the objects which passe by according to the lively per­spective. 100
  • Of Gallileus admirable Optick-Glasse, which helps one to see the beginning and end­ing of Eclipses, the spots in the Sunne, the Starres which move about the Planets, and perspicuously things far remote.
  • Of the parts of Gallileus his Glasse. 102
  • Prob. 67 Of the Magnes and Needles touched therewith.
  • [Page]How Rings of Iron may hang one by ano­ther in the aire. 103
  • Of Mahomets Tombe which hangs in the aire by the touch of the Magnes. 104
  • How by the Magnes only to finde out North and South 105
  • Of a secrecie in the Magnes, for discovering things farre remote. 106
  • Of finding the Poles by the Magnes 107
  • Prob. 68 Of the properties of Aeolipiles or Bowles to blow the fire. 108
  • Prob. 69 Of the Thermometer, or that which measures the degrees of heat and cold by the aire. 110
  • Of the proportion of humane bodies, of statues, of Colosses, or huge Jmages and monstrous Giants. 113
  • Of the commensuration of the parts of the bodie the one to the other in particular, by which the Lion was measured by his claw, the Giant by his thumbe, and Hercules by his foot. 115, 116
  • Of Statues or Colosses, or huge Images; that mount Athos metamorphosed by Dyno­crites into a statue, in whose hand was a Towne able to receive ten thousand men. 117
  • Of the famous Colossus at Rhodes which bad 70 cubits in height, and loaded 900. Camels, which weighed 1080000 l. 118
  • [Page]Of Nero his great Colossus which had a face of 12 foot large. 119
  • Of monstrous Giants
  • Of the Giant Og and Goliah. 119, 120
  • Of the Carkasse of a man found which was in length 49 foot; and of that monster found in Creet, which had 46. Cubits of height. 120
  • Of Campesius his relation of a monster of 300 foot found in Sicile, whose face according to the former proportion should be 30 foot in length. 121
  • Prob. 71 Of the game at the Palme, at Trap, at Bowles, Paile-maile, and others. 122
  • Prob. 72 Of the game of square formes. 124
  • Prob. 73 How to make the string of a Viol sensibly shake without any one touching it. 126
  • Prob. 74 Of a Vessell which containes 3 severall kindes of liquor, all put in at one bung-hole, and drawne out at one Tap seve­rally without mixture. 128
  • Prob. 75 Of burning-Glasses.
  • Archimedes his way of burning the ships of Syracuse. 129
  • Of Proclus his way, and of concave and sphericall Glasses which burne, the cause and demonstration of burning with Glasses. 131
  • Of Maginus his way of setting fire to Pow­der in a Mine by Glasses. 131
  • [Page]Of the examination of burning by Glasses. 133
  • Prob. 76 Of pleasant questions by way of Arithmetick.
  • Of the Asse and the Mule. 134
  • Of the number of Souldiers that fought be­fore old Troy. 135
  • Of the number of Crownes that two men had. 136
  • About the houre of the day. 137
  • Of Pythagoras Schollers. 137
  • Of the number of Apples given amongst the Graces and the Muses. 138
  • Of the testament or last will of a dying Fa­ther. 138
  • Of the cups of Croesus. 139
  • Of Cupids Apples. 139
  • Of a Mans Age. 140
  • Of the Lion of Bronze placed upon a foun­taine with his Epigram. ibid.
  • Prob. 77 In Opticks, excellent experi­ments.
  • Principles touching reflections. 141
  • Experiments upon flat and plaine Glasses. 142
  • How the Images seeme to sink into a plaine Glasse, and alwayes are seene perpendicular to the Glasse, an [...] also inversed. 143
  • [Page]The things which passe by in a street may by help of a plaine glasse be seen in a Chamber, and the height of a tower or tree observed. 143
  • How severall Candles from one Candle are repre­sented in a plaine Glasse, and Glasses alternately may be seene one within another, as also the back-parts of the body as well as the fore-parts are evidently repre­sented. 144
  • How an Image may be seene to hang in the aire by help of a Glasse: and writing read or easily understood. 146
  • Experiments upon Gibbous, or convex Sphericall Glasses.
  • How lively to represent a whole City, fortification, or Army, by a Gibbous Glasse. 147
  • How the Images are seen in Concave Glasses. 149
  • How the Images are transformed by approaching to the centre of the Glasse, or point of concourse; and of an exceeding light that a Concave Glasse gives by help of a Candle. 151
  • How the Images, as a man, a sword, or hand, doth come forth out of the Glasse. 152, 153
  • Of strange apparitions of Images in the aire, by help of sundry Glasses. 152, 154
  • Of the wonderfull augmentation of the parts of mans body comming neare the point of inflammation, or centre of the Glasse. 155
  • How writing may be reverberated from a Glasse upon a VVall, and Read. 156
  • How by help of a Concave Glasse to cast light in­to a Campe, or to give a perspective light to Pyoneers in a Mine, by one Candle only. 156
  • How excellently by help of a Concave Glasse and a [Page] Candle placed in the centre, to give light to read by. 157
  • Of other Glasses of pleasure. 158
  • Of strange deformed representations by Glasses; causing a man to have foure eyes, two Mouthes, two Noses, two heads.
  • Of Glasses which give a colour to the visage, and make the face seeme faire and foule. 160
  • Prob. 78 How to shew one that is suspicious, what is in another Chamber or Roome, notwithstanding the interposition of that wall. 160
  • Corolary, 1. To see the Besiegers of a place, upon the Rampa [...]t of a fortification 161
  • Corolary 2. and 3. Notwithstanding the interposi­tion of VValls and Chambers, by help of a Glasse things may be seen, which passe by. 162
  • Prob. 79 How with a Musket to strike a marke not looking towards it, as exactly as one aimed at it. 162
  • How exactly to shoot out of a Mu [...]ket to a place which is not seene, being hindred by some obstacle or other interposition. 163
  • Prob. 80 How to make an Image to be seen hang­ing in the aire, having his head downward. 164
  • Prob. 81. How to make a company of representa­tive souldiers seeme to be as a regiment, or how few in number may be multiplyed to seem to be many in number. 165
  • COROLARIE. Of an excellent delightfull Cabinet made of plaine Glasses. 165
  • Prob. 82 Of fine and pleasant Dyalls in Horo­logiographie.
  • Of a Dyall of herbs for a Garden. 166
  • Of the Dyall upon the finger and hand, to finde what [Page] of the Clock it is. 167
  • Of a Dyall which was about an Obelisk at Rome. 168
  • Of Dyals with Glasses. 168
  • Of a Dyall which hath a Glasse in the place of the stile. 169
  • Of Dyals with water, which the Ancients use [...] 171
  • Prob. 83 Of shooting out of Cannons or great Artillery.
  • How to charge a Cannon without powder. 173
  • To finde how much time the Bullet of a Cannon spends in the Aire before it falls to the ground. 174
  • How it is that a Cannon shooting upward, the Bullet flies with more violence, than being shot point blanke, or shooting downeward. 174
  • VVhether is the discharge of a Cannon so much the more violent, by how much it hath the more length? 176
  • Prob. 84 Of prodigious progressions, and mul­tiplications of creatures, plants, fruits, numbers, gold, silver, &c.
  • Of graines of Mustardseed, and that one graine being sowne, with the increase thereof for 20 yeares will produce a heap greater than all the earth a hun­dred thousand times. 178
  • Of Pigges, and that the great Turke with all his Revenne, is not able to maintaine for one yeare, a Sow with all her increase for 12 yeares. 179
  • [Page]Of graines of Corne, and that 1 graine with all its increase for 12 yeares, will amount to 244140625000000000000 graines, which exceeds in value all the treasures in the World. 183
  • Of the wonderfull increase af Sheepe. 182
  • Of the increase of Cod-fish. 182
  • Of the Progressive Multiplication of soules; that from one of Noahs Sonnes, from the flood unto Nimrods Monarchie, should be produced 111350 soules. 183
  • Of the increase of Numbers in double proportion, and that a pin being doubled as often as there are weekes in the yeare, the number of pinnes that should arise is able to load 45930 ships of a thou­sand Tunne apiece, which are worth more than tenne hundred thousand pounds a day. 183, 184
  • Of a man that gathered Apples, stones, or such like upon a condition. 185
  • Of the changes in Bells, in musicall in­struments, transmutation of Places, in Numbers, Letters, Men and such like▪ 185
  • [Page]Of the wonderfull interchange of the Let­ters in the Alphabet: the exceeding number of men, and time to expresse the words that may be made with these letters, and the num­ber of Books to comprehend them. 187, 188
  • Of a servant hired upon certaine condition, that he might have land lent him to sowe one graine of Corne with its increase for 8 yeares time, which amounted to more than four hun­dred thousand Acres of Land. 188
  • Prob. 85 Of Fountaines, Hydriatiques; Stepticks, Machinecks, and other experi­ments upon water, or other liquor.
  • First, how water at the foot of a Mountaine may be made to ascend to the top of it, and so to descend on the other side of it 190
  • Secondly, to finde how much Liquor is in a Vessell, onely by using the tap-hole. 191
  • Thirdly, how is it, that a Vessell is said to hold more water at the foot of a Mountaine, then at the top of it 191
  • 4 How to conduct water from the top of one Mountaine to the top of another 192
  • 5 Of a fine Fountaine which spouts water very high and with great violence, by turn­ing of a Cock 193
  • 6 Of Archimedes screw which makes wa­ter ascend by descending. 194
  • 7 Of a fine Fountaine of pleasure. 196
  • [Page] 8 Of a fine watering pot for Gardens. 197
  • 9 How easily to take Wine out of a Vessell at the bung hole without piercing a hole in the Vessell. 198
  • 10 How to measure irregular bodies by help of water. 198
  • 11 To finde the weight of water. 199
  • 12 To finde the charge that a vessell may carry, as Ships, Boats or such like. 200
  • 13 How comes it that a ship having safely sailed in the vast Ocean, and being come into the port or harbour, will sinke down right. 200
  • 14 How a grosse body of metall may swim upon the water. 201
  • 15 How to weigh the lightnesse of the aire. 203
  • 16 Being given a body, to mark it about, and shew how much of it will sink in the wa­ter, or swim above the water. 204
  • 17 To finde how much severall metalls or other bodies do weigh lesse in the water than in the aire. 204
  • 18 How is it that a ballance having like weight in each scale, and hanging in Aequi­librio in the aire, being removed from that place (without diminishing the weights in each balance, or adding to it) it shall cease to hang in Aequilibrio sensibly, yea by a great difference of weight. 205
  • [Page] 19 To shew what waters are heavier one than another, and how much. 206
  • 20 How to make a pound of water weigh as much as 10, 20, 30, or a hundred pound of Lead, nay as much as a thousand or ten thou­sand pound weight. 207
  • Prob. 86. Of sundry questions of Arith­metick, and first of the number of sands calculated by Archimedes and Clavius. 208
  • 2 Divers metalls being melted together in one body, to finde the mixture of them. 210
  • 3 A subtile question of three partners about equality of Wine and Vessels. 213
  • 4 Of a Ladder which standing upright a­gainst a wall of 10 foot high, the foot of it is pulled out 6 foot from the wall upon the pave­ment, how much hath the top of the Ladder descended. 214
  • Prob. 87 Witty suits or debates between Caius and Sempronius, upon the forme of fi­gures, which Geometricians call Isoperime­ter, or equall in circuit, or Compasse. 214
  • 1 Incident: of changing a field of 6 measures square, for a long rectrangled fiel of 9 mea­sures in length and 3 in breadth: both equall in circuit but not in quantity. 215
  • 2 Incident: about two sacks each of them ho [...]ding but a bushell, and yet were able to hold 4 bushels. 217
  • [Page] 3 Incident: sheweth the deceit of pipes which conveygh water, that a pipe of two inches dia­meter, doth cast out foure times as much wa­ter as a pipe of one such diameter. 218
  • 7 Heapes of Corne of 10 foot every way, is not as much as one heap of Corne of 20 foot every way. 218
  • Prob. 88 Of sundry questions in matter of Cosmographie, and Astronomy.
  • In what place the middle of the earth is sup­posed to be. 219
  • Of the depth of the earth, and height of the Heavens, and the compasse of the World, how much. 219
  • How much the starry Firmament, the Sun, and the Moone are distant from the centre of the earth. 220
  • How long a Mill-stone would be in falling to the centre of the earth from the superfi­cies, if it might have passage thither. 220
  • How long time a man or a bird may be in compassing the whole earth. 220
  • If a man should ascend by supposition 20 miles every day: how long it would be before he approach to the Moone. 221
  • The Sunne moves more in one day than the Moone in 20 dayes. 221
  • If a milstone from the orbe of the Sun should descend a thousand miles in an houre how long [Page] it would be before it come to the earth. 221
  • Of the Sunnes quick motion, of more than 7500 miles in one minute. 221
  • Of the rapt and violent motion of the starry Firmament, which if a Horseman should ride every day 40 miles, he could not in a thou­sand yeares make such a distance as it moves every houre. 221
  • To finde the Circle of the Sunne by the fin­gers. 223
  • Prob. 93 Of finding the new and full Moone in each moneth. 224
  • Prob. 94 To finde the latitude of Countreys. 225
  • Prob. 95 Of the Climates of Countreys, and how to finde them. 225
  • Prob. 96 Of longitude and latitude of the places of the earth, and of the Starres of the Heavens. 227
  • To finde the Longitude of a Countrey. 228
  • Of the Latitude of a Countrey. 229
  • To finde the Latitude of a Countrey. 230
  • To finde the distance of places. 230
  • Of the Longitude, Latitude, Declination, and distance of the starres. 231
  • How is it that two Horses or other creatures comming into the World at one time, and dy­ing at one and the same instant, yet the one of them to be a day older than the other? 232
  • [Page] Certaine fine Observations.
  • In what places of the World is it that the needle hangs in Aequilibrio, and verticall? 233
  • In what place of the world is it the sun is East or West but twice in the yeare? 233
  • In what place of the World is it that the Sunnes Longitude from the Equinoctiall paints and Altitude, being equall, the Sunne is due East or West?
  • That the sunne comes twice to one point of the Compasse in the forenoone or afternoone. 233
  • That in some place of the World there are but two kindes of winde all the yeare. 233
  • Two ships may be two leagues asunder un­der the equinoctiall, and sayling North at a certaine parallell they will be but just halfe so much. 233
  • To what inhabitants, and at what time the sunne will touch the north-part of the Hori­zon at midnight. 234
  • How a man may know in his Navigation when he is under the Equinoctiall. 234
  • At what day in the yeare the extremitie of the styles shadow in a Dyall makes a right line. 234
  • What height the Sunne is of, and how far from the Zenith, or Horizon, when a mans [Page] shadow is as long as his height. 234
  • Prob. 97 To make a Triangle that shall have three right Angles. 234
  • Prob. 98 To divide a line in as many parts as one will, without compasses or without see­ing of it. 235
  • Prob. 99 To draw a line which shall incline to another line, yet never meet against the Axiome of Parallells. 236
  • Prob. 100 To finde the variation of the Compasse by the Sunne shining. 237
  • Prob. 101 To know which way the winde is in ones Chamber without going abroad. 238
  • Prob. 102 How to draw a parallel sphaeri­call line with great ease. 239
  • Prob. 103 To measure an height onely by help of ones Hat. 240
  • Prob. 104 To take an height with two strawes. 240
  • In Architecture how statues or other things in high buildings shall beare a proportion to the eye below either equall, double, &c. 242
  • Prob. 106 Of deformed figures which have no exact proportion, where to place the eye to see them direct. 243
  • Prob. 107 How a Cannon that hath shot may be covered from the battery of the Ene­my. 244
  • Prob. 108 Of a fine Lever, by which one [Page] man alone may place a Cannon upon his Carriage. 245
  • Prob. 109 How to make a Clock with one wheele 246
  • Of Water-workes.
  • Prob. 110 How a childe may draw up a Hogshead of water with ease. 247
  • Prob. 111 Of a Ladder of Cords to cary in ones pocket, by which he may mount a wall or Tower alone. 248
  • Prob. 112 Of a marvelous Pump which drawes up great quantity of water. 249
  • Prob. 113 How naturally to cause water to ascend out of a Pit. 250
  • Prob. 114 How to cast water out of a fountaine very high. 252
  • Prob. 115 How to empty the water of a Pit by help of a Cisterne. 253
  • Prob. 116 How to spout out water very high. 253
  • Prob. 117 How to re-animate simples though brought a thousand miles. 255
  • Prob. 118 How to make a perpetuall motion. 255
  • Prob. 119 Of the admirable invention of making the Philosophers Tree, which one may see to grow by little and little. 256
  • Prob. 120 How to make the representation of the great world 257
  • Prob. 121 Of a Cone, or Pyramidall figure that moves upon a Table 258
  • Prob. 122 How an Anvill may be cleaved by the blow of a Pistoll. 258
  • Prob. 123 How a Capon may be rosted in a mans travells at his sa [...]le-bowe. 259
  • Prob. 124 How a Candle may be made to burne three times longer than usually it doth 259
  • Prob. 125 How to draw Wine out of water 260
  • [Page] Prob. 126 Of two Marmouzets, the one of which lights a Candle, and the other blowes it out. 261
  • Prob. 127 How to make Wine fresh without Ice or Snow in the height of Summer. 262
  • Prob. 128 To make a Cement which lastes as marble, resisting aire and water. 262
  • Prob. 129 How to melt metall upon a shell with little fire. 263
  • Prob. 130 Of the hardning of Iron and steele. 263
  • Prob. 131 To preserve fire as long as you will, imitating the inextinguible fire of the Vestales. 264

Ad Authorem D.D. Henricum Van Etenium, Alumnum Academiae Ponta Mousson.

ARdua Walkeri sileant secreta profundi,
Desinat occultam carpere Porta viam.
Itala Cardani mirata est Lampada docti
Terra, Syracusium Graecia tota senem:
Orbi terrarum, Ptolemaei Clepsydra toti,
Rara dioptra Procli, mira fuêre duo,
Anglia te foveat doctus Pont-Mousson alum­num:
Quidquid naturae, qui legis, hortus habet.
Docta, coronet opus doctum, te sit tua docto
Digna, Syracusii, arca, corona, viri.
Arca Syracusiis utinam sit plumbea servis,
Aurea sed dominis, aurea tota suis.


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.


The Number thought upon
The Quadruple of it
Put 8 unto it, makes
The halfe of it is
[Page 2]Take away halfe the number added from it, viz 4, the rest is
The double of the number thought upon, viz.

Another way to finde what Number was thought upon.

BId him which thinketh double his Number, and unto that double adde 4, and bid him multiply that same product by 5, and unto that product bid him adde 12, and multiply that last number by 10 (which is done easily by setting a Cypher at the end of the number) then ask him the last number or product, and from it secretly subtract 320, the remainder in the hundreth place, is the number thought upon.


The number thought upon
7 For which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon.
His double
14 For which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon.
To it add 4, makes
18 For which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon.
Which multiplyed by 5 makes
90 For which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon.
To which add 12 makes
102 For which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon.
This multiplyed by 10 which is only by adding a Cypher to it, makes
1020 For which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon.
From this subtract
320 For which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon.
700 For which 700 account onely but the number of the hundreds viz. 7. so have you the number thought upon.

To finde numbers conceived upon, otherwise than the former.

BId the party which thinks the number, that he triple his thought, and cause him to take the half of it: (if it be odde take the least half, and put one unto it:) then will him to triple the half, and take half of it as before: lastly, ask him how many nines there is in the last half, and for every nine, account four in your memo­ry, for that shall shew the number thought up­on, if both the triples were even: but if it be odde at the first triple, and ev [...]n at the second, for the one added unto the least halfe keep one in memory: if the first triple be even, and the second odde, for the one added unto the least halfe keepe two in memory; lastly, if at both times in tripling, the numbers be odde, for the two added unto the least halfes, keep three in memory, these cautions observed, and added un­to as many fours as the party sayes there is nines contained in the last halfe, shall never fail you to declare or discern truly what number was thought upon.


The number thought upon
4 or 7
The triple
12 or 21
The half thereof 6 or 10, one put to it makes
The triple of the halfe
18 or 33
The halfe 9 or 1 [...], one put to it makes
The number of nines in the last halfe
1 or 1

[Page 4]The first 1. representeth the 4. number thought upon, and the last 1. with the caution makes 7. the other number thought upon.


Order your method so that you be not disco­vered, which to help, you may with dexterity and industry make Additions▪ Substractions, Mul­tiplications, Divisions, &c. and instead of asking how many nines there is, you may ask how ma­ny eights tens, &c. there is, or subtract 8.10. &c. from the Number which remains, for to finde out the number thought upon.

Now touching the Demonstrations of the former directions, and others which follow, they depend upon the 2, 7, 8, and 9, Books of the Elements of Euclide: upon which 2. Book & 4. proposition this may bee extracted, for these which are more learned for the finding of any number that any one thinketh on.

Bid the party that thinks, that he break the number thought upon into any two parts, and unto the Squares of the parts, let him adde the double product of the parts, then ask what it amounteth unto, so the root Quadrat shall be the number thought upon.

The number thought upon 5, the parts suppose 3 and 2.
[Page 5]The square of 3 makes
9 the sum of these three nūbers 25, the squa [...]e Root of which is 5, the number thought upon
The square of 2 makes
4 the sum of these three nūbers 25, the squa [...]e Root of which is 5, the number thought upon
The product of the parts. viz. 3 by 2 makes 6, which 6 doubled makes
12 the sum of these three nūbers 25, the squa [...]e Root of which is 5, the number thought upon

Or more compendiously it may be delivered thus.

Break the number into two parts, and to the product of the parts, adde the square of half the difference of the parts, then the Root Quadrat of the aggregate is halfe the number conceived.


THe Problems which concern Arithme­tick, we examine not, for these are easie to any one which hath read the grounds and prin­ciples of Arithmetick, but we especially touch upon that, which tends to the speculations of Physick, Geometry, and Optickes, and such others which are of more difficulty, and more principally to be examined and conside­red.

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 wch 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.


THe species being pressed together or con­tracted doth not perform it upon a wall, for the species of any thing doth represent it selfe not only in one hole of a window, but in infinite holes, even unto the whole Sphere, or at least unto a Hemisphere (intellectuall in a free medium) if the beams or reflections be not interposed, and by how much the hole is made less to give passage to the species, by so much the more lively are the Images formed.

In convexe, or concave Glasses the Ima­ges will be disproportionable to the eye, by how much they are more concave, or convexe, & by how much the parts of the image comes [Page 9] neer to the Axis, for these that are neer are better proportioned then these which are far­ther off.

But to have them more lively and true, ac­cording to the imaginary conicall section, let the hole be no greater than a pins head made upon a piece of thin brasse, or such like, which hole represents the top of the Cone, and the Base thereof the term of the species: this pra­ctice is best when the sun shines upon the hole, for then the objects which are opposite to that plaine will make two like Cones, and will lively represent the things without in a per­fect inversed perspective, which drawn by the Pensill of some artificiall Painter, turn the paper upside down, and it will be direct and to the life.

But the apparences may be direct, if you place another hole opposite unto the former, so that the spectator be under it; or let the spe­cies reflect upon a concave Glass, and let that glas reflect upon a paper or some white thing.

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.


THis Problem was extracted from Sca­liger, who had it from Aristotle, but somwhat refractory compiled, & the strength of the effect he says depends only in the vio­lence of the motion; then would it follow that a little light hammer upon a piece of wood being quickly caused to smite, would give a greater blow, and do more hurt than a great sledge striking soft; this is absurd, and con­trary to experience: therefore it consists not totally in the motion, for if two severall hammers, the one being 20 times heavier than the other, should move with like quick­ness, the effect would be much different, there is then some thing else to be considered besides the Motion which Scaliger understood not, for if one should have asked him, what is the reason that a stone falling from a win­dow to a place neer at hand, is not so forceable as if it fell farther [...] when a bullet flying out of a peece and striking the mark neer at hand [...] not make such an effect as striking [...] that Scaliger and [...] this subiect▪ would not be less troubled to re­solve this, than they have been in that.

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.


IT were necessary here to note, that this thing may be experimented, first, without Glasses, in placing a small slender staffe up­on two props, and then making tryall upon it, by which you may see how the Staffe will either break, bow, or depart from his props, and that either directly or obliquely: But why by this violence, that one Staffe striking another, (which is supported by two Glasses) will be broken without offending the Glasses, is as great a difficulty to be resolved as the former.

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.

Populeam virgam mater regina ferebat.

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 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.


THis Proposition seems doubtfull; for to imagine absolutely, that a Pike, or such like, armed with two Knives, or other things, shall stand upright in the Aire, and so re­main without any other support, seeing that all the parts have an infinite difference of propensity to fall; and it is without question that a staff so accommodated upon his Centre of gravity, but that it may incline to some one part without some remedy be applied, and such as is here specified in the Probleme will not warrant the thing, nor keep it from fal­ling; and if more Knives should be placed about it, it should cause it to fall more swift­ly, forasmuch as the superiour parts (by rea­son of the Centricall motion) is made more ponderous, and therefore lesse in rest.

To place therefore this prop really, let the two Knives, or that which is for counterpoise, be longer always then the staffe, and so it will hang together as one body: and it will appear admirable if you place the Centre of gravity, neer the side of the top of the finger or point; for it will then hang Horizontall, and seem to hang onely by a touch, yet more strange, if you turn the point or top of the finger upside down.

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.


NOw although it be thus delivered, yet here may be noted, that a ponderosity in his own medium is not weighty: for things are said to be weighty, when they are out of their place, or medium, and the difference of such gravity, is according to the motion: the smoak therefore certainly is light being in its true medium (the aire,) if it should change his medium, then would we change our discourse.

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.


NOw without some experience of this Probleme, a man would acknowledge either a possibility or impossibity; there­fore it is that very touchstone of knowledge in any thing, to discourse first if a thing be possible in nature, and then if it can be brought to experience and under sence with­out seeing it done. At the first, this proposition seems to be absurd, and impos­sible. Notwithstanding, being supported [Page 32] with two sticks, as the figure declareth, it is made facile: for the Horizontall line to the edge of the Table, is the Centre of motion; and passeth by the Centre of gravity, which necessarily supporteth it.

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 upon.
The number thought
The halfe of it
The summe
The halfe of it
The summe of it
The double of the number,
Which taken away, rests
The halfe of it
For which account
and 1 put to it because the
could not be halfed, makes
this multiplied by 4 makes
The number thought
The greatest halfe
40 3
The summe
The greatest halfe of which is
60 2
The summe of it is
The double of 79 is
Which taken from it, rests
The lesser half 10. wch halve:
The halfe of this is 5 which makes
The half of this is 2 wch 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.


YOu may note upon the last two Problemes farther, that if a Cone be cut Ecliptick-wise, it may passe through an Issoc [...] Tri­angle through many Scalen Triangles, and through an Ellipsis; and if there be a Cone cut scalen-wise, it will passe through all the former, only for the Ellipsis place a Circle: and further, if a solid colume be cut Eclip­tick-wise it may fill a Circle, a Square, di­vers Parallelogrammes, and divers Ellipses, which have different Diameters.

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 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 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.


THe cause of this, is not in the rarefa­ction of the parts of the Glasse, but it is rather in the quick locall motion of the fin­ger, for reason sheweth us that by how much a Body draweth nearer to a quality, the lesse is it subject or capable of another which is contrary unto it? now condensation, and ra­refaction are contrary qualities, and in this Probleme there are three bodies considered, the Glasse, the Water, and the Aire, now it is e­vident that the Glasse being the most solid, and impenitrable Body, is lesse subject and capable of rarefaction than the water, the water is lesse subject than the Aire, and if there be any rarefaction, it is rather consi­derable in the Aire then in the Water, which is inscribed by the Glasse, and above the Wa­ter, and rather in the Water then in the Glasse: the agitation, or the trembling of the parts of the Glasse to the sense appeares not: for it is a continued body; if in part, why then not in the whole? and that the Water turnes in the Glasse, this appeares not, but only the [Page 56] upper contiguous parts of the Water: that at the bottome being lesse subiect to this agi­tation, and it is most certaine that by how much quicker the Circular motion of the fin­ger upon the edge of the Glasse is, by so much the more shall the Aire be agitated, and so the water shall receive some apparant affe­ction more or lesse from it, according to that motion: as we see from the quicknesse of winde upon the Sea, or c [...]lme thereof, that there is a greater or lesser agitation in the water; and for further examination, we leave it to the search of those which are cu­rious.

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▪


HEre is to be noted, that if the face of the water without be in one and the same plaine, with that which is within, though the outtermost Pipe be ten times greater than that which is within; the water naturally will not runne, but if the plaine of the wa­ter without be any part lower then that which is within, it will freely runne: and here may be noted further, that if the mouth of the Pipe which is full of water, doth but only touch the superficies of the water within, although the other end of the pipe without be much lower than that within, the water it will not run at all: which contradicts the first ground; hence we gather that the pressure or ponderosity of the water within, is the cause of running in some respect.

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.


IF the two Boxes being of equall magnitude weighed in the aire be found to be of equall weight, they shall necessarily take up like place in the water, and therefore weigh also one as much as another: hence there is no possibilitie to finde the inequalitie of the met­talls which are inclosed in these Boxes in the water: the intention of Archimedes was not upon contrary mettalls inclosed in [...] Boxes, but consisted of comparing metta [...], simple in the water one with another: there­fore the inference is false and absurd.

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.


HEre is observable, that the moisture which aideth to the growth of the tree, is dila­ted and rarefied by the Meridionall heat, and contracted by the Septentrionall cold: this rarefaction works upon the part of the hu­mour or moisture that is more thinne, which doth easily dissipate and evaporate: which [Page 77] evaporation carries a part of the salt with it; and because that solidation or condensation, so that there is left but a part of the nourish­ment which the heat bakes up and consumes: so contrarily on the other side the condensati­on and restrictive quality of the moisture causeth lesse evaporation and perdition: and so consequently there remaines more nourish­ment, which makes a greater increase on that side than on the other side: for as trees have their growth in winter, because of their pores and these of the earth are shut up: so in the spring when their pores are open, and when the sappe and moisture is drawne by it, there is not such cold on the North-side that it may be condensed at once: But contrarily to the side which is South, the heat may be such, that in little time by continuance, this moisture is dissipated greatly: and cold is nothing but that which hardneth and con­tracteth the moisture of the tree, and so con­verteth it into wood.

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 9th 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.


IT is certaine, that not onely Glasses dyed green, but all other Glasses coloured, yield the app [...]arances of objects strong or weak in colour according to the quantity of the dye, more or lesse, as one being very yellow, [Page 99] another a pale yellow; now all colours are not proper to Glasses to give colour, hence the de­fect is not that they want facultie to receive light, or resist the penetration of the beams; for in the same Glasses those which are most dyed, give alwayes the objects more high coloured and obscure, and those which are lesse dyed give them more pale and cleare: and this is daily made manifest by the paint­ing of Glasse, which hinders more the pene­tration of the light than dying doth, where all the matter by fire is forced into the Glasse, leaving it in all parts transparent.

Spectacles of Crystall cut with divers Angles diamond-wise do make a marvellous multiplication of the appearances, for looking towards a house it becomes as a Towne, a Towne becomes like a Citie, an armed man seems as a whole company caused solely by the diversity of refractions, for as many plaines as there are on the outside of the spectacle, so many times will the object be multiplied in the appearance, because of diverse Images cast into the eye. These are pleasurable spectacles for avaricious persons that love Gold and silver, for one piece will seeme many, or one heap of money will seeme as a treasury: but all the mischiefe is, he will not have his end in the enjoying of it, for indeavouring to take [Page 100] it, it will appeare but a deceitfull Image, or delu­sion of nothing. Here may you note that if the finger be directed by one and the same ray or beam, which pointeth to one and the same object, then at the first you may touch that visible object without being de­ceived: otherwise you may faile often in touching that which you see. Againe, there are Spectacles made which do diminish the thing seen very much, and bring it to a faire perspective forme, espe­cially if one look upon a faire Garden plat, a greater walk, a stately building, or great Court, the indu­stry of an exquisite Painter cannot come neare to expresse the lively forme of it as this Glasse will re­present it; you will have pleasure to see it really ex­perimented, and the cause of this is, that the Glasses of th [...]se Spectacles are hollow and thinner in the middle, than at the edges by which the visuall Angle is made lesser: you may observe a further secret in these Spectacles, for in placing them upon a window one may see those that passe to and fro in the streets, without being seen of any, for their property is to raise up the objects that it lookes upon.

Now I would not passe this Probleme without saying something of Galileus admirable Glasse, for the common simple perspective Glasses give to aged men but the eyes or sight of young men, but this of Galileus gives a man an Eagles eye, or an eye that pierceth the heavens: first it discovereth the spottie and shadowed opacous bodies that are found about the Sunne, which darknet and diminisheth the splendor of that beautifull and shining Lumina­ry: secondly, it shewes the new Planets that ac­company [Page 101] Saturne and Jupiter: thirdly, in Venus is seen the new, full, and quartill in­crease; as in the Moon by her separation from the Sunne: fourthly, the artificiall structure of this instrument helpeth us to see an innumerable number of stars, which other­wise are obscured, by reason of the naturall weaknesse of our sight, yea the starres in via lactea are seen most apparantly; where there seem no starres to be, this Instrument makes apparantly to be seen, and further delivers them to the eye in their true and lively co­lour, as they are in the heavens: in which the splendor of some is as the Sunne in his most glorious beauty. This Glasse hath also a most excellent use in observing the body of the Moone in time of Eclipses, for it aug­ments it manifold, and most manifestly shewes the true forme of the cloudy substance in the Sunne; and by it is seene when the shadow of the earth begins to eclipse the Moon, & when totally she is over shadowed: besides the celestiall uses which are made of this Glasse, it hath another noble property; it farre exceedeth the ordinary perspective Glasses, which are used to see things remote upon the earth, for as this Glasse reacheth up to the heavens and excelleth them there in his performance, so on the earth it claimeth [Page 102] preheminency, for the objects which are far­thest remote, and most obscure, are seen plain­er than those which are neere at hand, scorn­ing as it were all small and triviall services, as leaving them to an inferiour help: great use may be made of this Glass in discovering of Ships, Armies, &c. Now the apparell or parts of this instrument or Glasse, is very meane or simple, which makes it the more ad­mirable (seeing it performes such great ser­vice) having but a convex Glasse thickest in the middle, to unite and amasse the rayes, and mak the object the greater: to the augmenting the visuall Angle, as also a pipe or trunk to amasse the Species, and hinder the greatness of the light which is about it: (to see well, the object must be well inlightened, and the eye in obscurity;) then there is adjoyn­ed unto it a Glasse of a short sight to distin­guish the rayes, which the other would make more confused if alone. As for the proportion of those Glasses to the Trunk, though there be certaine rules to make them, yet it is often by hazard that there is made an excellent one there being so many difficulties in the action, therefore many ought to be tryed, seeing that exact proportion, in Geometricall calculati­on cannot serve for diversity of sights in the observation.

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.


THe experimentall difference of rejecti­on, and attraction proceeds not from the different nature of Stones, but from the qua­lity of the Iron; and the vertue of the stone consisteth only, and especially in his poles, which being hanged in the Aire, turnes one of his ends alwayes naturally towards the South, and the other towards the North: but if a rod of Iron be touched with one of the ends thereof, it hath the like property in [Page 107] turning North and South, as the magnes hath: notwithstanding the end of the Iron Rod touched, hath a contrary position, to that end of the stone that touched it; yet the same end will attract it, and the other end reject it: and so contrarily this may easily be experi­mented upon two needles touched with one or different stones, though they have one and the same position; for as you come unto them apply one end of the magnes neare unto them, the North of the one will abhorre the North of the other, but the North of the one will alwayes approach to the South of the other: and the same affection is in the stones themselves. For the finding of the Poles of the magnes, it may be done by holding a small needle between your fingers softly, and so moving it from part to part over the stone untill it be held perpendicular, for that shall be one of the Poles of the stone which you may marke out; in like manner finde out the other Pole: Now to finde out which of those Poles is North or South, place a needle being touched with one of the Poles upon a smooth convex body, (as the naile of ones finger or such like,) and marke which way the end of the needle that was touched turn­eth: if to the South, then the point that [Page 100] [...] [Page 101] [...] [Page 102] [...] [Page 103] [...] [Page 104] [...] [Page 105] [...] [Page 106] [...] [Page 107] [...] [Page 108] touched it was the South-Pole, &c. and it is most certain and according to reason and ex­perience: that if it be suspended in aequi­librio in the aire, or supported upon the wa­ter, it will turne contrary to the needle that toucheth it; for then the pole that was marked for the South shall turne to the North, &c.

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.

Of Statues, of Colossus, or huge images.

VItruvius relates in his second book, that the Architect Dinocrates was desirous to put out to the world some notable thing, went to Alexander the great, and proposed unto him a high and speciall piece of work which he had projected: as to figure out the mount Athos in forme of a great Statue, which should hold in his right hand a Towne capable to receive ten thousand men: and in his left hand a vessell to receive all the water that floweth from the Mountaine, which with an ingine should cast into the Sea. This is a pretty project, said Alex­ander, but because there was not field-roome thereabout to nourish and reteine the Citizens of that place, Alexander was wise not to enter­taine the designe.

Now let it be required of what greatnesse this Statue might have been, the Towne in his right hand, and the receiver of water in his left hand if it had been made.

For the Statue, it could not be higher than the Mountaine it selfe, and the Mountaine was about a mile in height plumb or perpendicular; [Page 118] therefore the hand of this Statue ought to be the 10th part of his height, which would be 500 foot, and so the breadth of his hand would be 250 foot, the length now multiplyed by the breadth, makes an hundred twenty five thousand square feet, for the quantitie of his hand to make the towne in, to lodge the said 10 thou­sand men, allowing to each man neere about 12 foot of square ground: now judge the ca­pacitie of the other parts of this Collossus by that which is already delivered.

Secondly, Plinie in his 34 book of his natural History, speakes of the famous Colossus that was at Rhodes, between whose legges a Shippe might passe with his sailes open or displayed, the Statue being of 70 cubits high: and other Histories report that the Sarasens having bro­ken it, did load 900 Camels with the mettal of it, now what might be the greatnesse and weight of this Statue?

For answer, it is usually allowed for a Ca­mels burthen 1200 pound weight, therefore all the Collosus did weigh 1080000 pound weight, which is ten hundred and fourescore thousand pound vveight.

Novv according to the former rules, the head being the tenth part of the body, this Statues head should be of 7 cubits, that is to say, 10 foot and a halfe, and seeing that the Nose, the brovv, and the thumbe, are the third part of the face, his Nose vvas 3 foot and a halfe long, and so much also vvas his thumbe in length: novv the thicknesse being alvvayes the third part of [Page 119] the length, it should seem that his thumb was a foot thick at the least.

Thirdly, the said Plinie in the same place reports that Nero did cause to come out of France into Italy, a brave and bold Statue-ma­ker called Zenodocus, to erect him a Colossus of brasse, which was made of 120 foot in height, which Nero caused to be painted in the same height. Now would you know the greatnesse of the members of this Colossus, the breadth would be 20 foot, his face 12 foote, his thumb and his nose 4 foot, according to the proportion before delivered.

Thus I have a faire field or subject to extend my selfe upon, but it is upon another occasion that it was undertaken, let us speak therefore a word touching the Giants, and then passe away to the matter.

Of monstrous Giants.

YOu will hardly beleeve all that which I say touching this, neither will I beleeve all that which Authors say upon this subject: not­withstanding you nor I cannot deny but that long ago there have been men of a most pro­digious greatnesse; for the holy vvritings vvitnesse this themselves in Deut, Chap. 3. that there vvas a certaine Giant called Og, of the Town of Rabath, vvho had a bed of Iron, the length thereof vvas 9 cubits, and in breadth 4 cubits.

So in the first of Kings Chap. 17. there is mention made of Goliah, vvhose height vvas a [Page 120] palme and 6 cubits, that is more then 9 foot, he was armed from the head to the foot, and his Curiat onely with the Iron of his lance, weighed five thousand and six hundred shekels, which in our common weight, is more than 233 pound, of 12 ounces to the pound: Now it is certaine, that the rest of his armes taking his Target, Helmet, Bracelets, and other Ar­mour together, did weigh at the least 5 hun­dred pound, a thing prodigious; seeing that the strongest man that now is, can hardly beare 200 pound, yet this Giant carries this as a ve­sture without paine.

Solinus reporteth in his 5 Chap. of his Histo­rie, that during the Grecians warre after a great overflowing of the Rivers, there was found up­on the sands the carcase of a man, whose length was 33 Cubits, (that is 49 foot and a halfe) therefore according to the proportion delivered, his face should be 5 foot in length, a thing prodigious and monstrous.

Plinie in his 7. book and 16 Chap. saith, that in the Isle of Crete or Candie, a mountaine be­ing cloven by an Earth-quake, there was a bo­dy standing upright, which had 46 Cubits of height: some beleeve that it was the body of Orion or Othus, (but I think rather it was some Ghost or some delusion) whose hand should have beene 7 foot, and his nose two foot and a half long. But that which Plutarch in the l [...]fe of Sertorius reports of, is more strange, who saith, that in Timgy a Morative Towne, where [Page 121] it is thought that the Giant Antheus was buri­ed, Sertorius could not beleeve that which was reported of his prodigious greatnesse, caused his sepulchre to be opened, and found that his body did containe 60 Cubits in length, then by proportion he should be 10 Cubits or 15 foot in breadth; 9 foot for the length of his face, 3 foot for his thumb, which is neare the capa­citie of the Colossus at Rhodes.

But behold here a fine fable of Symphoris Campesius, in his book intituled Hortus Galli­cus, who sayes that in the Kingdome of Sicilie, at the foot of a mountaine neare Trepane, in opening the foundation of a house, they found a Cave in which was [...]aid a Giant, which held in stead of a staffe a great post like the mast of a Ship: and going to handle it, it mouldered all into ashes, except the bones which remained of an exceeding great measure, that in his head there might be easily placed 5 quarters of corn, and by proportion it should seeme that his length was 200 cubits, or 300 foot: if he had said that he had been 300 cubits in length, then he might have made us beleeve that Noahs Ark was but great enough for his sepulchre.

Who can believe that any man ever had 20 cubits, or 30 foot in length for his face, and a nose of 10 foot long? but it is very certaine that there have been men of very great stature, as the holy Scriptures before witnesse, and many Authours worthy of beliefe relate: Jo­sephus Acosta in his first book of the Indian History, Chap. 19. a late writer, reporteth, that [Page 122] at Peru was found the bones of a Giant, which was 3 times greater than these of ours are, that is 18 foot, for it is usually attributed to the tallest ordinary man in these our times but 6 foot of length; and Histories are full of the description of other Giants of 9, 10, and 12 foot of height, and it hath been seen in our times some which have had such heights as these.

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.


IN this Examination we have something else to imagine, than the bare sympathie of the Cords one to another: for first there ought to be considered the different effect that it produceth by extention upon one and the same Cord in capacitie: then what might be produced upon different Cords of length and bigness to make them accord in a unisone or octavo, or some consort intermediate: this being naturally examined, it will be facill to lay open a way to the knowledge of the true and immediate cause of this noble and admi­rable Phaenomeny. Now this will sensibly appeare when the Cords are of equall length and greatnesse, and set to an unisone; but when the Cords differ from their equalitie, it will be lesse sensible: hence in one and the same Instrument, Cords at a unisone shall ex­cite or shake more than that which is at an octavo, and more than those which are of an intermediate proportionall consort: as for the other consorts they are not exempted, though the effect be not so sensible, yet more in one than in another: and the experiment will seem more admirable in taking 2 Lutes, Viols, &c. & in setting them to one tune: for then in touching the Cord of the one, it will [Page 128] give a sensible motion to the Cord of the o­ther: and not onely so but also a harmony.

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.


IT is certaine in the first part of this Pro­bleme that Conicall, [...]oncave and sphericall Glasses, of what matter soever, being placed to receive the beames of the Sun will excite heat, and that heat is so much the greater, by how much it is neere the point of conc [...]rse or inflamatio [...]. But that Archimedes or Proclus d [...]d fire or burne Shipps with such Glasses, the ancient Histories are silent, yea the selves say nothing: besides the great difficultie that doth oppose it in remotenesse, and the matter that the effect is to work upon: Now [Page 133] by a common Glasse we fire things neare at hand, from which it seems very facil to such which are lesse read, to do it at a farre greater distance, and so by re [...]ation some de­liver to the World by supposition that which never was done in action: this we say the rather, not to take away the most excellent and admirable effects which are in Burning-glasses, but to shew the variety of Antiquity, and truth of History: and as touching to burne at a great distance, as is said of some, it is absolutely impossible; and that the Pa­rabolicall and Ovall Glasses were of Ar­chimedes and [...]roclus invention is much uncertaine: for besides the construction of such Glasses, they are more difficult than the obtuse concave ones are; and further, they cast not a great heat but neere at hand; for if it be cast farre off, the effect is little, and the heat weake, or otherwise such Glasses must be greatly extended to contract many beames to amasse a sufficient quantity of beames in Parabolicall and Conicall Glasses, the point of inflammation ought to concur in a point, which is very difficult to be done in a due pro­portion. Moreover if the place be farre re­mote, as is supposed before, such a Glasse cannot be used but at a great inclination of [Page 134] the Sunne▪ by which the eff [...]ct of [...]urning is d [...]min [...]shed, by reason of the weaknesse of the Sunne-beames.

And here may be noted in the last part of this Probleme, that by r [...]ason of obstacles if one plaine Glasse be not sufficient, a second Glasse may be applyed to help it: that so if by one simple reflection it cannot be done, yet by a double reflection the Sun-beames may be [...]ast into the said Caverne or Mine, and though the reflected beams in this case be weak▪ yet upon a [...] c [...]mbustible matter it will not faile to do the effect.

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

Of the [...] the Mule.

JT [...] [...]hat [...]he Mule and the Asse upon a day [...] a voyage each of them carried [Page 135] a Barrell full of Wine: now the las [...]e Asse f [...]lt her selfe over-loaden, complained and bowed under her burthen; which th [...] Mule seeing said unto her being angry, (for it was in the time when beasts spake) Thou great Asse, wherefore complainest thou? if I had but onely one mea­sure of that which thou carriest, I should be loaden twice as much as thou art, and if J should give a measure of my loading to thee, yet my burthen would be as much as thine.

Now how many measures did each of them carry? Answer, the Mule did carry 7 measures, and the Asse 5 measures: for if the Mule had one of the measures of the Asses loading, then the Mule would have 8 measures, which is double to 4, and giving one to the Asse, each of them would have equall burthens: to wit, 6 measures apiece.

Of the number of Souldiers that fought before old Troy.

HOmer being asked by He [...]iodus how many Grecian Souldiers came against Troy? who answered him thus; The Grecians, said Homer, made 7 fires, or had 7 Kitchins, and before eve­ry fire, or in every Kitchin there were 50 broa­ches turning to rost a great quantitie of flesh, and each broach had meat enough to satisfie 900 men: now judge how many men there might be. Answer, 315000. that is, three hun­dred and fifteen thousand men, which is cleare by multiplying 7 by 50, and the product by 900 makes the said 315000.

Of the number of Crownes that two men had.

JOhn and Peter had certaine number of crowns: John said to Peter, If you give me 10 of your crownes, I shall have three times as much as you have: but Peter said to J [...]hn, If you give me 10 of your crownes I shall have 5 times as much as you have: how much had each of them? An­swere, John had 15 crownes and 5 sevenths of a crowne, and Peter had 18 crownes, and 4 se­venths of a crowne. For if you adde 10 of Pe­ters crownes to those of Johns, then should John have 25 crownes and 5 sevenths of a crowne, which is triple to that of Peters, viz. 8▪ and 4 sevenths: and John giving 10 to Peter, Peter should have then 28 crownes, and 4 sevenths of a crowne, which is Quintupla, or 5 times as much as John had left, viz. 5 crownes and 5 se­venths.

In like manner two Gamesters playing toge­ther, A and B ▪ after play A said to B, Give me 2 crownes of thy money, and I shall have twice as much as thou hast: and B said to A, Give me 2 crownes of thy money, and I shall have 4 times as much as thou hast: now how much had each? Answer, A had 3 and 5 se­venthes, and B had 4 and 6 seventhes.

About the houre of the day.

SOme one asked a Mathemacian what a clocke it was; who answered that the rest of the day is foure thirds of that which is past: now judge what a clock it is. Answer, if the day were according to the Jewes and ancient Romanes, which ma [...]e it alwayes to be 12 houres, it was then the [...] houre, and one se­venth of an hou [...]e, so there remained of the whole day 6, that is, 6 houres, and 6 sevenths of an hour. Now if you take the 1/ [...] of 5 [...]/7 it is [...]2/7 or [...] and [...] 7, which multipled by 4 makes 6 and 6/7, which is the remainder of the day, as before: but if the day had been 24 houres, then the houre had been 10 of the clock▪ and two seventhes of an houre, which is found▪ out by dividing 12, or 24 by [...].

There might have been added many curious propositions in this kinde, but they vvould be too difficult for the most part of people▪ there­fore I have omi [...]ted them▪

Of Pythagoras his Schollers.

PYthagoras being asked what number of Schollers he had, ansvvered, that halfe of them studied Mathematickes, the fourth part Physick, the seventh part Rethorick, and be­sides he had 3 vvomen: novv judge you saith he, hovv many Schollers I have. Ansvver, he had in all 28, the halfe of vvhich is 14, the quarter [Page 138] of which is 7, and the seventh part of which is which 14, 7, and 4, makes 25, and the other 3 to make up the 28, were the 3 women.

Of the number of Apples given amongst the Graces and the Muses.

THe three Graces carrying Apples upon a day, the one as many as the other, met with the 9 Muses, who asked of them some of their Apples; so each of the Graces gave to each of the Muses alike, and the distribution being made, they found that the Graces & the Muses had one as many as the other: The que­stion is how many Apples each Grace had, and how many they gave to each Muse? [...]o an­svver the qeustion, joyne the number of Graces and Muses together vvhich makes 12, and so many Apples had each Grace: Novv may you take the double, triple, &c. of 12 that is 24, 36, &c. conditionally, that if each Grace had but 12, then may there be allotted to each Muse but one onely; if 24, then to each 2 Apples, if [...]6, then to each Muse 3 Apples, and so the di­stribution being made, they have a like number, that is one as many as the other.

Of the Testament or last Will of a dying Father.

A Dying Father left a thousand Crovvnes a­mongst his tvvo children; the one being legitimate, and the other a Bastard, conditio­nally [Page 139] that the fifth part which his legittimate Sonne should have, should exceed by 10, the fourth part of that which the Bastard should have: what was each [...] part? Answer, the legitimate Sonne had 577 crownes and 7/ [...], and the Bastard 42 [...] crownes and 2/9 now the fifth part of 577 and 7 ninthes is 1 [...]5, and 5/9, and the fourth part of 422 and [...] is 105 and [...] which is lesse then [...]15 [...] by 10, according to the Will of the Testator.

Of the Cups of Croesus.

CRoesus gave to the Temple of the [...]ods six Cups of Gold▪ which weighed together [...]00 Drammes, but each cup was heavier one than another by one Dram: how much did each of them therefore weigh? Answer, the first weigh­ed 102 Drammes and a halfe; the second 101 Drammes and a halfe, the third 100 Drammes and [...], the fourth 99 a & halfe, the fifth 98 & a halfe; and the sixt Cup weighed 97 Drammes and a halfe▪ which together makes 600 Drams as before.

Of Cupids Apples.

CVpid complained to his mother that the Muses had taken away his Apples, Clio, said he, took from me the fifth part, Euterp the twelfth part, Thalia the eighth part, M [...]lp [...]meno the twentieth part, Erates the seventh part▪ Terpomene the fourth part, Polyhymnia took a­way 30, Vrania 220, and Calliope 300. so [Page 140] there vvere left me but 5 Appls, hovv many had he in all at the first? I ansvver 3 [...]60.

There are an infinite of such like questions a­mongst the Greek Epigrams: but it would be un­pleasant to expresse them all: I will onely adde one more, and shew a generall rule for all the rest.

Of a Mans Age.

A Man vvas said to passe the sixth part of his life in childe-hood, the fourth part in his youth, the [...]hird part in Manhood, and 18 yeares besides in old age: what might his Age be? the ansvver is, 72 yeares: vvhich and all o­thers is thus resolved: multiply 1/ [...] ▪¼ and ⅓▪ toge­ther, that is, 6 by 4 makes 24, and that againe by 3 makes 72, then take the third part of 72, vvhich is 24, the fourth part of it, vvhich is 18, and the sixth part of it vvhich is 12, these added together make 54, vvhich taken from 72, rests 18 this divided by 18 (spoken in the Question) gives 1, which multiplied by the summe of the parts, viz. 72, makes 72, the Ansvver as before.

Of the Lion of Bronze placed upon a Foun­taine with this Epigramme.

OVt of my right eye if I let vvater passe, I can fill the Cisterne in 2 dayes: if I let it passe out of the left eye, it vvill be filled in 3 dayes: if it passe out of my feet, the Cistern vvill be 4 dayes a [...]filling; but if I let the vvater passe out of my mouth, I can fill the Cistern then in 6 [Page 141] houres: in vvhat time should I fill it, if I poure forth the vvater at all the passages at once?

The Greeks (the greatest talkers in the vvorld) variously apply this question to divers statues, and pipes of Fountaines: and the solu­tion is by the Rule of [...], by a generall Rule, or by [...]lgebra. They have also in their Antholo­gie many other questions, but because they are more proper to exercise, than to recreate the spi­rit, I passe them over (as before) with silence.

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.

Experiment upon flat and plaine Glasses.

FIrst, a man cannot see any thing in these Glasses, if he be not directly and in a perpen­dicular line before it, neither can he see an ob­ject in these Glasses, if it be not in such a place, that makes the Angle of incidence equall to the Angle of reflexion: therefore when a Glasse stands upright, that is, perpendicular to the Horizon, you cannot see that which is above, except the Glasse be placed down flat: and to see that on the right hand, you must be on the left hand, &c.

Secondly, an image cannot be seen in a Glass if it be not raised above the surface of it; or place a Glasse upon a wall, you shall see no­thing which is upon the plaine of the wall, and place it upon a Table or Horizontal Plaine, you shall see nothing of that which is upon the Table.

[Page 143]Thirdly, in a plaine Glasse all that is seene appeares or seemes to sink behinde the Glasse, as much as the image is before the Glasse, as before is said.

Fourthly, (as in water) a Glasse lying downe flat, or Horizontall, Towers, Trees, Men, or any height doth appeare, inversed or upside downe; and a Glasse placed upright, the right hand of the Jmage seems to be the left, and the left seems to be the right. Fifthly, will you see in a Chamber that which is done in the street, without being seen▪ then a Glasse must be dis­posed, that the line upon which the Jmages come on the Glasse, make the Angle of inci­dence equall to that Angle of reflexion.

Sixtly, an height as suppose DE.) may be measured by a plaine Glasse, as let the Glasse be G. placed downe upon the ground, and let the eye be at C. so farre


removed from the Glasse, that the eye at C. may see the toppe of the Tower E in the Angle or edge of the Glass at A, but in the line of reflexion CA, then measure the di­stance between your foot B, and the point A, & also the distance betweene the Glasse A ▪ and the foot of the Tower D, viz. AD. Now as often as AB is found in AD, so often doth the height of ther [Page 144] Tower ED contain the distance from your eye to the foot, viz. CB for the Triangles A, B, C, and A, D, E, are equal Triangles: therefore as BA. to AD, so CB, to ED, or alternately as BA. to BC, so AD. to D [...].

Seventhly, present a Candle upon a plaine Glasse, and look flaunting upon it, so that the Candle and the Glasse be neere in a right line, you shall see 3, 4, 5, &c. images, from one and the same Candle.

Eightly, take tvvo plaine Glasses, and hold them one against the other, you shall alternately see them oftentimes one vvithin the other, yea vvithin themselves, againe and againe.

Ninthly, if you hold a plaine Glasse behinde your head, and another before your face, you may see the h [...]nder part of your head, in that Glasse vvhich you hold before your face.

Tenthly, you may have a fine experiment if you place tvvo Glasses together, that they make an acute angle, and so the lesser the angle is, the more apparances you shall see, the one di­rect, the other inversed, the one approaching, and the other retiring.

Eleventhly, it is a vvonder & astonishment to some, to see within a Glasse an Image vvithout knovving from vvhence it came, and it may be done many vvayes: as place a Glass higher than the eye of the beholder, and right against it is some Image; so it resteth not upon the behol­der, but doth cast the Image upvvards. Then place another object, so that it reflect, or cast [Page 145] the Image downeward to the eye of the specta­tor▪ without perceiving it being hid behinde something, for then the Glasse will represent a quite contrary thing, either that which is before the Glasse, or that which is about it, to wit, the other hidden object.

Twelfthly, if there be ingraved behinde the backside of a Glasse, or drawne any Image up­on it, it will appeare before as an Image, with­out any appearance: o [...] portraicture to be per­ceived.


THis 12 Article of ingraving an Image behinde the Glasse, will be of no great consequence▪ be­cause the lineaments will seem so obscure, but if there were painted some Image, and then that covered ac­cording to the usuall covering of Glasses behinde, and so made up like an ordinary looking-Glasse ha­ving an Image in the middle, in this respect it would be sufficiently pleasant: and that which would ad­mire the ignorant, and able to exercise the most subtillest, and that principally if the Glasse be in an obscure place, and the light which is given to it be somewhat farre off.

PLace a Glasse neare the floor of a Chamber, & make a hole through the place under the Glasse, so that those which are below may not perceive it, and dispose a bright Image under [Page 146] the hole so that it may cast his species upon the Glasse, and it will cause admiration to those which are below that know not the cause; The same may be done by placing the Image in a Chamber adjoyning, and so make it to be seen upon the side of the Wall.

14 In these Channel-Images which shew one side a deaths head, & another side a faire face: and right before some other thing: it is a thing evident, that setting a plaine Glasse sidewise to this Image you shall see it in a contrary thing, then that which was presented before sidewise.

15 Lastly, it is a fine secret to present unto a plaine Glasse writing with such industry, that one may read it in the Glasse, and yet out of the Glasse there is nothing to be known, which will thus happen, if the writing be writ back­ward: but that which is more strange, to shew a kinde of writing to a plaine Glasse, it shall ap­pear another kinde of writing both against sense and forme, as if there were presented to the Glasse WEL it would shew it MET; if it were written thus MIV, and presented to the Glasse, it would appeare thus VIM; for in the first, if the Glasse ly flat, then the things are inversed that are perpendicular to the Glass, if the Glass and the object be upright, then that on the right hand, is turned to the left, as in the latter.

And here I cease to speak further of these plaine Glasses, either of the Admirable multi­plications, or appearances, which is made in a great number of them; for to content the sight [Page 147] in this particular, one must have recourse to the Cabinets of great Personages who inrich themselves with most beautifull ones.

Experiments upon Gibbous, or convex Sphericall Glasses.

IF they be in the forme of a Bowle, or part of a great Globe of Glasse, there is singular con­tentment to contemplate on them.

First, because they present the objects lesse and more gracious, and by hovv much more the Images are separated from the Glasse, by so much the more they diminish in Magnitude.

Secondly, they that shew the Images plait­ing, or foulding, which is very pleasant, es­pecially when the Glasse is placed downe, and behold in it some Blanching, feeling, &c. The upper part of a Gallerie, the porch of a Hall, &c. for they will be represented as a great vessel having more belly in the middle then at the two ends, and Posts, and Joists of Timber will seeme as Circles.

Thirdly, that which ravisheth the spirits, by the eye, and which shames the best perspective Painting that a Painter can make, is the beau­tifull contraction of the Images, that appeare within the sphericity of these small Glasses: for present the Glasse to the lower end of a Gal­larie, or at the Corner of a great Court full of People, or towards a great street, Church, for­tification, an Army of men, to a whole Cittie; all the faire Architecture, and appearances will [Page 148] be seene contracted within the circuit of the Glasse with such varietie of Colours, and di­stinctions in the lesser parts, that I know not in the world what is more agreeable to the sight, and pleasant to behold, in which you will not have an exact proportion, but it will be va­riable, according to the distance of the Object from the Glasse.

Exptriments upon hollow, or Concave sphericall Glasses.

I Have heretofore spoken how they may burne, being made of Glasse, or Metall, it re­maines now that I deliver some pleasant uses of them, which they represent unto our sight, and so much the more notable it will be, by how much the greater the Glasse is, and the Globe from whence it is extracted for it must in proportion as a segment of some be made circle or orbe.


IN this we may observe that a section of 2.3. or 4. Inches in diameter, may be segments of spheres of 2.3. or 4. foot [...] nay of so many fadome, for it is certaine that amongst those which comprehend a great portion of a lesser sphere, and those which comprehend a little segment of a great spheere, whe­ther they be equall or not in section, there will hap­pen an evident difference in one and the same expe­riment, [Page 149] in the number, situation, quantitie, and figure of the Images of one or many different objects, and in burning there is a great difference.

MAginus, in a little Tractate that he had up­on these Glasses, witnesseth of himselfe that he hath caused many to be polished for sundry great Lords of Italy, and Germanie, which were segments of Globes of 2.3. and 4. foot diameter; and I wish you had some such like to see the experiments of that which fol­loweth; it is not difficult to have such made, or bought here in Town, the contentment here­in would beare with the cost.


TOuching Maginus he hath nothing ayded us to the knowledge of the truth by his extract out of Vitellius, but left it: expecting it from others, ra­ther than to be plunged in the search of it himselfe, affecting rather the forging of the matter, and composition of the Glasses, than Geometrically to establish their effects.

FIrst therefore in concave Glasses, the Images are seene sometimes upon the surface of the Glasses, sometimes as though they were with­in it and behinde it, deeply sunk into it, some­times they are seene before, and without the Glasse, sometimes between the object and the [Page 150] Glasse; sometimes in the place of the Eye, some­times farther from the Glasse then the object is: which comes to passe by reason of the di­vers concourse of the beames, and change of the place of the Images in the line of reflection.


THe relation of these appearances passe current amongst most men, but because the curious may not receive prejudice in their experiments, something ought to be said thereof to give it a more lively touch: in the true causes of these appearances, in the first place it is impossible that the Image can be upon the surface of the Glasse, and it is a prin­cipall point to declare truly in which place the I­mage is seen in the Glasse those that are more learned in Opticall knowledge affirme the contrary, and nature it selfe gives it a certaine place accord­ing to its position being alwayes seen in the line of reflection which Alhazen, Vitellius, and others full of grea [...] knowledge, have confirmed by their writings: but in their particular they were too much occupied by the authority of the Ancients who were not s [...]fficiently ci [...]cumspect in experience upon which the principles of this sub [...]ect ought to be built, an [...] searched not fully into the true cause of these appearances, seeing they leave unto posterities many [...] in their writings, [...]nd those that followed [Page 151] them for the most part fell into the like errors.

As for the Jmages to bid [...] in the eye▪ it cannot be but is imp [...]rtinent and absurd; but it followeth that, by how much neerer the ob [...]ect appro [...]cheth to the Glasse, by so much the more the appearances seem to come to the eye: and if the eye be without the point of concourse, and the object also; as long as the object approacheth thereto, the representation of the Image cometh neere the eye, but passing the point of concourse it goes back againe: these ap­pearances thus approaching do not a little asto­nish those which are ignorant of the cause: they are inversed, if the eye be without the point of con­course untill the object be within, but contrarily if the eye be between the point of concourse and the Glasse, then the Jmages are direct: and if the eye or the object be in the point of concourse, the Glasse will be enlightened and the Jmages confu­sed, and if there were but a spark of fire in the said point of concourse, all the Glasse would seeme a burning fire-brand, and we dare say it would oc­curre without chance, and in the night be the most certaine and subtilest light that can be, if a candle were placed there. And whosoever shall enter into the search of the truth of new experiments in this subject without doubt he will confirme what we here speak of: & will finde new lights with a con­veniable position to the Glasse, he will have reflecti­on of quantities, of truth, and fine secrets in nature, yet not known, which he may easily comprehend [Page] if he have but an indifferent sight, and may assure himselfe that the Images cannot exceed the fight, nor trouble it, a thing too much absurd to nature.

And it is an absolute verity in this science, that the eye being once placed in the line of reflection of any object, and moved in the same line: the obect is seene in one and the same place immutable; or if the Image and the eye move in their owne lines, the re­presentation in the Glasse seemes to invest it selfe continually with a different figure.

NOw the Image comming thus to the eye, those which know not the secret, draw their sword when they see an Image thus to issue out of the Glasse, or a Pistoll which some one holds behinde: and some Glasses will shew a sword wholly drawne out, sepa [...]ated from the Glasse, as though it were in the aire: and it is daily exercised, that a man may touch the I­mage of his hand or his face out of the Glasse, which comes out the farther, by how much the Glasse is great and the Centre remote.


NOw that a Pistoll being presented to a Glasse be­hinde a man, should come out of the Glasse, and make him afraid that stands before, seeming to shoot at him, this cannot be: for no object whatsoever [Page 153] presented to a concave Glasse, if it be not neerer to the G [...]asse then the eye is it comes not out to the sight of the party; therefore he needs not feare that which is said to be behinde his back, and comes out of the Glasse; for if it doth come out, it must then necessarily be before his face, so in a concave Glasse whose Centre is farre remote of a sword, stick, or such like be presented to the Glasse, it shall totally be seen to come forth of the Glasse and all the hand that holds it. And here generally note that if an Image be seen to issue out of the Glasse to come towards the face of any one that stands by, the object shall be likewise seen to thrust towards that face in the Glass and may easily be knowne to all the standers by: so many persons standing before a Glasse, if one of the company take a sword, and would make it issue forth towards any o [...]her that stands there: let him chuse his Image in the Gl [...]sse and carry the sword right towards it and the effect will follow. In like man­ner ones hand being presented to the Glosse as it is thrust towards the Centre, s [...] the representation of it comes towards it, and so the hands will seeme to be united, or to touch one another.

FRom which may be concluded, if such a Glasse be placed at the seeling or planching of a Hall, so that the face be Horizontall and look downward; one may see under it as it were a man hanging by the feet, and if there were many placed so, one could not enter into that place without great feare or scaring: for [Page 154] one should see many men in the aire as if they were hanging by the feet.


TOuching a Glasse tyed at a seeling or plan­ching, that one may see a man hang by the feet in the aire, and so many Glasses, many men may be seen: without caution this is very absurd for if the Glasse or Glasses be not so great that the Centre of the sphere upon which it was made, ex­tend not neere to the head of him that is under it, it will not pleasantly appeare, and though the Glasse should be of that capacity that the Centre did ex­tend so farre, yet will not the Images be seene to them which are from the Glasse but on [...]y to those which are under it, or neere unto it: and to them it will not ably appeare, and it would be most admi­rable to have a Gallerie vaulted over with such Glasses which would wonderfully astonish any one that enters into it: for a [...]l the things in the Gallery would be seen to hang in the aire, and you could not walk without incountering airie apparitions.

SEcondly, in flat or plaine Glasses the Image is seen equall to his object, and to repre­sent a whole man, there ought to be a Glasse as great as the Image is: In convex Glasses the I­mages are seen alwayes lesse, in concave Glasses [Page 155] they may be seen greater or lesser, but not truly proportionable, by reason the diverse reflexi­ons which contracts or inlargeth the Species: when the eye is between the Centre and the surface of the Glasse; the Image appeares some­times very great and deformed, and those which have but the appearance of the beginning of a beard on their chinne, may cheare up them­selves to see they have a great beard; those that seeme to be faire will thrust away the Glasse with despight, because it will transforme their beauty: those that put their hand to the Glasse vvill seeme to have the hand of a Giant, and if one puts his finger to the Glasse it vvill be seen as a great Pyramide of flesh, inversed against his finger.

Thirdly, it is a thing admirable that the eye being approached to the point of concourse of the Glasse, there vvill be seen nothing but an intermixture or confusion: but retiring back a little from that point, (because the rayes do there meet▪) he shall see his Image inversed, having his head belovv and his feet above.

Fourthly, the divers appearances caused by the motion of objects, either retiring or ap­proaching: whether they turne to the right hand or to the left hand, whether the Glasse be hung against a wall, or whether it be placed upon a Pavement, as also what may be repre­sented by the mutuall aspect of concave Glasses with plaine and convex Glasses but I will with silence passe them over, only say something of two rare experiments more as followeth.

[Page 156]The first is to represent by help of the Sun, such letters as one would upon the front of a house: so that one may read them: Maginus doth deliver the way thus. Write the Letters, saith he, sufficiently bigge, but inversed upon the surface of the Glasse, with some kinde of colour, or these letters may be written with wax, (the easier to be taken out againe:) for then placing the Glasse to the Sunne, the letters which are written there will be reverberated or reflected upon the Wall: hence it was perhaps that Pythagoras did promise with this invention to write upon the Moone.

In the second place, how a man may sundry wayes help himselfe with such a Glasse, with a lighted Torch or Candle, placed in the point of concourse or inflammation, which is neare the fourth part of the Diameter: for by this meanes the light of the Candle will be reverbe­rated into the Glasse, and vvill be cast back a­gaine very farre by parrallel lines, making so great a light that one may clearly see that vvhich is done farre off, yea in the camp of an Enemie: and those which shall see the Glasse a farre off, will think they see a Silver Basin in­lightened, or a fire more resplendent then the Torch. It is this way that there are made cer­taine Lanthorns which dazell the eyes of those which come against them; yet it serves singu­lar well to enlighten those which carry them, accommodating a Candle with a little hollow Glasse, so that it may successively be applyed to the point of inflammation.

[Page 157]In like manner by this reflected light, one may reade farre off, provided that the letters be indifferent great, as an Epitaph placed high, or in a place obscure; or the letter of a friend which dares not approach without perill or sus­picion.


THis will be scarce sensible upon a wall remote from the Glasse, and but indifferently seen upon a wall which is neare the Glasse, and withall it must be in obscuritie or shadowed, or else it will not be seen. To cast light in the night to a place remote, with a Candle placed in the point of concourse or in­flammation, is one of the most notablest properties which can be shewne in a concave Glasse: for if in the point of inflammation of a parabolicall section, a Candle be placed, the light will be reflected by pa­rallel lines, as a columne or Cylinder; but in the sphericall section it is defective in part, the beames being not united in one point, but somewhat scatte­ring: notwithstanding it casteth a very great beau­tifull light.

Lastly, those which feare to hurt their sight by the approach of Lampes or Candles, may by this artifice place at some corher of a Chamber, a Lamp with a hollow Glasse behinde [Page 158] it, which will commodiously reflect the light upon a Table, or to a place assigned: so that the Glasse be somewhat raised to make the light to streeke upon the Table with sharp Angles, as the Sunne doth when it is but a little elevated above the Horizon, for this light shall exceed the light of many Candles placed in the Roome, and be more pleasant to the sight of him that useth it.

Of other Glasses of pleasure.

FIrst, the Columnary and Pyramidall Glasses that are contained under right lines, do represent the Images as plaine Glasses do; and if they be bowing, then they represent the I­mage, as the concave and convex Glasses do.

Secondly, those Glasses which are plaine, but have ascents of Angels in the middle, will shew one to have foure Eyes, two Mouthes, two Noses, &c.


TH [...]se experiments will be found different ac­cording to the diverse meeting of the Glasses, which commonly are made scuing-wise at the end, [...] which there will be two divers superficies in the Glasse, making the exteriour Angle somewhat raised, at the interiour onely one superficies, which [Page 159] may be covered according to ordinary Glasses to c [...]use a reflexion, and so it will be but one Glasse, which by refraction according to the different thick­nesse of the Glasse, and different Angles of the scu­ing forme, do differently present the Images to the eye, as foure eyes, two mouthes, two noses; sometimes three eyes one mouth, and one nose, the one large and the other long, sometimes two eyes onely: with the mouth and the nose deformed, which the Glasse (impenetrable) will not shew. And if there be an in­teriour solid Angle, according to the difference of it (as if it be more sharp) there will be represented two distinct double Images, that is, two entire vi­sages and as the Angle is open, by so much the more the double Images will reunite and enter one within another, which will present sometimes a whole visage extended at large, to have foure eyes, two noses, and two mouthes: and by moving the Glasse the Angle will vanish, and so the two su­perficies will be turned into one, and the duplicity of Images will also vanish and appeare but one one­ly: and this is easily experimented with two little Glasses of steel, or such like so united, that they make divers Angles and inclinations.

THirdly, there are Glasses which make men seeme pale, red, and coloured in diverse man­ners, which is caused by the dye of the Glasse, or the diverse refraction of the Species: and those which are made of Silver, Latine, Steele, &c. do give the Images a diverse colour also. [Page 160] In which one may see that the appearances by some are made fairer, younger or older than they are; and contrarily others will make them foule and deformed: and give them a contrary visage: for if a Glasse be cut as it may be, or if many pieces of Glasse be placed together to make a conveniable reflexion: there might be made of a Mole (as it were) a mountaine, of one Haire a Tree, a Fly to be as an Elephant, but I should be too long if I should say all that which might be said upon the property of Glasses. I will therefore conclude this discourse of the properties of these Glasses with these foure re­creative Problemes following.

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.

Corolarie. 1.

BY this invention of reflections the besie­gers of a Towne may be seene upon the Rampart: notwithstanding the Parapet, which the besieged may do by placing a Glasse in the hollow of the Ditch, and placing another up­on the toppe of the wall, so that the line of in­cidence comming to the bottom of the Ditch, make an Angle equall to the Angle of reflexi­on, then by this situation and reflexion, the Image of the besiege [...] [...] will be seen to him is upon the Rampart

Corolarie 2.

BY which also may be inferred, that the same reflexions may be seen in a Regular Polygon, and placing as many Glasses as there are sides, counting two for one; for then the object being set to one of the Glasses, and the eye in the other, the Jmage will be seen easily.

Corolarie 3.

FArther, notwithstanding the interposition of many Walls, Chambers, or Cabinets, one may see that which passeth through the most remotest of them, by placing of many Glasses as there are openings in the walls, making them to receive the incident angles equall: that is, placing them in such sort by some Geometricall assistant, that the incident points may meet in the middle of the Glasses: but here all the de­fect will be, that the Jmages passing by so ma­ny reflexions, will be very weak and scarce ob­servable.

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.

Corolaries. From which may be gathered, that one may exactly shoot out of a Musket to a place which is not seen, being hindered by some obstacle, or other interposition.

AS let the eye be at M, the mark C, and the wall which keeps it from being seene, ad­mit to be QR, then


set up a plaine Glass as AB, and let the Musket by GH, pla­ced upon his rest PO. Now because the marke C is seen at D, move the Musket to and fro, untill it doth agree with the line of reflection MB, [Page 164] which suppose at LI, so shall it be truly placed, and giving fire to the Musket, it shall not faile to strike the said mark at C.

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.


BY this invention you may make a little Cabinet of foure foot long, and tvvo foot large, (more or lesse) vvhich being filled vvith [Page 166] Rockes or such like things, or there being put into it Silver, Gold, Stones of luster, Jewels, &c. and the walls of the said Cabinet being all covered, or hung with plaine glasse; these vi­sibles will appeare manifoldly increased, by rea­son of the multiplicitie of reflexions, and at the opening of the said Cabinet, having set some­thing which might hide them from being seen, those that look into it will be astonished to see so few in number which before seemed to be so many.

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.

Of a Dyall of hearbes.

CAn you have a finer thing in a Garden, or in the middle of a Compartemeet, than to see the lines and the number of houres repre­sented with little bushie hearbes, as of Hysope [Page 167] or such which is proper to be cut in the bor­ders; and at the top of the style to have a Fanne to shew which way the winde b [...]oweth? this is very pleasant and useful.

Of the Dyall upon the fingers and the hand.

IS it nor a commoditie very agreeable, when one is in the fie [...]d or in some vil [...]age vvithout any other Dyall, to see onely by the hand what of the clock it is? vvhich gives it very neare; and may be practised by the left hand, in this manner.

Take a stravv or like thing of the length of the Index or the second finger, hold this straw very right betvveen the thumb and the fore­finger, then stretch forth the hand▪ and turne your back, and the palm of your hand tovvards the Sunne; so that the shadovv of the muscle vvhich is under the Thumb, touch the line of life, vvhich is betvveen the middle of the tvvo other great lines, vvhich is seen in the palme of the hand, this done, the end of the shadovv vvill shevv vvhat of the clock it is: for at the end of the first finger it is 7 in the morning, or 5 in the evening, at the end of the Ring-finger it is 8 in the morning, or 4 in the evening, at the end of the little finger or first joynt, it is 9 in the morning, or 3 in the after-noone, 10 & 2 at the second joynt, 11 and 1 at the third joynt, and midday in the line follovving, vvhich comes from the end of the Index.

Of a Dyall which was about an Obe­liske at Rome.

WAs not this a pretty fetch upon a pave­ment, to choose an Obeliske for a Dyall, having 106 foot in height, without removing the Basis of it? Plinie assures us in his 26 book and 8 Chap. that the Emperour Augustus ha­ving accom [...]odated in the field of Mars an O­beliske of this height, he made about it a pave­ment,


and by the industry of Man [...]lius the Mathematician, there were enchaced markes of Copper upon the Pavement, and placed also an Apple of Gold up­on the toppe of the said Obeliske, to know the houre and the course of the Sunne, with the increase and de­crease of dayes by the same shadow: and in the same manner do some by the shadow of their head or other style, make the like experi­ments in Astronomie.

Of Dyals with Glasses.

PT [...]lomie w [...]ites, as Cardanus reports, that long ago there were Glasses which served for Dyals, and presented the face of the be­holder [Page 169] as many times as the houre ought to be, twice if it were 2 of the clock, 9 if it were 9, &c. But this was thought to be done by the help of water, and not by Glasses, which did leake by little and little out of the vessell, dis­covering anon one Glasse, then anon two Glas­ses, then 3, 4, 5 Glasses, &c. to shew so many faces as there were houres, which was onely by leaking of water.

Of a Dyall which hath a Glasse in the place of the Style.

WHat will you say of the invention of Ma­thematicians, which finde out daily so many fine and curious novelties? they have now a way to make Dyals upon the wainscot or seeling of a Chamber, and there where the Sunne can never shine, or the beames of the Sunne cannot directly strike: and this is done in placing of a little Glasse in the place of the style which reflecteth the light, with the same condition that the shadow of the style sheweth the houre: and it is easie to make experiment upon a common Dyall, changing only the dis­position of the Dyall, and tying to the end of the style a piece of plaine Glasse. The Almaines use it much, who by this way have no greater trouble, but to put their Noses out of their beds and see what a clock it is, which is reflected by a little hole in the Window upon the wall or seeling of the Chamber.


IN this there are two experiments consider­able, the first is with a very little Glasse placed so that it may be open to the beames of the Sunne, the other hath respect to a spaci­ous or great Glasse placed to a very little hole so that the Sun may shine on it, for then the shadow which is cast upon the Dyall is con­verted into beames of the Sunne, and will re­flect and becast upon a plain opposite: and in the other it is a hole in the window or such like, by which may passe the beames of the Sun, which represent the extreamity of the style, & the Glasse representeth the plaine of the Dyall, upon which the beames being in manner of shadowes reflect cast upon a plaine opposite: and it is needfull that in this second way the Glasse may be spacious, as be­fore, to receive the delineaments of the Dyall.

Otherwise you may draw the lineaments of a Dyall upon any plaine looking-glasse which reflecteth the Sunne-beames, for the apply­ing a style or a pearle at the extreamitie of it: and placed to the Sunne, the reflexion will be answerable to the delineaments on the Glasse: but here note, that the Glasse ought to be great, and so the delineaments thereon.

[Page 170]But that which is most noble, is to draw houre-lines upon the outside of the Glasse of a window, and placing a style thereto upon the outside, the shadow of the style will be seen within, and so you have the hour, more cer­taine without any difficulty.

Of Dyals with water.

SVch kinde of Dyals were made in ancient times, and also these of sand: before they had skill to make Sun-dyals or Dyals with wheeles; for they used to fill a vessell with wa­ter, and having experience by tryall thar it would runne out all in a day, they did marke within the vessell the houres noted by the run­ning of the water; and some did set a piece of light board in the vessell to swimme upon the top of the water, carrying a little statue, which with a small stick did point out the houre upon a columne or wall, figured with houre-notes, as the vessell was figured within.

Vitruvius writes of another manner of water-Dyal more difficult;


and B [...]ptista à Porta amongst his naturall secrets, delivers this invention following. Take a vessel full of water like a caldron, & another vessell of glasse like unto a Bell, (with which some accustome to cover Melo [...]s:) and let this [Page 172] vessell of Glasse be almost as great as the Cal­dron, having a small hole at the bottome, then when it is placed upon the vvater, it vvill sink by little and little: by this one may marke the houres on the surface of the Glasse to serve an­other time. But if at the beginning one had drawn the water vvithin the same vessell of Glasse in sucking by the little hole, the vvater vvould not fall out, but as fast as the aire vvould succeed it, entering slovvly at the little hole: for contrarily the houres may be distinguished by diminution of water, or by augmentation.

Novv it seemes a safer vvay that the vvater passe out by drop and drop, and drop into a Cylindricall Glasse by help of a Pipe: for ha­ving marked the exterior part of the Cylinder in the houre notes, the vvater it selfe vvhich falls vvithin it, vvill shevv vvhat of the clock it is, farre better than the running of sand, for by this may you have the parts of the houres most accurate, vvhich commonly by sand is not had: and to vvhich may be added the houres of o­ther Countreys vvith greater ease. And here note, that as soone as the vvater is out▪ of one of the Glasses, you may turne it over into the same againe out of the other, and so let it runne a­nevv.

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

The first is, how to charge a Cannon without powder?

THis may be done vvith aire and vvater, only having throvvn cold vvater into the Can­non, vvhich might be squirted forceably in by the closure of the mouth of the Piece, that so by this pressure the aire might more condense; then having a round piece of vvood very just, and oiled vvell for the better to slide, and thrust the Bullet vvhen it shall be time: This piece of vvood may be held fast vvith some Pole, for feare it be not thrust out before his time: then let fire be made about the Trunion or hinder part of the Piece to heat the aire and vvater, and then vvhen one vvould shoot it, let the pole be quickly loosened, for then the aire searching a greater place, and having vvay novv offered, vvill thrust out the vvood and the bullet very quick: the experiment vvhich vve have in long trunkes shooting out pellats vvith aire only, shevveth the verity of this Probleme.

2 In the second question it may be demanded, how much time doth the Bull [...] of a Cannon spend in the aire before i [...] falls to the ground?

THe resolution of this Question depends upon the goodnesse of the Piece & charge thereof, seeing in each there is great difference. It is reported, that Tich [...] Bra [...]e, and the Lands­grave did make an experiment upon a Can­non in Germany, which being charged and


shot off; the Bullet spent two minutes of time in the aire be­fore it fell: and the distance was a Ger­mane mile, which di­stance proportiona­ted to an hours time, makes 120 Italian miles.

3. In the third question it may be asked, how it comes to passe, that a Cannon shooting up­wards, the Bullet flies with more vio­lence than being shot point-blanke, or shooting downeward?

IF we regard the effect of a Cannon when it is to batter a wall, the Question is false, seeing it is most evident that the blowes which fall Perpendicular upon a wall, are more violent than those which strike byas-wise or glaunsing­ly.

[Page 175]But considering the strength of the blow on­ly, the Question is most true, and often ex­perimented to be found true: a Piece mounted at the best of the Randon, which is neare halfe of the right, conveyes her Bullet with a farre greater violence then that which is shot at point blanke, or mounted parallel to the Hori­zon. The common reason is, that shooting high, the fire carries the bowle a longer time in the aire, and the aire moves more [...]acill up­wards, than dovvnevvards, because that the airy circles that the motion of the bullet makes, are soonest broken. Hovvsoever this be the gene­rall tenet, it is curious to finde out the inequality of moving of the aire; vvhether the Bullet fly upvvard, dovvnevvard, or right forvvard, to produce a sensible dfference of motion; & some think that the Cannon being mounted, the Bullet pressing the povvder maketh a greater resistance, and so causeth all the Povvder to be inflamed before the Bullet is throvvne out, vvhich makes it to be more violent than other­vvise it vvould be. When the Cannon is other­vvise disposed, the contrary arives, the fire leaves the Bullet, and the Bullet rolling from the Povvder resists lesse: and it is usually seene, that shooting out of a Musket charged onely vvith Povvder, to shoot to a marke of Paper placed Point blanke, that there are seene many small holes in the paper, vvhich cannot be o­ther than the graines of Powder which did not take fire: but this latter accident may happen [Page 176] from the over-charging of the Piece, or the length of it, or windy, or dampenesse of the Powder.

From which some may think, that a Cannon pointed right to the Zenith, should shoot with greater violence, then in any other mount or forme whatsoever: and by some it hath beene imagined, that a Bullet shot in this fashion hath been consumed, melted, and lost in the Aire, by reason of the violence of the blow, and the activity of the sire, and that sundry experi­ments have been made in this nature, and the Bullet never found. But it is hard to believe this assertion: it may rather be supposed that the Bullet falling farre from the Piece cannot be discerned where it falls: and so comes to be lost.

4. In the fourth place it may be asked, whether the discharge of a Cannon b [...] so much the greater, by how much it is longer?

IT seemeth at the first to be most true, that the longer the Piece is, the more violent it shoots: and to speak generally, that which is direction by a Trunke, Pipe, or other concavi­tie, is conveyed so much the more violent, or better, by how much it is longer, either in re­spect of the Sight, Hearing, Water, Fire, &c. & the reason seems to hold in Cannons, because in those that are long, the fire is retained a lon­ger time in the concavitie of the Piece, and so [Page 177] throwes out the Bullet with more violence; and experience lets us see that taking Cannons of the same boare, but of diversitie of length from 8 foot to 12, that the Cannon of 9 foot long hath more force than that of 8 foot long, and 10 more than that of 9, and so unto 12 foote of length. Now the usuall Cannon carries 600 Paces, some more, some lesse, yea some but 200 Paces from the Piece, and may shoot into soft earth 15 or 17 foot, into sand or earth which is loose, 22 or 24 foot, and in firme ground, about 10 or 12 foot, &c.

It hath been seen lately in Germany, where there were made Pieces from 8 foot long to 17 foot of like boare, that shooting out of any piece which was longer than 12 foot; the force was diminished, and the more in length the Piece increaseth, the lesse his force was: there­fore the length ought to be in a meane measure, and it is often seene, the greater the Cannon is, by so much the service is greater: but to have it too long or too short, is not convenient, but a meane proportion of length to be taken, o­therwise the flame of the fire will be over-pres­sed with Aire: whic hinders the motion in re­spect of substance, and distance of getting out.


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,

Of graines of Mustard-seed.

FIrst, therefore it is certaine that the increase of one graine of Mustard-seed for 20 yeares space, cannot be contained within the visible world, nay if it were a hundred times greater than it is: and holding nothing besides from the Centre of the earth even unto the firma­ment, but only small grains of Mustard-seed: Now because this seems but words, it must be proved by Art, as may be done in this wise, as suppose one Mustard-seed sowne to bring forth a tree or branch, in each extendure of which might be a thousand graines: but we will sup­pose onely a thousand in the whole tree, and let us proceed to [...]0 yeares, every seed to bring forth yearely a thousand graines, now multiply­ing alwayes by a thousand, in lesse then 17 years [Page 179] you shall have to many graines which will sur­passe the sands, which are able to fill the whole firmament: for following the supposition of Archimedes, and the most probable opinion of the greatness of the firmament which [...]i [...]ho Bra­he hath left us; the number of graines of sand will be sufficiently expressed with 49 Ciphers, but the number of graines of Mustard-seed at the end of 17 yeares will have 52 Ciphers: and moreover, graines of Mustard-seed, are farre greater than these of the sands: it is therefore evident that at the seventeenth yeare, all the graines of Mustard-seed which shall successive­ly spring from one graine onely, cannot be con­tained within the limits of the whole firma­ment; what should it be then, if it should be multiplied againe by a thousand for the [...]8 yeare: and that againe by a thousand for every yeares increase untill you come to the 20 yeare, it's a thing as cleare as the day, that such a heap of Mustard-seed would be a hundred thousand times greater than the Earth: and bring onely but the increase of one graine in 20 yeares.

Of Pigges.

SEcondly, is it not a strange proposition, to say that the great Turke with all his Reve­nues, is not able to maintaine for one yeares time, all the Pigges that a Sow may pigge with all her race, that is, the increase with the increase unto 12 years: this seemes impossible, yet it is most true, for let us suppose and put, the case, [Page 180] that a Sow bring forth but 6, two Males, and 4 Females, and that each Female shall bring forth as many every yeare, during the space of 12 yeares, at the end of the time there will be found above 3 [...] millions of Pigges: now allowing a crowne for the maintenance of each Pigge for a yeare, (which is as little as may be, being but neare a halfe of a farthing allowance for each day;) there must be at the least so many crownes to maintaine them, one a year, viz. 33 millions, which exceeds the Turkes revenue by much.

Of graines of Corne.

THirdly, it will make one astonished to think that a graine of Corne, with his increase successively for the space of 12 yeares will pro­duce in grains 24414062 [...]000000000000, which is able to load almost al the creatures in the World.

To open which, let it be supposed that the first yeare one graine being sowed brings forth 50, (but sometimes there is seen 70, sometimes 100 fold) which graines sowen the next yeare, every one to produce 50, and so consequently the whole and increase to be sowen every yeare, until 12 yeares be expired, there will be of in­crease the aforesaid prodigious summe of graines, viz. 244140625000000000000, which will make a cubical heap of 6258522 graines every way, which is more than a cubicall body of 31 miles every way: for allowing 40 graines [Page 181] in length to each foot, the Cube would be 156463 foot every way: from which it is evi­dent that if there were two hundred thousand Cities as great as London, allowing to each 3 miles square every way, and 100 foot in height, there would not be sufficient roome to con­taine the aforesaid quantitie of Corne: and sup­pose a bushel of Corne were equal unto two Cubicke feet, which might containe twenty hundred thousand graines then would there be 122070462500000. bushells, and allowing 30 bushels to a Tunne, it would be able to load 81380 [...]0833 vessels, which is more than eight thousand one hundred and thirty eight millions, ship loadings of [...]00 Tunne to each ship a: quantity so great that the Sea is scarce able to beare, or the universal world able to finde vessels to carry it. And if this Corne should be valued at halfe a crown the bushel, it would amount unto 15258807812500 pounds sterling, which I think exceeds all the Treasures of all the Princes, and of other particular men in the whole world: and is not this good husbandry to sowe one grain of Corne; and to continue it in sowing, the increase only for 12 yeares to have so great a profit?

Of the increase of Sheep.

FOurthly, those that have great flocks of Sheep may be quickly rich, if they would preserve their Sheep without killing or selling of them: so that every Sheep produce one each [Page 182] yeare, for at the end of 16 yeares, 100 Sheepe will multiply and increase unto 6553600, which is above 6 millions, 5 hundred 53 thousand Sheep: now supposing them worth but a crown a piece, it would amount unto 1638400 pounds sterling, vvhich is above 1 million 6 hundred 38 thousand pounds, a faire increase of one Sheep: and a large portion for a Childe if it should be allotted.

Of the increase of Cod-fish, Carpes, &c.

FIfthly, if there be any creatures in the vvorld that do abound vvith increase or fertilitie, it may be rightly attributed to fish; for they in their kindes produce such a great multitude of Eggs, and brings forth so many little ones, that if a great part vvere not destroyed continually, vvithin a [...]ittle vvhile they vvould fill all the Sea, Ponds, and Rivers in the vvorld; and it is easie to shevv hovv it vvould come so to passe, onely by supposing them to increase without taking or destroying them for the space of 10 or 12 yeares: having regard to the soliditie of the waters which are allotted for to lodge and containe these creatures, as their bounds and place of rest to live in.

Of the increase and multiplication of men.

SIxthly, there are some that cannot conceive how it can be that from eight persons (which [Page 183] were saved after the deluge or Noahs flood) should spring such a world of people to begin a Monarchie under Nimrod, being but 200 yeares after the flood, and that amongst them should be raised an army of two hundred thou­sand fighting men: But it is easi [...]y proved if vve take but one of the Children of Noah, and suppose that a nevv generation of people begun at every 30 yeares, and that it be continued to the seventh generation vvhich is 200 yeares; for then of one only family there vvould be pro­duced one hundred and eleven thousand soules, three hundred and five to begin the vvorld: though in that time men lived longer, and vvere more capable of multiplication and increase: vvhich number springing onely from a simp [...]e production of one yearly, vvould be farre great­er, if one man should have many vvives, vvhich in ancient times they had: from vvhich it is al­so that the Children of Israel, vvho came into Egypt but onely 70 soules, yet after 210 yeares captivity, they came forth vvith their hostes, that there vvere told six hundred thousand fighting men, besides old people, women and children; and he that shall separate but one of the fami­lies of Joseph, it would be sufficient to make up that number: how much more should it be then if we should adjoyne many families together?

Of the increase of numbers.

SEventhly, what summe of money shall the City of London be worth, if it should be sold, and the money be paid in a yeare after this [Page 184] manner: the first week to pay a pinne, the se­cond week 2 pinnes, the third week 4 pinnes, the fourth week 8 pinnes, the fifth week 16 pinnes. and so doubling untill the 52 weeks, or the yeare be expired.

Here one would think that the value of the pinnes would amount but to a small matter, in comparison of the Treasures, or riches of the whole City: yet it is most probable that the number of pinnes would amount unto the sum of 4519599628681215, and if we should al­low unto a quarter a hundred thousand pinnes, the whole would contain ninetie eight millions, foure hundred thousand Tunne: which is able to load 45930 Shippes of a thousand Tunne a­piece: and if we should allow a thousand pins for a penny, the summe of money would a­mount unto above eighteen thousand, eight hundred and thirty millions of pounds sterling, an high price to sell a Citie at, yet certain, ac­cording to that first proposed. So if 40 Townes were sold upon condition to give for the first a penny, for the second 2 pence, for the third 4 pence, &c. by doubling all the rest unto the last, it would amount unto this number of pence, 109951 [...]62 [...]76, which in pounds is 4581298444, that is foure thousand five hundred and fourescore millions of pounds and more.

Of a man that gathered up Apples, Stones, or such like upon a condition.

EIghtly, admit there were an hundred Apples, Stones, or such like things that were plac'd in a straight line or right forme, a Pace one from another, and a basket being placed a Pace from the first: how many paces would there be made to put all these Stones into the basket, by fetching one by one? this would require near halfe a day to do it, for there would be made ten thousand and ninety two paces before he should gather them all up.

Of Changes in Bells, in musicall Instruments, transmutation of places, in numbers, letters, men or such like.

NInethly, is it not an admirable thing to con­sider how the skill of numbers doth easily furnish us with the knowledge of mysterious and hidden things? which simply looked into by others that are not versed in Arithmetick, do present unto them a world of confusion and difficultie.

As in the first place, it is often debated a­mongst our common Ringers, what number of Changes there might be made in 5, 6, 7, 8, or more Bells: who spend much time to answer their owne doubts, entring often into a La­byrinth in the search thereof: or if there were 10 voyces, how many severall notes might there [Page 186] be? These are propositions of such facility, that a childe which can but multiply one number by another, may easily resolve it, which is but only to multiply every number from the unite suc­cessively in each others product, unto the terme assigned: so the 6 number that is against 6 in the Table, is 720, and so many (hanges may be made upon 6 Bells, upon 5 there are 120, &c.

In like manner against 10 in the Table is 3618800, that is, three millions, six hundred twenty eight thousand, eight hundred: which shews that 10 voices may have so many con­sorts, each man keeping his owne note, but on­ly altering his place; and so of stringed Instru­ments, and the Gamat may be varied accord­ing to which, answerable to the number against X, viz. 1124001075070399680000 notes, from which may be drawne this, or the like proposi­tion. Suppose that 7 Schollers were taken out of a free Schoole to be sent to an Vniversitie, there to be entertained in some Colledge at com­mons for a certaine summe of money, so that each of them have two meales daily, and no longer to continue there, then that sitting all together upon one bench or forme at every meale, there might be a divers transmutation of place, of account in some one of them, in comparison of another, and never the whole company to be twice alike in situation: how long may the Steward entertaine them? (who being not skilled in this fetch may answere un­advisedly.) It is most certaine that there will be [Page 187] five thousand and forty several

7 [...]0f6
403 [...]0h8
6 [...]0448593438860623360000z24

positions or changings in the seatings, which maks 14 years time wanting 10 weeks and 3 dayes. Hence from this mutability of transmuta­tion, it is no marvell tha [...] by 24 letters there aris­eth and is made such variety of languages in the world, & such infinite number of words in each lan­guage; seeing the diversity of syl­lables produceth that effect, and also by the in­terchanging & placing of letters a­mongst the vowels, & amongst themselves maketh these syllables: vvhich Al­phabet of 24 letters may be varied so many times, viz. 620448593438860623360000 vvhich is six hundred tvventy thousand, foure hundred forty eight millions of millions of mil­lions five hundred ninety three thousand, foure hundred thirty eight milions of milions, & more.

Novv allovving that a man may reade or speak one hundred thousand vvords in an houre vvhich is tvvice more vvords than there are con­teined [Page 188] in the Psalmes of David, (a taske too great for any man to do in so short a time) and if there were foure thousand six hundred and fifty thousand millions of men, they could not speak these words (according to the hourely proportion aforesaid) in threescore and ten thousand yeares; which variation & trans­mutation of letters, if they should be written in bookes, allowing to each leaf 28000 words, (which is as many as possibly could be insert­ed,) and to each book a reame or 20 quire of the largest and thinnest printing paper, so that each book being about 15 inches long, 12 broad, and 6 thick: the books that would be made of the transmutation of the 24 letters aforesaid, would be at least 38778037089928788: and if a Library of a mile square every way, of 50 foot high, were made to containe 250 Galle­ries of 20 foot broad apiece, it would con­taine foure hundred mill [...]ons of the said books: so there must be to containe the rest no lesse than 9 [...]945092 such Libraries; and if the books were extended over the surface of the Globe of the Earth, it would be a decuple covering unto it: a thing seeming most incredible that 24 let­ters in their transmutation should produce such a prodigious number, yet most certaine and in­fallible in computation.

Of a Servant hired upon certaine conditions.

A Servant said unto his Master, that he would dvvell vvith him all his life-time, if [Page 189] he would but onely lend him land to sowe one graine of Corne with all his increase for 8 years time; how think you of this bargaine? for if he had but a quarter of an inch of ground for each graine, and each graine to bring forth yearely of increase 40 graines, the whole sum would amount unto, at the terme aforesaid, 6553600000000 graines: and seeing that three thousand and six hundred millions of inches do but make one mile square in the superficies, it shall be able to receive foureteene thousand and foure hundred millions of graines, which is 14400000000. thus dividing the aforesaid 6553600000000, the Quotient will be 455, and so many square miles of land must there be to sowe the increase of one graine of Corne for 8 yeares, which makes at the least foure hun­dred and twenty thousand Acres of land, which rated but at five shillings the Acre per Annum, amounts unto one hundred thousand pound; which is twelve thousand and five hundred pound a yeare, to be continued for 8 yeares; a pretty pay for a Masters Servant 8 yeares ser­vice.

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

1. First how to make water at the foot of a mount­aine to ascend to the top of it, and so to de­scend on the other side?

TO do this there must be a Pipe of lead, which may come from the fountaine A, to the top of the Mountaine B; and so to descend on the other side a little lower then the Foun­taine, as at C. then make a hole in the Pipe at


the top of the Moun­taine, as at B, and stop the end of the Pipe at A and C; and fill this Pipe at B with water: & close it very carefully a­gaine at B, that no aire get in: then un­stop the end at A, & at C; then will the water perpetually runne up the hill, and de­scend on the other side, which is an invention of great consequence to furnish Villages that want water.

2. Secondly, how to know what wine or other liquor there is in a vessell without opening the bung-hole, and without making any other hole, than that by which it runnes out at the top?

IN this problem there is nothing but to take a bowed pipe of Glasse, and put it into the faucets hole, and stopping it close about: for then you shall see the wine or liquor to ascend in this Pipe, untill it be just even with the liquor in the vessel; by which a man may fill the ves­sel, or put more into it: and so if need were, one may empty one vessel into another without opening the bung-hole.

3. Thirdly, how is it that it is said that a vessell holds more water being placed at the foot of a Mountaine, than standing upon the top of it?

THis is a thing most certaine, because that water and all other liquor disposeth it selfe sphericaliy about the Centre of the earth; and by how much the vessel is nearer the Centre, by so much the more the surface of the water makes a lesser sphere, and therefore every part more gibbous or swelling, than the like part in a greater sphere: and therefore when the same vessell is farther from the Centre of the earth, the surface of the water makes a greater sphere, and therefore lesse gibbous, or swelling over the [Page 192] vessell: from whence it is evident that a vessell near the Centre of the Earth holds more water than that which is farther remote from it; and so consequently a vessel placed at the bottome of the Mountaine holds more water, than be­ing placed on the top of the Mountaine. First, therefore one may conclude, that one and the


same vessel will al­wayes hold more: by how much it is nearer the Centre of the earth. Secondly, if a vessell be very neare the Centre of the earth, there will be more water above the brims of it, than there is within the vessel. Thirdly, a vessel full of water comming to the Centre wil spherically increase, and by little and little leave the vessel; and passing the Centre, the vessel will be all emp­tied. Fourthly, one cannot carry a Paile of wa­ter from a low place to a higher, but it will more and more run out and over, because that in ascending it lies more levell, but descending it swels and becomes more gibbous.

4. Fourthly, to conduct water from the top of one Mountaine, to the top of another.

AS admit on the top of a Mountaine there is a spring, and at the toppe of the other [Page 193] Mountaine there are Inhabitants which want water: now to make a bridge from one Mount­aine to another, were difficult and too great a charge; by way of Pipes it is easie and of no great price: for if at the spring on the top of the Mountaine be placed a Pipe, to descend in­to the valley, and ascend to the other Mount­laine, the water will runne naturally, and con­tinually, provided that the spring be somewhat higher than the passage of the water at the In­habitants.

5. Fifthly, of a fine Fountaine which spouts water very high, and with great violence by turning of a Cock.

LEt there be a vessell as AB, made close in all his parts, in the middle of which let CD be a Pipe open at D neare the bottome, and then with a Squirt squirt in the water at C, stopped above by the cock or faucet C, vvith as great violence as possible you can; and turne the cock immediatly.


Novv there being an indifferent quantity of vvater and aire in the vessel, the vvater keeps it selfe in the bottome, and the aire vvhich vvas greatly pressed, seeks for more place, that [Page 194] turning the Cock the water issueth forth at the Pipe, and flyes very high, and that especially if the vessell be a little heated: some make use of this for an Ewer to wash hands withall, and therefore putting a moveable Pipe above C, such as the figure sheweth: which the water will cause to turne very quick, pleasurable to behold.

6. Sixtly, of Archimedes screw, which makes water ascend by descending.

THis is nothing else but a Cylinder, about the which is a Pipe in form of a screw, and when one turnes it, the water descends alwayes in respect of the Pipe: for it passeth from one part which is higher to that which is lower, and at the end of the Engine the water is found higher than it was at the spring. This great En­giner admirable in all Mathematicall Arts in­vented this Instrument to wash King Hieroies


great vessells, as some Authors saye, also to water the fields of Egypt, as Diodo­rus witnesseth: and Cardanus reporteth that a Citizen of Milan having made the like Engine, thinking himselfe to be the first inventer, con­ceived such exceeding joy, that he be came mad, foll. 2.

Againe a thing may ascend by descending, [Page 195] if a spiral line be made having many circulations or revolutions; the last being alwayes les­ser than the first, yet higher than the Plaine sup­posed it is most certaine that then putting a ball into it, and turning the spirall line so, that the first circulation may be perpendicular, or touch alwayes the supposed Plain: the ball shall in descending continually ascend, untill at last it come to the highest part of the spirall line, & so fall out. And here especially may be noted, that a moving body as water, or a Bullet, or such like, will never ascend if the Helicall re­volution of the screw be not inclining to the Horizon: so that according to this inclination the ball or liquor, may descend alwayes by a continuall motion and revolution. And this ex­periment may be more usefull, naturally made with a thred of [...]ron, or Latine turned or bow­ed Helically about a Cylinder, with some di­stinction of distances between the Heli [...]es, for then having drawn out the Cylinder, or having hung or tyed some weight at it in such sort, that the water may easily drop if one lift up the said thred: these Helices or revolutions, notwithstan­ding will remaine inclining to the Horizon, and then turning it about forward, the said weight will ascend, but backward it will descend. Now if the revolutions be alike, and of equallity a­mongst themselves, and the whirling or turn­ing motion be quicke, the sight vvill be so de­ceived, that producing the action it vvill seeme to the ignorant no lesse than a Miracle.

7. Seventhly, of another fine Fountaine of pleasure.

THis is an Engine that hath two wheeles with Cogges, or teeth as AB, which are placed within an Ovall CD, in such sort, that the teeth of the one, may enter into the notches of the other; but so just that neither aire nor water may enter into the Ovall coffer, either by the middle or by the sides, for the wheele must joyne so neare to the sides of the coffer, that there be no vacuitie: to this there is an axeltree


with a handle to each wheele, so that they may be turned, and A being turned, that turneth the o­ther wheele that is opposite: by which motion the aire that is in E, & the water that is carried by the hollow of the wheeles of each side, by continu­all motion, is constrained to mount and flie out by the funnell F: now to make the water runne what way one would have it, there may be ap­plied upon the top of the Pipe F, two other moveable Pipes inserted one within another; as the figure sheweth. But here note, that there may acrue some inconveniency in this Machine seeing that by quick turning the Cogges [Page 197] or teeth of the wheeles running one against an­other, may neare break them, and so give way to the aire to enter in, which being vio­lently inclosed vvill escape to occupie the place of the vvater, vvhose vveight makes it so quick: hovvsoever, if this Machine be curiously made as an able vvorkeman may easily do, it is a most sovereigne Engine, to cast vvater high and farre off for to quench fires. And to have it to raine to a place assigned, accommodate a socket having a Pipe at the middle, vvhich may point tovvards the place being set at the top thereof, and so having great discretion in turning the Axis of the vvheele, it may vvork exceeding vvell, and continue long.

8. Eightly, of a fine watering pot for gardens.

THis may be made in forme of a Bottle ac­cording to the last figure or such like, ha­ving at the bottome many small holes, and at the neck of it another hole somevvhat greater than those at the bottome, vvhich hole at the top you must unstop vvhen you vvould fill this vvatering pot, for then it is nothing but putting the lovver end into a paile of vvater, for so it vvill fill it selfe by degrees: and being full, put your thumb on the hole at the neck to stop it, for then may you carry it from place to place, and it vvill not sensibly runne out, som [...] ­thing it vvill, and all in time (if it vvere never so close stopped) contrary to the ancient tenet in Philosophy, that aire will not penetrate.

9. Ninthly, how easily to take wine out of a vessell at the bu [...]g-hole, without piercing of a hole in the vessell?

IN this there is no need but to have a Cane or Pipe of Glasse or such like, one of the ends of which may be closed up almost, leaving some small hole at the end; for then if that end be set into the vessell at the bung-hole, the whole


Cane or Pipe will be filled by little and little; and once being full, stop the other end which is without and then pull out the Cane or Pipe, so will it be ful of wine, then opening a little the top above, you may fill a Glasse or other Pot with it, for as the Wine issueth out, the aire commeth into the Cane or Pipe to supply vacuity.

10. Tenthly, how to measure irregular bodies by help of water?

SOme throw in the body or magnitude into a vessell, and keep that which floweth out over, saying it is alwayes equal to the thing cast into the water: let i [...] is more nea [...]er this way to poure into a vessell such a quantity of water, [Page 199] which may be thought sufficient to cover the body or magnitude, and make a marke how high the water is in the vessell, then poure out all this water into another vessell, and let the body or magnitude be placed into the first vessel; then poure in water from the second vessell, until it ascend unto the former marke made in the first vessell, so the vvater vvhich remaines in the second vessel is equall to the body or magni­tude put into the water: but here note that this is not exact or free from error, yet nearer the truth than any Geometrician can otherwise possibly measure, and these bodies that are not so full of pores are more truly measured this way, than others are.

11. To finde the weight of water.

SEeing that 574/1000 part of an ounce weight, makes a cubicall inch of water: and every pound weight Haverdepoize makes 27 cubicall inches, and 1 9/ [...]; fere, and that [...] Gallons and a halfe wine measure makes a foot cubicall, it is easie by inversion, that knowing the quantity of a vessel in Gallons, to finde his content in cubicall feet or weight: and that late famous Geometrician Master Brigs found a cubical foot of vvater to vveigh neare 62 pound vveight Haverdepoize But the late learned Simon Stevin found a cubicall foot of vvater to vveigh 65 pound, vvhich difference may arise from the in­equalitie of vvater; for some vvaters are more ponderous than others, and some difference [Page 200] may be from the weight of a pound, and the measure of a foot: thus the weight and quan­titie of a solid foot settled, it is easie for Arith­meticians to give the contents of vessells or bo­dies which containe liquids.

12. To finde the charge that a vessell may carry as Shippes, Boates, or such like.

THis is generally conceived, that a vessell may carry as much weight as that water weigheth, which is equall unto the vessell in bignesse, in abating onely the weight of the ves­sell: we see that a barrel of wine or water cast into the water, will not sink to the bottome, but swim easily, and if a ship had not Iron and other ponderosities in it, it might swim full of water without sinking: in the same manner if the vessell were loaden with lead, so much should the watter weigh: hence it is that Mar­riners call Shippes of 50 thousand Tunnes, be­cause they may containe one or two thousand Tunne, and so consequently carry as much.

13. How comes it that a Shippe having safely sayl­ed in the vast Ocean, and being come into the Port or harbour, without any tempest will sink down right?

THe cause of this is that a vessel may carry more upon some kinde of water than upon other; now the water of the Sea is thicker and heavier than that of Rivers, Wels, or Fountains; [Page 201] therefore the loading of a vessell which is ac­counted sufficient in the Sea, becomes too great in the hurbour or sweet water. Now some think that it is the depth of the water that makes vessells more easie to swimme, but it is an abuse; for if the loading of a Ship be no heavier than the water that would occupie that place, the Ship should as easily swim upon that water, as if it did swim upon a thousand fa­thom deep of water, and if the vvater be no thicker than a leafe of paper, and weigheth but an ounce under a heavy body, it vvill support it, as vvell as if the vvater under it vveighed ten thousand pound vveight: hence it is if there be a vessell capable of a little more than a thousand pound vveight of vvater, you may put into this vessell a piece of vvood, vvhich shall vveigh a thousand pound vveight; (but lighter in his kinde than the like of magnitude of vvater:) for then pouring in but a quart of vvater or a very little quantitie of vvater, the vvood vvill svvim on the top of it, (provided that the vvood touch not the sides of the vessell:) vvhich is a fine ex­periment, and seems admirable in the perform­ance.

14. How a grosse body of mettle may swimme upon the water?

THis is done by extending the mettle into a thin Plate, to make it hollovv in forme of a vessel; so that the greatnesse of the vessell which the aire vvith it containeth, be equal to [Page 203] the magnitude of the vvater, vvhich vveighes as much as it, for all bodies may svvim vvithout sinking, if they occupie the place of vvater equal in vveight unto them, as if it vveighed 12 pound it must have the place of 12 pound of vvater: hence it is that vve see floating upon the vvater great vessells of Copper or Brasse, vvhen they are hollovv in forme of a Caldron. And how can it be otherwise conceived of Islands in the Sea that swim and float? is it not that they are hollow and some part like unto a Boat, or that their earth is very light and spongeous, or ha­ving many concavities in the body of it, or much wood within it?

And it would be a pretty proposition to shew how much every kinde of metall should be inlarged, to make it swim upon the water: which doth depend upon the proportion that is between the vveight of the vvater and each metall. Novv the proportion that is betvveene metalls and water of equall magnitude, accord­ing to some Authors, is as followeth.

A magnitude of 10 pound weight of water will require for the like magnitude of

187 ½
116 ½

From which is inferred, that to make a piece of Copper of [...]0 pound weight to swimme, it must be so made hollow, that it may hold 9 times that weight of water and somewhat more, that is to say, 91 pound: seeing that Cop­per [Page 203] and water of like magnitudes in their pon­derosities, are as before, as [...]0 to 91.

15. How to weigh the lightnesse of the aire?

PLace a Ballance of wood turned upside downe into the water, that so it may swim, then let water be inclosed within some body, as within a Bladder or such like, and suppose that such a quantitie of aire should weigh one pound, place it under one of the Ballances, and place under the other as much weight of light­nesse as may counter-ballance and keep the other Ballance that it rise not out of the water: by which you shall see how much the lightnesse is.

But without any Ballance do this; take a Cubicall hollow vessell, or that which is Cylin­dricall, which may swimme on the water, and as it sinketh by placing of weights upon it, marke hovv much, for then if you vvould ex­amine the vveight of any body, you have no­thing to do but to put it into this vessell, and marke hovv deep it sinkes, for so many pound it vveighes as the vveights put in do make it so to sinke.


16. Being given a body, to marke it about, and shew how much of it will sink in the water, or swim above the water.

THis is done by knovving the vveight of the body vvhich is given, and the quantity of vvater, vvhich vveighes as much as that body; for then certainly it vvill sink so deep, untill it occupieth the place of that quantitie of vvater.

17. To finde how much severall mettle or other bodies doe weigh lesse in the water than in the aire:

TAke a Ballance, & vveigh (as for example) 9 pound of Gold, Silver, Lead, or Stone in the aire, so it hang in aequilibrio; then comming to the vvater, take the same quantity of Gold Silver, Lead, or Stone, and let it softly dovvne into it, and you shall see that you shall need a lesse counterpoise in the other Ballance to counter-ballance it: vvherefore all solids or bodies vveigh lesse in the vvater than in the aire, and so much the lesse it vvill be, by hovv much the vvater is grosse and thick, because the vveight findes a greater resistance, and therefore the vvater supports more than aire; and further, because the vvater by the ponderositie is dis­pleased, and so strives to be there againe, pres­sing to it, by reason of the other vvaters that are about it, according to the proportion of [Page 205] his weight. Archimedes demonstrateth, that all bodies weigh lesse in the water (or in like liquor) by how much they occupie place: and if the water weigh a pound weight, the magni­tude in the water shall weigh a pound lesse than in the aire.

Now by knowing the proportion of water and mettles, it is found that Gold loseth in the water the 19 part of his weight, Copper the 9 part, Quicksilver the 15 part, Lead the 12 part, Silver the 10 part, Iron the 8 part, Tinne the 7 part and a little more: wherefore in ma­teriall and absolute weight, Gold in respect of the water that it occupieth weigheth 18, and ¾ times heavier than the like quantitie of water, that is, as 18 ¾ to the Quicksilver 15 times, Lead 11 and ⅗, Silver 10 and ⅔, Copper 9 and 1/10, Iron 8 and ½, and Tinne 8 and 1/ [...]. Contrarily in re­spect of greatnesse, if the water be as heavy as the Gold, then is the water almost 19 times greater than the magnitude of the Gold, and so may you judge of the rest.

18. How is it that a ballance having like weight in each scale, and hanging in aequilibrio in the aire, being placed in another place, (with­out removing any weight) it shall cease to hang in aequilibrio sensibly: yea by a great difference of weight?

THis is easie to be resolved by considering different mettles, which though they [Page 206] vveigh equall in the aire, yet in the vvater there vvill be an apparant difference; as suppose so that in the scale of each Ballance be placed 18 pound vveight of severall metalls, the one Gold, and the other Copper, vvhich being in aequilibrio in the aire, placed in the vvater, vvill not hang so, because that the Gold los [...]eth neare the 18 part of his vveight, vvhich is about 1 pound, and the Copper loseth but his 9 part, vvhich is 2 pound: vvherefore the Gold in the vvater vveigheth but 17 pound, and the Copper 16 pound, vvhich is a difference most sensible to confirme that point.

19. To shew what waters are heavier one than another, and how much.

PHysicians have an especiall respect unto this, judging that vvater vvhich is lightest is most healthfull and medicinall for the body, & Sea-men knovv that the heaviest vvaters do beare most, and it is knovvne vvhich water is heaviest thus. Take a piece of wax, and fasten Lead unto it, or some such like thing that it may but precisely swimme, for then it is equal to the like magnitude of water, then put it into ano­ther vessell which hath contrary water, and if it sinke, then is that water lighter than the other: but if it sinke not so deep, then it argueth the water to be heavier or more grosser than the first water, or one may take a piece of vvood, and marke the quantitie of sinking of it into severall waters, by vvhich you may judge [Page 207] which is lightest or heaviest, for in that which it sinkes most, that is infallibly the lightest, and so contrarily.

20. How to make a Pound of water weigh as much as 10, 2 [...], [...]0, or a hundred pound of Lead; nay as much as a thousand, or ten thousand and pound weight?

THis proposition seems very impossible, yet water inclosed in a vessell, being con­strained to dilate it selfe, doth weigh so much as though there were in the concavitie of it a solid body of water.

There are many wayes to experiment this proposition, but to verifie it, it may be suffici­ent to produce two excellent ones onely: which had they not been really acted, little credit might have been given unto it.

The first way is thus. Take a Magnitude which takes up as much place as a hundred or a thousand pound of water, and suppose that it were tied to some thing that it may hang in the aire; then make a Ballance that one of the scales may inviron it, yet so that it touch not the sides of it: but leave space enough for one pound of water: then having placed 100 pound weight in the other scale, throw in the water about the Magnitude, so that one pound of water shall weigh downe the hundred pound in the other Ballance.

The second vvay is yet more admirable: take a common Ballance that is capable to receive [Page 208]


10 or 20 pound of water, then put into it a magnitude which may take up the place of 9 or 19 pound of water, which must be hung at some Iron or beame which is placed in a wall; so that it hang quiet: (now it is not materiall whether the magnitude be hollow or massie) so that it touch not the Ballance in which it is put, for then having put the Lead or weight into the other Ballance, poure in a pound of water into the Ballance where the magnitude is, and you shall see that this one pound of water shall counterpoise the 10 or 20 pound of Lead which is set in the o­ther Ballance.

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.

2. Divers metalls being melted together in one body, to finde the mixture of them.

THis wat a notable invention of Archimedes, related by Vitrivius in his Architecture, where he reporteth that the Gold-smith which King Hiero imployed for the making of the Golden Crowne, which was to be dedicated to the gods, had stolen part of it, and mixed Silver in the place of it: the King suspicious of the work proposed it to Archimedes, if by Art he could discover without breaking of the Crowne, if there had been made mixture of any other metall with the Gold. The way which he found out was by bathing himselfe; for as he entred into the vessell of water, (in which he bathed himselfe) so the water ascended or flew out over it, and as he pulled out his body the water descended: from which he gathered that if a Bowle of pure Gold, Silver, or other metall were cast into a vessell of water, the water pro­portionally according to the thing cast in would ascend; and so by way of Arithmetick the que­stion lay open to be resolved: who being so in­tensively taken with the invention, leapes out of the Bath all naked, crying as a man trans­ported, I have found, I have found, and so disco­vered it.

Now some say that he took two Masses, the one of pure Gold, and the other of pure Silver; each equall to the weight of the Crowne, and therefore unequall in magnitude or greatnesse; [Page 211] and then knowing the severall quantities of wa­ter which was answerable to the Crown, and the severall Masses, he subtilly collected, that if the Crowne occupied more place within the water than the Masse of Gold did: it appeared that there was Silver or other metall melted with it. Now by the rule of position, suppose that each of the three Masses weighed 18 pound a piece, and that the Masse of Gold did occupie the place of one pound of water, that of Silver a pound and a halfe▪ and the Crown one pound and a quarter only: then thus he might operate the Masse of Silver which weighed 18 pounds, cast into the water, did cast out halfe a pound of water more then the Masse of Gold, which weighed 18 pound, and the Crowne which weighed also 18 pound, being put into a vessell full of water, threw out more water than the Masse of Gold by a quarter of a pound, (be­cause of mixt metall which was in it:) there­fore by the rule of proportion, if halfe a pound of water (the excesse) be answerable to 18 pound of Silver, one quarter of a pound of ex­cesse shall be answerable to 9 pound of Silver, and so much was mixed in the Crowne.

Some judge the way to be more facill by weighing the Crowne first in the aire, then in the water; in the aire it weighed 18 pound, and if it were pure Gold, in the water it would weigh but 17 pound; if it were Copper it would weigh but 16 pound; but because vve vvill suppose that Gold and Copper is mixed together, it vvill vveigh lesse then 17 pound, [Page 212] yet more than 16 pound, and that according to the proportion mixed: let it then be supposed that it vveighed in the vvater 16 pound and 3 quarters, then might one say by proportion, if the difference of one pound of losse, vvhich is betvveen 16 and 17) be ansvverable to 18 pound, to vvhat shall one quarter of difference be ansvverable to, vvhich is betvveen 17 and 16 ¾, and it vvill be 4 pound and a halfe; and so much Copper vvas mixed vvith the Gold.

Many men have delivered sundry vvayes to resolve this proposition since Archimedes in­vention, and it vvere tedious to relate the diver­sities.

Baptista Benedictus amongst his Arithmeti­call Theoremes, delivers his vvay thus: if a Masse of Gold of equall bignesse to the Crovvne did vveigh 20 pound, and another of Silver at a ca­pacity or bignesse at pleasure, as suppose did vveigh 12 pound, the Crovvne or the mixt body would vveigh more than the Silver, and lesser than the Gold, suppose it vveighed 16 pound vvhich is 4 pound lesse than the Gold by 8 pound, then may one say, if 8 pound of differ­ence come from 12 pound of Silver, from vvhence comes 4 pound vvhich vvill be 6 pound and so much Silver vvas mixed in it, &c.

3. Three men bought a quantitie of wine, each paid alike, and each was to have alike; it happened at the last partition that there were 21 Barrells, of which 7 were full, 7 halfe full, and 7 empty, how must they share the wine and vessells, that each have as many vessells one as another, & as much wine one as another?

THis may be answered two wayes as follow­eth, and these numbers 2, 2, 3, or 3, 3, 1, may serve for direction, and signifies that the first person ought to have 3 Barrells full, and as many empty ones, and one which is halfe full; so he shall have 7 vessells and 3 Barrels, and a halfe of liquor; and one of the other shall in like manner have as much, so there will remaine for the third man 1 Barrell full, 5 which are halfe full, and 1 empty, and so every one shall have alike both in vessells and wine. And ge­nerally to answer such questions, divide the number of vessells by the number of persons, and if the Quotient be not an intire number, the question is impossible; but when it is an intire number, there must be made as many parts as there are 3 persons, seeing that each part is lesse than the halfe of the said Quotient: as divid­ing 21 by 3 there comes 7 for the Quotient, which may be parted in these three parts, 2, 2, 3, or 3, 3, 1, each of which being lesse than ha [...]fe of 7.

4. There is a Ladder which stands upright against a wall of 10 foot high, the foot of it is pulled out 6 foot from the wall upon the pave­ment: how much hath the top of the Ladder descended?

THe ansvver is, 2 foot: for by Pythagoras rule the square of DB, the Hypothenusal is


equall to the square of DA 6, & AB 10. Novv if DA be 6 foot, and AB 10 foot, the squares are 36 and 100, vvhich 36 taken from 100 rests 64, vvhose Roote-quadrate is 8 so the foot of the Ladder being novv at D, the toppe vvill be at C, 2 foot lovver than it vvas vvhen it vvas at B.

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.

1. Incident.

CAius had a field which was directly square, having 24 measures in Circuit, that was 6 on each side: Sempronius desiring to fit himselfe, prayed Caius to change with him for a field which should be equivalent unto his, and the bargaine being concluded, he gave him for counterchange a piece of ground which had just as much in circuit as his had, but it was not square, yet Quadrangular and Rectangled, having 9 measures in length for each of the two longest sides, and 3 in breadth for each shorter side: Now Caius which was not the most sub­tillest [Page 216] nor wisest in the world accepted his bar­gaine at the first, but afterward [...] having con­ferred with a Land-measures and Mathematici­an, found that he was over-reached in his bar­gaine, and that his field contained 36 square measures, and the other field had but 27 mea­sures, (a thing easie to be knowne by multiply­ing the length by the breadth:) Sempronius con­tested with him in suite of Law, and argued that figures which have equall Perimeter or cir­cuit, are equall amongst themselves: my field, saith he, hath equall circuit with yours, there­fore it is equall unto it in quantitie. Now this was sufficient to delude a Judge which was ig­norant in Geometricall proportions, but a Ma­thematician will easily declare the deceit, being assured that figures which are Isoperemiter, or equall in circuit, have not alwayes equall capa­citie or quantitie: seeing that with the same cir­cuit, there may be infinite figures made which shall be more and more capable, by how much they have more Angles, equall sides, and ap­proach nearer unto a circle, (which is the most capablest figure of all,) because that all his parts are extended one from anothes, and from the middle or Centre as much as may be: so we see by an infa [...]lible rule of experience, that a square is more capable of quantitie than a Tri­angle of the same circuit, and a Pentagone more than a square, and so of others, so that they be regular figures that have their sides equall, [Page 217] otherwise there might be that a regular Tri­angle, having 24 measures in circuit might have more capacitie than a rectangled Parallelo­gram, which had also 24 measures of circuit, as if it were 11 in length, and 1 inbreadth, the circuit is still 24, yet the quantitie is but 11. and if it had 6 every way, it gives the same Pe­rimeter, viz. 24. but a quantitie of 36 as before.

2. Incident.

SEmpronius having borrowed of Caius a sack of Corne, which was 6 foot high and 2 foot broad, and when there was question made to repay it, Sempronius gave Caius back two sacks full of Corne, which had each of them 6 foot high & 1 foot broad: who beleeved that if the sackes were full he was repaid, and it seems to have an appearance of truth barely looked on. But it is most evident in demonstration, that the 2 sacks of Corn paid by Sempronius to Caius, is but halfe of that one sack which he lent him: for a Cylinder or sack having one foot of dia­meter, and 6 foot of length, is but the 4 part of another Cylinder, whose length is 6 foot, and his diameter is 2 foot: therefore two of the lesser Cylinders or sackes, is but halfe of the greater; and so Caius was deceived in halfe his Corne.

3. Incident.

SOme one from a common Fountaine of a City hath a Pipe of water of an inch diame­ter; [Page 218] to have it more commodious, he hath leave to take as much more water, whereupon he gives order that a Pipe be made of two inches diameter. Now you will say presently that it is reason to be so bigge, to have just twice as [...] before: but if the Ma­gistrate


of the Citie understood Geome­tricall proportions, he would soon cause it to be amended, & shew that he hath not only taken twice as much water as he had before, but foure times as much: for a Circular hole which is two inches diameter is foure times greater than that of one inch, and therefore vvill cast out four times as much vva­ter as that of one inch, and so the deceit is double also in this.

Moreover, if there vvere a heap of Corne of 20 foot every vvay, vvhich vvas borrovved to be paid next yeare▪ the party having his Corne in heapes of 12 foot every vvay, and of 10 foot every vvay, proffers him 4 heapes of the greater or 7 heaps of the lesser, for his ovvne heap of 20 every vvay, vvhich vvas lent: here it seems that the proffer is faire, nay vvith advantage, yet the losse vvould be neare 1000 foot. Infi­nite of such causes do arise from Geometricall figures, vvhich are able to deceive a Judge or [Page 219] Magistrate, vvhich is not somevvhat seene in Mathematicall Documents.

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.

2. Secondly, how much is the depth of the earth, the height of the heavens, and the compasse of the world?

FRom the surface of the earth unto the Cen­tre according to ancient traditions, is 3436. miles, so the vvhole thicknesse is 6872 miles, [Page 220] of which the whole compasse or circuit of the earth is 21600 miles.

From the Centre of the earth to the Moone there is neare 56 Semidiameters of the earth, which is about 192416 miles. unto the Sunne there is 1142 Semidiameters of the earth, that is in miles 3924912; from the starry firma­ment to the Centre of the earth there is 14000 Semidiameters, that is, 48184000 miles, accord­ing to the opinion and observation of that learned Ticho Brahe.

From these measures one may collect by A­rithmeticall supputations, many pleasant pro­positions in this manner.

First, if you imagine there were a hole through the earth, and that a Milstone should be let fall down into this hole, and to move a mile in each minute of time, it would be more than two dayes and a halfe before it would come to the Centre, and being there it would hang in the aire.

Secondly, if a man should go every day 20 miles, it would be three yeares wanting but a fortnight, before he could go once about the earth; and if a Bird should fly round about it in two dayes, then must the motion be 450 miles in an houre.

Thirdly, the Moone runnes a greater com­passe each houre, than if in the same time she should runne twice rhe Circumference of the whole earth.

Fourthly, admit it be supposed that one [Page 221] should go 20 miles in ascending towards the heavens every day, he should be above 15 years before he could attaine to the Orbe of the Moone.

Fifthly, the Sunne makes a greater way in one day than the Moone doth in 20 dayes, be­cause that the Orbe of the Sunnes circumfe­rence is at the least 20 times greater than the Orbe of the Moone.

Sixthly, if a Milstone should descend from the p [...]ace of the Sunne a thousand miles every houre, (which is above 15 miles in a minute, farre beyond the proportion of motion) it would be above 163 dayes before it would fall dovvne to the earth.

Seventhly, the Sunne in his proper sphere moves more than seven thousand five hundred and seventy miles in one minute of time: novv there is no Bullet of a Cannon, Arrovv, Thun­derbolt, or tempest of vvinde that moves vvith such quicknesse.

Eightly, it is of a farre higher nature to con­sider the exceeding and unmoveable quicknesse of the starry firmament, for a starre being in the Aequator, (which is just between the Poles of the world) makes 12598666 miles in one houre which is two hundred nine thousand nine hun­dred and seventy foure miles in one minute of time: & if a Horseman should ride every day 40 miles, he could not ride such a compasse in a thousand yeares as the starry firmament moves in one houre, which is more than if one [Page 222] should move about the earth a thousand times in one houre, and quicker than possible thought can be imagined: and if a starre should flye in the aire about the earth with such a prodigious quicknesse, it would burne and consume all the world here below. Behold therefore how time passeth, and death hasteth on: this made Coper­nicus, not unadvisedly to attribute this motion of Primum mobile to the earth, and not to the starry firmament; for it is beyond humane sense to apprehend or conceive the rapture and vio­lence of that motion being quicker than thought; and the word of God testifieth that the Lord made all things in number, measure, weight, and time.

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.

To finde the Circle of the Sun by the fingers.

LEt 123, 24, 25, 26, or 27, be divided by 28, (which is the Circle of the Sunne or whole revolution of the Dominicall letters) and that which remaines is the number of joynts, which is to be accounted upon the fingers by Filius esto Dei, coelum bonus accipe gratis: and where the number ends, that finger it sheweth the yeare which is present, and the words of the verse shew the Dominicall letter.


DIvide 123 by 28 for the yeare (and so of o­ther yeares) and the Quotient is 4, and there remaineth 11, for which you must account 11 words: Filius esto Dei, &c. upon the joynts beginning from the first joynt of the Index, and you shall have the answer.

For the present to know the Dominicall let­ter for each moneth, account from January un­to the moneth required, including January, and if there be 8, 9, 7, or 5, you must begin upon the end of the finger from the thumbe and ac­count, Adam degebat, &c. as many words as there are moneths, for then one shall have the letter which begins the moneth; then to know what day of the moneth it is, see how many times 7 is comprehended in the number of dayes, and take the rest, suppose 4, account upon the first finger within & without by the joynts, [Page 242] unto the number of 4, which ends at the end of the finger: from whence it may be inferred that the day required was Wednesday, Sunday being attributed to the first joynt of the first finger or Index: and so you have the present yeare, the Dominicall letter, the letter which begins the Moneth, and all the dayes of the Moneth.

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.


THat the Epact is made alwayes by adding 11 unto 30, and if it passe 30, sub­tract 30, and adde 11 to the remainder, and so ad infinitum: as if the Epact were 12, adde 11 to it makes 23 for the Epact next year, to vvhich adde 11 makes 34, subtract 30, rests 4 the Epact for the yeare after, and 15 for the yeare follovving that, and 26 for the next, and 7 for the next, &c.

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.


That the first Meridian may be taken at plea­sure upon the Terrestriall Globe or Mappe, for that some of the ancient Astronomers would have it at Hercules Pillars, which is at the straights at Gibraltar: Ptolomy placed it at the Canary Isl [...]nds, but now in these latter times it is held to be neare the Azores. But why it was first placed by Ptolomy at the Canary Islands, were because that in his time these Islands were the farthest westerne parts of the world that vvas then discovered. And vvhy it reteines his place novv at Saint Michaels neare the [Page 228] Azores, is that because of many accurate ob­servations made of late by many expert Navi­gators and Mathematicians, they have found the Needle there to have no variation, but to point North and South: that, is to each Pole of the world: and why the Longitude from thence is accounted Eastwards, is from the mo­tion of the Sunne Eastward, or that Ptolomy and others did hold it more convenient to be­gin from the Westerne part of the world and so account the Longitude Eastward from Countrey to Countrey that was then knowne; till they came to the Easterne part of Asia, ra­ther than to make a beginning upon that which was unknowne: and having made up their ac­count of reckoning the Longitude from the Westerne part to the Eastern part of the world knowne, they supposed the rest to be all sea, which since their deaths hath been found al­most to be another habitable world.

To finde the Longitude of a Countrey.

IF it be upon the Globe, bring the Countrey to the Brasen Meridian, and whatsoever degree that Meridian cuts in the Equinoctiall, that de­gree is the Longitude of that Place: if it be in a Mappe, then mark what Meridian passeth over it, so have you the Longitude thereof, if no Meridian passe over it, then take a paire of Compasses, and measure the distance betweene the Place and the next Meridian, and apply it [Page 229] to the divided parallel or Aequator, so have you the Longitude required.

Of the Latitude of Countreys.

LAtitude of a Countrey is the distance of a Countrey from the Equinoctiall, or it is an Arke of the Meridian conteined between the Zenith of the place and the Aequator; which is two-fold, viz. either North-Latitude or South-Latitude, either of which extendeth from the Equinoctiall to either Pole, so the greatest Latitude that can be is but 90 degrees: If any Northern Countrey have the Artick Circle verticall, which is in the Latitude of 66. gr. 30. m. the Sun will touch the Horizon in the North part thereof, and the longest day will be there then 24 houres, if the Countrey have lesse Latitude than 66. degrees 30. m. the Sun will rise and set, but if it have more Latitude than 66. gr. 30 m. it will be visible for many dayes, and if the Countrey be under the Pole, the Sun will make a Circular motion above the Earth, and be visible for a half yeare: so under the Pole there will be but one day, and one night in the whole yeare.


To finde the latitude of Countreys.

IF it be upon a Globe, bring the place to the Brasen Meridian, and the number of degrees


which it meeteth therewith, is the La­titude of the place. Or with a paire of Compasses take the distance between the Countrey and the E­quinoctiall, which applied unto the E­quinoctiall will shew the Latitude of that Countrey; which is equall to the Poles height; if it be upon a Mappe. Then mark what parallel passeth over the Countrey and where it crosseth the Meridian, that shall be the Latitude: but if [...]o parallel passeth over it, then take the distance betweene the place and the next parallel, which applied to the divided Meridian from that pa­rallel will shew the Latitude of that place.

To finde the distance of places.

IF it be upon a Globe: then with a paire of Compasses take the distance betweene the two Places, and apply it to the divided Meridian or Aequator, and the number of degrees shall shew [...]e distance; each degree being 60. miles. [...]f it be in a Mappe (according to Wrights pro­ [...]ection) [Page 231] take the distance with a paire of Com­ [...]asses between the two places, and apply this distance to the divided Meridian on the Mappe right against the two places; so as many degrees as is conteined between the feet of the Compasses so much is the distance between the two places. If the distance of two places be required in a particular Map then with the Compasses take the distance between the two places, and apply it to the scale of Miles, so have you the distance, if the scale be too short, take the scale between the Compasses, and apply that to the two places as often as you can, so have you the distance required.

Of the Longitude, Latitude, Declinati­on, and distance of the Starres.

THe Declination of a starre is the nearest distance of a Star from the Aequator; the Latitude of a Starre is the nearest distance of a Sarre from the Ecliptick: the Longitude of a Starre is an Ark of the Ecliptick conteined be­tween the beginning of Aries, and the Circle of the Starres Latitude, which is a circle drawne from the Pole of the Ecliptick unto the starre, and so to the Ecliptick. The distance between two Sarres in Heaven is taken by a Crosse-staffe or other Instrument, and upon a Globe it is done by taking between the feet of the Compasses the two Starres, and applying it [Page 232] to the Aequator, so have you the distance be­tweene those two starre [...].

How is it that two Horses or other creatures being foaled or brought forth into the world at one and the same time, that after certaine dayes travell the one lived more dayes than the other, not­withstanding they dyed together in one and the sam [...] moment also?

THis is easie to be answered: let one of them travell toward the West and the other to­wards the East: then that which goes towards the West followeth the Sunne: and therefore shall have the day somewhat longer than if there had been no travell made, and that which goes East by going against the Sunne, shall have the day shorter, and so in respect of tra­vell though they dye at one and the selfe same houre and moment of time, the one shall be older than the other.

From which consideration may be inferred that a Christian, a Jew, and a Saracen, may have their Sabbaths all upon one and the same day though notwithstanding the Saracen holds his Sabath upon the Friday, the Jew upon the Saturday, and the Christian upon the Sunday: For being all three resident in one place, if the Saracen and the Christian begin their travell upon the Saturday, the Christian going West, and the Saracen Eastwards, shall compasse the [Page 233] Globe of the earth, thus the Christian at the conclusion shall gaine a day, and the Saracen shall lose a day, and so meet with the Jew every one upon his owne Sabbath.

Certaine fine observations.

1 UNder the Equinoctiall the Needle hangs in equilibrio, but in these parts it inclines under the Horizon, and being under the Pole it is thought it will hang verticall.

2 In these Countreys which are without the Tropicall Circles, the Sunne comes East and West every day for a halfe yeare, but being un­der the Equinoctiall the Sun is never East, nor West▪ but twice in the yeare, to wit, the 10. of March and the 13 of September.

3 If a ship be in the Latitude of 23 gr. 30 m. that is, if it have either of the Tropicks verti­call: then at what time the Sunnes Altitude is equall to his distan [...]e from any of the Equi­noctiall points, then t [...]e Sunne is due East or West.

4 If a ship be betweene the Equinoctiall and either of the Tropicks, the Sunne will come twice to one point of the compasse in the fore­noone, that is, in one and the same position.

5 Vnder the Equinoctiall neare Guinea there is but two sorts of windes all the year, 6 moneths a Northerly winde, and 6 moneths a Southerly winde, and the flux of the Sea is accordingly.

6 If two ships under the Equinoctiall be 100. leagues asunder, and should sayle Northerly [Page 234] untill they were come under the Articke circle, they should then be but 50 leagues asunder.

7 Those which have the Artick circle, verti­call: when the Sunne is in the Tropick of Can­cer, the Sun setteth not, but toucheth the western part of the Horizon.

8 If the complement of the Sunnes height at noon be found equall to the Sunnes Declina­tion for that day, then the [...]quinoctiall is ver­ticall: or a shippe making such an observation, the Equinoctiall is in the Zenith, or direct over them, by which Navigators know when they crosse the line, in their travels to the Indies, or other parts.

9 The Sunne being in the Equinoctiall, the ex­tremity of the stile in any Sunne-dyall upon a plaine, maketh a right line, otherwise it is Eclip­ticall, Hyperbolicall, &c.

10 When the shadow of a man, or other thing upon a Horizontall [...] is equall unto it in length, then is the Sunne in the middle point between the Horizon and the Zenith, that is, 45 degrees high.

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.


TAke an ordinary


square wch Car­penters or other workemen use, as HKL. and placing H. to the eye so that HK. be levell, go back or come nearer untill that by it you may see the top M. for then the distance from you to the height is equall to the 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.

Artificiall fire-Workes: Or the manner of making of Rockets and Balls of fire, as well for the Wa­ter, as for the Aire; with the com­position of Starres, Golden-rain, Serpen [...]s, Lances, Whee [...]s of fire and such like, pleasant and Recreative.

Of the composition for Rockets.

IN the making of Rockets, the chiefest thing to be re­garded is the composition that they ought to be fil­led with; forasmuch as that which is proper to Rockets which are of a lesse sort is very improper to those which are of a more greater forme; for the fire being lighted in a great concave, which is filled with a quick composition, burnes with great violence; contrarily, a weak composition being in a small concave, makes no effect: therefore we shall here deliver in the first place rules and directions, which may serve for the true composition, or matter with which you may charge any Rocket, from Rockets which [Page 266] are charged but with one ounce of Powder un­to great Rockets which requireth for their charge 10 pound of Powder, as followeth.

For Rockets of one ounce.

Vnto each pound of good musket Powder smal beaten, put two ounces of smal Cole dust, and with this composition charge the Rocket.

For Rockets of 2 or 3 ounces.

Vnto every foure ounces and a halfe of pow­der dust, adde an ounce of Salt-peter, or to eve­ry 4 ounces of powder dust, adde an ounce of Cole dust.

For Rockets of 4 ounces.

Vnto every pound of Powder dust adde 4 ounces of Salt peter & one ounce of Cole dust: but to have it more slow, unto every 10. oun­ces of good dust powder adde 3 ounces of Salt-peter, and 3 ounces of Cole dust.

For Rockets of 5 or 6 ounces.

Vnto every pound of Powder dust, adde 3 ounces and a halfe of Salt peter, and 2 ounces and a halfe of Coledust, as also an ounce of Sulphur, and an ounce of fyle dust.

For Rockets of 7 or 8 ounces.

Vnto every pound of Powder dust adde 4 ounces of Salt peter, and 3 ounces of Sulphur.

Of Rockets of 10 or 12 ounces.

Vnto the precedent composition adde halfe an ounce of Sulphur, and it will be sufficient.

For Rockets of 14 or 15 ounces.

Vnto every pound of Powder dust adde 4 ounces of Salt peter, or Cole dust 2 ¼ ounces of [Page 267] Sulphur and file dust of 1 ¼ ounce.

For Rockets of 1, pound.

Vnto every pound of Powder dust adde 3 ounces of Cole dust, and one ounce of Sulphur.

Of Rockets of 2, pound.

Vnto every pound of Powder dust adde 9 ½ ounces of Salt peter, of Cole dust 2 1/ [...] ounces, filedust 1 [...]/2 ounce, and of Sulphur ¾ of ounce.

For Rockets of 3, pound.

Vnto every pound of Salt peter adde 6 ounces of Cole dust, and of Sulpher 4, ounces.

For Rockets of 4, 5, 6, or 7, pound.

Vnto every pound of Salt peter adde 5 oun­ces of Cole dust, and 2 ½ ounces of Sulphur.

For Rockets of 8, 9, or 10 pound.

Vnto every pound of Salt peter, adde 5 ½ ounces of Cole dust, and of Sulphur 2 ½ ounces.

Here note that in all great Rockets, there is no Powder put, because of the greatnesse of the fire which is lighted at once, which causeth too great a violence, therefore ought to be filled with a more weaker composition.

Of the making of Rockets and other Fireworkes.

FOr the making of Rockets of sundry kindes, divers moulds are to be made, with their Rolling pins, Breaths, Chargers, &c. as may be seen here in the figure. And having rolled a Case of paper upon the Rolling pin for your mould, fill it with the composition be­longing to that mould as before is delivered: [Page 268] now may you load it on the top, with Ser­pents, Reports, Stars, or Golden Raine: the Ser­pents are made about the bignesse of ones lit­tle finger, by rolling a little paper upon a small stick, and then tying one end of it, and filling it with the mixt composition somewhat close, and then tying the other end. The re­ports are made in their paper-Cases as the Serpents, but the Paper somewhat thicker to give the greater report. These are filled with


graine-Powder or halfe Powder and halfe com­position, and tying both ends close, they are finished. The best kinde of starres are made with this mixture following; unto every 4 ounces of Salt-peter, adde 2 ounces of Sul­phur, [Page 269] and to it put 1. ounce of Powder-dust, and of this composition make your starres, by putting a little of it within a small quantity of


towe; and then tying it up in the form of a ball as great as an Hasel-Nut or a little Wal-nut, through which there must be drawne a little Primer to make it take fire. Touching the ma­king of the Golden Raine, that is nothing but filling of Quilles with the composition of your Rockets somewhat hard. Now if the head of a Rocket be loaded with a thousand of those Quilles, its a goodly sight to see how pleasant­ly they [...]pread themselves in the Aire, and come downe like streames of Gold much like the falling downe of Snow being agitated by some turbulent winde.

Of recreative fires.

1 PHil [...]strates saith, that if wine in a platter be placed upon a receiver of burning Coles, to exhale the spirit of it, and be inclosed within a Cupboard or such like place, so that the Aire may not go in, nor out, and so being shut up for 30 yeares, he that shall open it, having a wax Candle lighted, and shall put it into the Cubboard there will appeare unto him the figure of many cleare starres.

2 If Aquavitae have Camphire dissolved in it; and be evaporated in a close Chamber, where there is but a Charcole fire, the first that enters into the Chamber with a Candle lighted, will be extremely astonished, for all the Chamber will seeme to be full of fire very subtile, but it will be of little continuance.

3 Candles which are deceitful are made of halfe powder, covered over with Tallow, and the other halfe is made of cleane Tallow, or Waxe, with an ordinary week; this Candle be­ing lighted, and the upper halfe consumed, the powder will take fire, not without great noise and astonishment to those which are ignorant of the cause.

4 A dozen or twenty smal Serpents placed se­cretly under a Candlestick that is indifferent big, which may have a hole passe through the socket of it to the Candle, through which a piece of primer may be placed, and setting a smal C [...]ndle in the socket to burne according [Page 271] to a time limited: which Candlestick may be set on a side Table without suspition to any; then when the Candle is burned, that it fires the primer, that immediately will fire all the Serpents, which overthrowing the Candlestick will flye here and there, intermixing them­selves, sometimes in the Aire, sometimes in the Planching, one amongst another, like the crawling of Serpents, continuing for a pretty while in this posture, and in extinguishing every one will give his report like a Pistoll; This will not a little astonish some, thinking the house will be fired, though the whole powder together makes not an ounce, and hath no strength to do such an effect.

How to make fire run up and downe, forward and backward

TAke small Rockets, and place the taile of one to the head of the other upon a Cord according to your fancie, as admit the Cord to be ABCDEFG. give fire to the Rocket at A, which will flye to B, which will come back againe to A, and fire another at C, that will flie at D, which will fire another there, and fl [...]e to E, and that to F, and so from F, to G, and at G, may be placed a pot of fire, viz. GH. which fired will make good sport▪ bec [...]u [...]e the Serpents which are in it will variously [...]nter­mix themselves in the Aire, and upon the ground, and every one will extinguish with a report: and here may you note that upon the [Page 272] Rockets may be placed fierie Dragons, Comba­tants, or such like to meet one another, having


lights placed in the Concavity of their bodies which will give great grace to the action.

How to make Wheels of fire.

TAke a Hoop, and place two Lath [...] a­crosse one the other; upon the crossing of which make a hole, so that it may be placed upon a pin to turne easily, as the figure Q. sheweth upon the sides of which hoope or round Circle place your Rockets, to which you may place Lances of fire between each [Page 273] Rocket: let this wheele be placed upon a stan­dard as here is represented, and place a piece of Primer from one Lance to another, then give fire at G, which will fire F, that B, that will fire


D, that C, and that will fire the Rocket at A ▪ then immediatly the wheel will begin to move, and represent unto the spectators a Circle of changeable fire, and if pots of fire be tied to it, you will have fine sport in the turning of the wheele and casting out of the Serpents.

Of night-Combatants.

CLubbes, Targets, Faulchons, and Maces charged with severall fires, do make your night-Combatants, or are used to make place amongst a throng of people. The Clubbes at the ends are made like a round Panier with [Page 274] small sticks, filled with little Rockets in a spi­rall forme glu [...]d and so placed that they fire but one after another; the Ma [...]es are of di­vers fashions, some made oblong at the end, some made of a sp [...]rall forme, but all made hol­low to put in several composition, and are boar­ed in divers places, which are for sundry Roc­kets, and Lances of weak composition to be fired at pleasure: The Faulchons are made of wood in a bowing forme like the figure A, having their backes large to receive many Roc­kets, the head of one neare the neck of ano­ther, glued and fastned well together, so that one being spent another may be fired. [...] Tar­gets


are made of wooden thinne boards, which are channeled in spiral lines to containe primer [Page 275] to fire the Rockets one after another, which is all covered with thinne covering of wood, or Pastboard, boared with holes spirally also; which Rockets must be glued and made fast to the place of the Channels: Now if two men, the one having a Target in his hand, and the other a Falchon, or Mace of fire, shall begin to fight, it will appeare very pleasant to the Spectators: for by the motion of fighting, the place will seem to be ful of streames of fire: and there may be adjoyned to each Target a Sunne or a burning Comet with Lances of fire, which will make them more beautifull and re­splendent in that acti [...]n.

Of standing Fires.

SVch as are used for recreation, are Collossus, Statues, Arches, Pyramides, Chariots,


[Page 276] Chaires of triumph and such like, which may be accommodated with Rockets of fire, and beau­tified with sundry other artificiall fires, as pots of fire for the Aire which may cast forth se­veral figures, Scutchions, Rockets of divers sorts, Starres, Crownes, Leaters, and such like, the borders of which may be armed with sundry Lances of fire, of small flying Rockets with re­ports, flames, of small birds of Cypres, Lan [...] ­hornes of fire, Candles of divers uses, and co­lours in burning: and whatsoever the fancie of an ingenious head may allude unto.

Of Pots of fire for the Aire, which are throwne out of one Case one after another of a long continuance.

MAke a long Trunk as AG, and by the side AH, let there be a Channel which may be fiered with slow primer or compositi­on; then having charged the Trunk AG, with the Pots of fire for the Aire at IGEC, and make the Trunk AG, very fast unto a Post as IK, give fire at the top as at A, which burning downewards will give fire to C, and so throw out that Pot in the Aire, vvhich being spent, in the meane time the fire vvil-burne from B to D, and so fire E, and throvv it out also into the Ayre, and so all the rest one after another vvill be throvvne out: and if the Pots of fire for the Aire vvhich are cast out, be filled vvith diverse Fire-vvorkes, they vvill be so much the [Page 277] more pleasant to the


beholders. These Trunks of fire doe greatly adorne a Firevvorke, and may conveniently be pla­ced at each angle of the vvhole vvorke.

Of Pots of fire for the ground.

MAny Pots of fire being fired together do give a fine representation, and recreation to the spectators, and


cause a vvonderfull shout amongst the common people vvch are standers by; for those Pots being fil­led vvith Balles of fire and flying Ser­pents for the Aire, they vvill so intermix one vvithin another, in flying here and there a little above the ground, and giving such a volley of reports that the Aire vvill rebound vvith their noise, and the vvhole place be filled vvith sundry [Page 278] streames of pleasant fire; which serpents will much occupie these about the place to defend themselves in their upper parts, when they will no lesse be busied by the balls of fire, which seeme to annoy their feet.

Of Balles of f [...]re.

THese are very various according to a mans fancy; some of which are made with very small Rockets, the head of one tyed to the neck of another: the ball being made may be cove­red over with pitch except the hole to give fire to it; this Ball will make fine sport amongst the standers by, which will take all a fire, and rolle sometimes this way, sometimes that way, be­tween the legs of those that are standers by▪ if they take not heed, for the motion will be very irregular, and


in the motion will cast forth severall fires with reports. In the second kind there may be a channell of Iron placed in divers places in spirall man­ner, against which may be placed as many small petards of paper as possible may be, the Channell must be full of slow comp [...]sition, and may be covered a [...] the former, and made fit with his Rockets in the middle: this Ball may be shot out of a [Page 279] morter Peece, or charged on the top of a Roc­ [...]et: for in its motion it will flye here and there, and give many reports in the Aire: because of the discharge of the petards.

Of fire upon the Water.

Places which are [...] upon Rivers or great Ponds, are proper to make Recreative fres on: and if it be required to make some of consequence, such may conveniently be made upon two Bo [...]ts, upon which may be built two Beasts, Turrets, Pagins, Castles, or such like,


to receive or hold the diversity of Fire workes that may be made within it, in which may play [...] fires, Petards, &c. and cast out many simple Granadoes, Balls of fire to burne in the [Page 280] water-Serpents and other things, and often times these boates in their incounters may hang one in another, that so the Combatants with the Targets, and Maces may fight; which will give great▪ content to the eyes of those which are lookers on, and in the conclusion fire one another, (for which end they were made:) by which the dexterity of the one may be knowne in respect of the other, and the tri­umph and victory of the fight gotten.

Of Balles of fire which move upon the water.

THese may be made in forme of a Ball stuf­fed with other little Balls, glued round about and filled with composition for the wa­ter, which fiered, will produce marvellous and admirable effects, for which there must be had little Cannons of white Iron, as the ends of small funnels; these Iron Cannons may be pier­ced in sundry places, to which holes, may be set small Balles ful of composition for the water which small Balls must be peirced deep and large, and covered with Pitch, except the hole: in which hole must


be first placed a little quantitie of grain-Powder; and the rest of the hole filled up with compo­sition; and note further that these Iron Cannons, must [Page 281] be filled with a slow composition; but such which is proper to burne in the water: then must these Cannons with their small Balls be put so together that it may make a Globe, and the holes in the Cannons be answerable to the hollow Balls, and all covered over with Pitch and Tallow; afterwards pierce this Ball against the greatest Cannon (to which all the lesser should answer) unto the composition, then fire it, and when it begins to blow, throw it in­to the water, so the fire comming to the holes will fire the graine Powder, the which will cause the Balls to separate and fly here and there, sometimes two at a time, sometimes three, sometimes more, which will burne with­in the water with great astonishment and con­tent to those which see it.

Of Lances of fire.

STanding Lances of fire, are made commonly with hollow wood, to containe sundry Petards, or Rockets, as the figure here shew­eth, by which is easie to invent others occord­ing to ones fancy. These Lances have wood­en handles, that so they may be fastned at some Post, so that they be not overthrowne in the flying out of the Rockets or Petards: there are lesser sorts of Lances whose cases are of three or foure fouldings of Paper of a foote long, and about the bignesse of ones finger, which are filled with a composition for Lances. But if these Lances be filled with a compositi­on, [Page 282] then (unto every 4 ouncs of powder add [...] 2 ounces of Salt-Peter, and unto that adde 1


ounce of Sulphur) it will make a brick fire red before it be halfe spent, if the Lance be fiered and held to it: and if 20 such Lances were placed about a great Rocket and shot to a house or ship, it would produce a mischie­vous effect.

How to shoot a Rocket Horizontall, or otherwise.

VNto the end of the Rocket place an Arrow which may not be too heavy, but in stead of the feathers let that be of thinne white


tinne plate, and place it upon a rest, as here you may see by the Figure, then give fire unto it, and you may see how ser­viceable it may be. To the head of such Rockets, may be pla­ced Petards, Balls of fire, Granadoes, &c. and so may be applyed to warlike affaires.

How a Rocket burning in the water for a certaine time, at last shall fly up in the Aire with an exceeding quickness.

TO do this, take two Rockets, the one equall to the other, and joyne them one unto ano­ther in the middle at C. in such sort that the fire may easily passe from one to another: it be­ing thus done, tye the two Rockets at a stick in D, and let it be so long and great that it may make the Rockets in the water hang, or lye up­right: then take a pack-thread and tye it at G. and let it come double about the stick DM. at [...] and at that


point hang a Bullet of some weight as K. for then giving fire at A. it will burne to B. by a small serpent filled there and tyed at the end, and covered so that the water injure it hot, which will fire the Rocket BD, and so mount­ing quick out of the water by the loose tying at C. and the Bullet at the pack-thread, will leave the other Rocket in the vvater: and so ascend like a Rocket in the Aire, to the admiration of such as knovv not the secrecie.

Of the framing of the parts of a Fire-Worke, together, that the severall workes may fire one after another.

CAuse a frame to be made as ABCD. of tvvo foot square every vvay, or thereabouts (according to the quantity of your severall vvorkes) then may you at each angle have a great Lance of fire to stand, vvhich may cast out Pots of fire as they consume: upon the ledges AB.BC. and CD. may be placed small Lances of fire about the number of 30 or 60, some sidevvise, and others upright, betvveen these Lances may be placed Pots of fire sloping outvvards, but made very fast, and covered ve­ry close, that they chance not to fire before they should; then upon the ledges RE.FG.HI. and AD may be placed your soucisons, and behinde all the vvork may be set your Boxes of Rockets, in each of vvhich you may place 6, 9, [...]2. or 20 small Rockets: Novv give fire at A. (by help of a piece of primer going from one Lance to another) all the Lances vvill instant­ly at once be lighted, and as soone as the Lance at A is consumed, it vvill fire the Channell vvhich is made in the ledge of the frame vvhich runnes under the Pots of fire, and as the fire goes along burning, the Pots vvill be cast forth, and so the rank of Pots upon the sides of the frame AB.BC. and CD. being spent, the soucisons vvill begin to play being fiered also by a Channel vvhich runnes under them, upon [Page]


[Page 285] the ledges AD, HI [...]G, and RE. then when the Soucisons are spent upon the last ledge RE. there may be a secret Channel in the ledge CD which may fire the Box of Rockets at K. and may fire all the rest one after another, which Boxes may be all charged with severall Fire-Workes: for the Rockets of the first Box may be loaden with Serpents, the second with Stars, the third with Reports, the fourth with Gold­en raine, and the fifth with small flying Ser­pents; these mounting one after another and flying to and fro will much inlighten the Aire in their ascending, but when these Rockets dis­charge themselves above, then will there be a most pleasant representation, for these fires will dilate themselves in divers beautifull formes, some like the branching of Trees, others like fountaines of water gliding in the Aire, others like flashes of lightning, others like the glitte­ring of starres, giving great contentment, and delight to those which behold them; But if the worke be furnished also with Balons (which is the chiefest in recreative Fire-works) then shall you see ascending in the Aire but as it were onely a quill of fire, but once the Balon taking fire, the Aire will seeme more than 100. foot square full of crawling, and flying Serpents, which will extinguish with a volley of more than 500 reports: and so fill the Aire and Fir­mament with their rebounding clamour.

[Page 286]The making of which with many other rare and excellent Fire-workes, and other practises, not onely for recreation, but also for service: you may finde in a book intituled Artificiall Fire-workes, made by Mr. Malthas (a master of his knowledge) and are to be sold by VVilliam Leake, at the Crowne in Fleet-street, between the two Temple-Gates.


In this Booke we have no­thing omitted what was materi­all in the originall, but have a­bundantly augmented it in sun­dry experiments: And though the examinations are not so full, and manifold; yet (by way of brevitie) we have expressed ful­ly their substance, to avoid pro­lixitie, and so past by things rei­terated.


Printed or sold by William Leak, at the Crovvne in Fleetstreet neere the Temple, these Books following.

  • YOrk's Heraldry, Folio
  • A Bible of a very fair large Roman letter, 4 [...]
  • Orlando F [...]rios [...] Folio.
  • Callu learned Readings on the Scat. 21. Hen. 80. Cap 5 of Sewer [...]
  • Perkins on the Laws of England.
  • Wi [...]kinsons Office of She [...]fs.
  • Vade Mecum, of a Justice of Peace.
  • The book of Fees.
  • Peasons Law.
  • Mirrour of Just [...]ce.
  • Topicks in the Laws of England.
  • Sken de significatione Verborum.
  • Delaman's use of the Horizontal Quadrant.
  • Wilby's 2d set of Musique, 345 and 6 Parts.
  • Corderius in English.
  • D [...]ctor Fulk's Meteors.
  • Malthus Fire-workes.
  • Nyes Gunnery & Fire-workes
  • C [...]to Ma [...]or with Annotations, by Wil. Austin Esquire.
  • Mel Helliconium, by Alex. Ross [...]
  • Nosce teipsum, by Sr John Davis
  • Animadversions on Lil [...]i [...]s Grammer.
  • The History of Vienna, & Paris
  • Lazarillo de Tormes.
  • Hero and L [...]ander, by G. Chapman and Christoph. Marlow.
  • Al [...]ilia or Philotas loving folly.
  • Bishop Andrews Sermons.
  • Adams on [...]eter.
  • Posing of the Accidence.
  • Am [...]dis de Gaule.
  • Guillieliam's Heraldry.
  • Herberts Travels.
  • Bacc [...]s Tales.
  • Man become guilty, by John Francis Sen [...]t, and Englished by Henry Earl of Monmouth.
  • The Ideot in 4 books; the first and second of Wisdom; the third of the Mind, the fourth of S [...]tick Experiments of the Ballance.
  • The life and Reign of Hen. the Eighth, written by the L. Herbet
  • Cornwallis Essays, & Paradoxes.
  • Clenards greek G [...]ammar 80
  • A [...]laluci [...], or the house of light: A discourse written in the year 1651, by SN. a modern Speculator.
  • A Tragedy written by the most learned Hugo Grotius called, Christus Patience, and transla­ted into Engl. by George Sand
  • The Mount of Olives: or Solli­tary Devotions, by Henry Vaughan Silurist VVith an excellent discourse of Man in glory, written by the Reve­rend Anselm Arch Bishop of Canterbury.
  • The Fort Royall of Holy Scrip­tures by I. H.
  • Hen. the Fourth. Philaster.
  • The wedding. The Hollander.
  • Maids Tragedie. King & no K.
  • The gratefull Servant.
  • The strange Discovery.
  • Othello; the Moor of Venice.
  • The Merchant of Venice.

THE DESCRIPTION AND USE OF THE DOVBLE Horizontall Dyall. WHEREBY NOT ONELY THE Houre of the Day is shewn; but also the Meridian Line is found: And most ASTRONOMICALL Questions, which may be done by the GLOBE: are resolved.


Whereunto is added, The Description of the generall HOROLOGICALL RING.

LONDON, Printed for WILLIAM LEAKE, and are to be sold at his Shop at the signe of the Crown in Fleetstreet, between the two Temple Gates. 1652.

The description, and use of the double Horizontall Diall.

THere are upon the Plate two severall Dyals. That which is outermost, is an ordinary diall, divided in­to houres and quarters, and every quarter into three parts which are five minutes a piece: so that the whole houre is understood to contein 60 minutes. And for this dyall the shadow of the upper oblique, or slanting edge of the style, or cocke, doth serve.

The other diall, which is within, is the projection of the upper Hemisphaere, upon the plain of the Horizon: the Ho­rizon it self is understood to be the innermost circle of the limbe: and is divided on both sides from the points of East and West into degrees, noted with 10.20.30, &c. As far as need requireth: And the center of the In­strument is the Zenith, or Verticall point.

Within the Horizon the middle straight line pointing North and South upon which the style standeth, is the Meridian or twelve a clock line: and the other short arching lines on both sides of it, are the houre lines, di­stinguished accordingly by their figures: and are divi­ded into quarters by the smaller lines drawn between them: every quarter conteining 15 minutes.

The two arches which crosse the houre lines, meeting on both sides in the points of intersection of the sixe a clocke lines with the Horizon, are the two semicircles of the Ecliptick or annuall circle of the sun: the upper of which arches serveth for the Summer halfe yeere; and the lower for the Winter half yeer: and therefore divi­ded into 365 dayes: which are also distinguished into twelve moneths with longer lines, having their names set down: and into tenths and fifts with shorter lines: [Page] and the rest of the dayes with pricks as may plainly be seene in the diall.

And this is for the ready finding out of the place of the Sun every day: and also for the shewing of the Suns yeerely motion, because by this motion the Sun goeth round about the heavens in the compasse of a yeer, ma­king the four parts, or seasons thereof▪ namely, the Spring in that quarter of the Ecliptick which begins at the intersecti­on on the East side of the diall▪ and is therefore called the Vernall intersection. Then the Summer in that quarter of the Ecliptick which begin at the intersection with the Meridian in the highest point next the Zenith. After that, Autumne in that quarter of the Ecliptick which beginneth at the intersection on the West side of the di­all, and is therefore called the Au [...]umna [...]l intersection and lastly, the Winter in that quarter of the Ecliptic [...], which beginneth at the intersection, with the Meridian i [...] the lowest point next the Horizon.

But desides this yeerely motte [...], the Sun hath a diurnall, or daily motion, whereby it maketh day and night, with all the diversities and inaequalities thereof: which is ex­pressed by those other circles drawn crosse the houre lines; the middlemost whereof, being grosser then the rest, meeting with the Ec [...]iptick in the points of the Vernall, and Autumnall intersections ▪ is the Equinoctiall: and the rest on both sides of it are called the parallels, or diurnall arch of the Sun, the two outermost whereof are the Tropicks, because in them the sun hath his furthe [...]t digres­sion or Declination from the Aequinoctiall, which is de­grees 23 1/ [...] ▪ and thence beginneth againe to return to­wards the Equinoctiall. The upper of the two Tropicks in this nor Northerne Hemisphere is the Trop [...]ck of Can­cer, and the sun being in it, is highest into the North, ma­king [Page] the longest day of Summer: And the lower next the Horizon is the Tropick of Capricorne; and the sun be­ing in it, is lowest into the South, making the shortest day of winter.

Between the two Tropicks and the Aequinoctiall, infi­nite such parallel circles are understood to be conteined: for the sun, in what point soever of the Ecliptick it is car­ried▪ describeth by his Lation a circle parallel to the Aequi­noctiall: yet those parallels which are in the instrument, though drawn but to every second degree of Declina­tion, may be sufficient to direct the eye in imagining and tracing out through every day of the whole yeere in the Ecliptick, a proper circle which may be the diur­nall arch of the sun for that day. For upon the right estimation of that imaginary parallel doth the manifold use of this instument especially rely: because the true place of the sun all that day is in some part or point of that circle. Wherefore for the bet [...]er con­ceiving and bearing in minde thereof, every fift paral­lel is herein made a little g [...]osser then the rest.

For this inner diall serveth the shadow of the up­right edge of the style; which I therefore call the upright shadow.

And thus by the eye and view onely to behold and com­prehend the course of the sun▪ throughout the whole yeere both for his annuall and diurnall motion, may be the first use of this instrument.

II Use. To finde the declination of the sun every day.

Looke the day of the moneth proposed in the Ecliptick, and mark how many degrees the prick shewing that day, is distant from the Equinoctiall, either on the Summer or Winter side, viz. North or South.

[Page]Example 1. What will the Declination of the sun be upon the eleven [...]h day of August? look the eleventh day of August and you shall finde it in the sixth circle above the [...]quinoctiall: Now because each parallel standeth (as hath been said before) for two degrees, the sun shall that day decline Northwards 12. degrees.

Example 2. What declination hath the sun upon the 24 day of March? look the 24 day of March, and you shall finde it betweene the second and third northern parallels, as it were an half and one fift part of that di [...]tance from the second: Reckon therefore four de­grees for the two circles, and one de [...]ree for the halfe space: So shall the Suns declination be five degrees, and about one fift part of a degree Northward that same day.

Example 3. What declination hath the sun upon the 13 day of November? look the 13 day of November, and you shall finde it below the Equinoctiall ten parallels, and about one quarter which is 20 degrees and an halfe southward. So much is the declination. And according to these examples judge of all the rest.

III. Use. To finde the diurnall arch, or circle of the suns course every day.

The sun every day by his motion (as hath been said) describeth a circle parallel to the [...]quinoctiall, which is either one of the circles in the diall, or some-where [...]e­tween two of them. First, theref [...]re se [...]k the day of the moneth; and if it fall upon one of those parallels; that is the circle of the suns course that same day: But if it fall be­tweene any two of the parallels, imagine in your mind [...], and estimate with your eye, another parallel th [...]ough that point betweene those two parallels keeping still the same distance from each of them.

[Page]As in the first of the three former examples, The circle of the Suns course upon 11 of August▪ shal be the very sixt circle above the Equinoctiall toward the cente [...].

In [...]xample 2. The circle of the suns cou [...]se upon the 24 of March shall be an imaginary circle between the se­cond and third parallels still keeping an half of that space, and one fifth part more of the rest, from the second.

In example 3. The circle of the suns course upon the 13 of November: shall be an imaginary circle between the tenth and eleventh parallels below the Equinoctiall, still keeping one quarter of that space from the tenth.

IIII Use. To finde the r [...]sing and setting of the sun eve [...]yday.

[...] (as was last shewed) the imaginary circle or parallel of the suns course for that day, and marke the point where it meeteth with the horizon, both on the East and W [...]st sides, for that is the very point of the suns r [...]sing, and setting that same day, and the houre lines which are on both sides of it, by proportioning the di­stance reasonably, according to 15 minutes for the quarter of the houre, will shew the houre of the suns rising on the East side, and the suns setting on the West side.

V Use. To know the reason and manner of the In­creasing and decreasing of the dayes and nights [...]hroughout the whole yeere.

When the Sun is in the Equinoctiall, it riseth and set­teth at 6 a clock, for in the instrument the intersection of the Equinoctiall, and the Ecliptick with the Horizon is in the six a clocke circle on both sides. But if the sun be out of the Equinoctial, declining toward the North, the in­tersections of the parallel of the sun with the Horizon is [Page] before 6 in the morning, and after 6 in the evening: and the Diurnall arch greater then 12 houres; and so much more great, the greater the Northerne Declination is. Againe, if the sun be declining toward the South, the in­tersections of the parallel of the sun, with the Horizon is after 6 in the morning, and before 6 in the evening: and the Diurnall arch lesser then 12 houres; and by so much lesser, the greater the Southerne Declination is.

And in those places of the Ecliptick in which the sun most speedily changeth his declination, the length also of the day is most a [...]tered: and where the Ecliptick goeth most parallel to the Equinoctiall changing the declina­tion, but little altered. As for example, when the sun is neer unto the Equinoctiall on both sides, the dayes in­crease and also decrease suddenly and apace; because in those places the Ecliptick inclineth to the Equinoctiall in a manner like a streight line, making sensible declination. Again, when the sun is neere his greatest declination, as in the height of Summer, and the depth of Winter, the dayes keep for a good time, as it were, at one stay, because in these places the Ecliptick is in a manner parallel to the Equinoctiall, the length o [...] the day also is but little, scarce altering the declination: And because in those two times of the yeer, the sun standeth as it were still at one declinati­on, they are called the summer solstice, and winter solstice. And in the mean space the neerer every place is to the Equinoctiall, the greater is the diversity of dayes.

Wherefore, we may hereby plainly see that the com­mon received opinion, that in every moneth the dayes doe equally increase, is erroneous.

Also we may see that in parallels equally distant from the Equinoctiall, the day on the one side is equall to the night on the other side.

[Page]VI. Vse. To finde how far the sun riseth, and setteth from the true east and west points, which is called the suns Amp [...]itude ortive, and occasive.

Seek out (as was shewed in III Vse) the imagina­ry circle, or parallel of the suns course, and the points of that circle in the horizon, on the East and West sides cutteth the degree of the Amplitude ortive, and occasive.

VII Use. To finde the length of every day and night.

Double the houre of the sunnes setting, and you shal have the length of the day; & double the hour of the sunnes rising, and you shal have the length of the right.

VIII Vse. To finde the true place of the sun upon the dyall, that is, the point of the instrument which answereth to the place of the sun in the heavens at any time, which is the very ground of all the questions following.

If the dyall be fixed upon a post: Look what a clock it is by the outward dyall, that is, look what houre and part of houre the shadow of the slanting edge of the style sheweth in the outward limbe. Then behold the shadow of the upright edge, and marke what point thereof is upon that very houre and part in the inner dyall among the parallels, that point is the true place of the Sunne at the same instant.

If the dyal be not fixed, and you have a Meridian line no [...]ed in any window where the Sunne shineth: place the Meridian of your dyal upon the Meridian line gi­ven, so that the top of the stile may point into the north: and so the dyal is as it were fixed, wherefore by the former rule you may finde the place of the Sunne upon it.

If the dyal be not fixed, neither you have a Meridian line, but you know the true houre of the day exactly: hold the dyal even and parallel to the Horizon, moving [Page] it till the slanting edge of the stile cast his shadow justly upon the time or houre given; for then the dyal is truly placed, as upon a post. Seek therefore what point of the upright shadow falleth upon that very houre, and there is the place of the Sun.

But if your dyal be loose, and you know neither the Meridian nor the time of the day. First, by the day of the moneth in the Ecliptique, finde the su [...]s parallel, or d [...]urnall arch for that day▪ then holding the dyal level to the horizon, move it every way untill the slanting sha­dow of the style in the outward limbe, and the upright shadow in the Sunnes diurnal arch, both shew the very same houre and minute, for that very point of the Sunnes parallel, which the upright shadow cutteth, is the true place of the Sun on the dyal at that present.

But note that by reason of the thicknes of the style, and the bluntnesse of the angle of the upright edge, the Sun cannot come unto that edge for some space before and after noone. And so during the time that the Sunne shineth not on that upright edge, the place of the Sunne in the dyal cannot be found. Wherefore they that make this kinde of double dyal, are to be careful to file the up­right edge of the style as thinne and sharpe as possible may be.

That which hath here bin taught concerning the finding out the Suns true place in the dyal, ought perfect­ly to be understood, that it may be readily, and dexterious­ly practised, for upon the true performance thereof de­pendeth all that followeth.

IX Vse. To finde the houre of the day.

If the dyal be fastned upon a post, the houre by the outward dyal, or limbe, is known of every one, and the [Page] upri [...]ht shadow in the Suns parallel, or diurnal arch will also shew the very same houre.

But if the dyall be loose, either hold it or set it paral­lel to the Horizon, with the style pointing into the north and move it gently every way untill the houre shewed in both dialls exactly agreeth, or which is all one, finde out the true place of the Sun upon the dyall, as was taught in the former question, for that point among the houre lines sheweth the houre of the day.

X Vse. To finde out the Meridian, and other points of the Compasse.

First, you must seek the tru [...] houre of the day (by the last question) for in that situation the Meridian of the dyall standeth direct [...]y north and south: and the east pointeth into the east, and the west into the west, and the rest of the points may be given by allowing degrees 11. 1/ [...] unto every point of the compasse.

XI Vse. To finde out the Azumith of the sun, that is, the distance of the Verticall circle, in which the sun is at that present, from the Meridian.

Set your diall upon any plain or flat which is paral­lel to the horizon, with the Meridian pointing directly north or south, as was last shewed: then follow with your eye the upright shadow in a streight line, till it cutteth the horizon: for the degree in which the point of intersection is, shal shew how far the suns Azumith is distant from the east and west points, and the comple­ment thereof unto 90; shal give the distance thereof from the meridian.

XII Vse. To finde out the Declination of any Wall upon which the sun shineth, that is, how far that wall swer­veth from the north or south, either eastward or west­ward.

[Page]Take aboard having one streight edg▪ & a line strick­en perpendicular upon it; apply the streight edg unto the wall at what time the sun shineth upon it, holding the board parallel to the horizon: Set the dyal thereon, and move it gently every way, untill the same hour and mi­nute be shewed in both dyals: and so let it stand: then if the dyal have one of the sides parallel to the Meridian strike a line along that side upon the board, crossing the perpendicular, or else with a bodkin make a point upon the board, at each end of the meridian, and taking away the instrument from the board, and the board from the wall, lay a ruler to those two points, and draw a line crossing the perpendicular: for the angle which that line maketh with the perpendicular, is the angle of the decli [...]nation of the wall. And if it be a right angle, the wall is exactly east or west: but if that line be parallel to the perpendicular, the wall is direct north or south without any declination at all.

You may also finde out the declination of a wall, if the dial be fixed on a post not very far from that wall; in this manner. Your board being applyed to the wall, as was shewed, hang up a thred with a plummet, so that the shadow of the thred may upon the board crosse the perpendicular line: make two pricks in the shadow and run instantly to the dyal and look the horizontal distance of the suns Azumith, or upright shadow from the meri­dian. Then through the two pricks draw a line crossing the perpendicular: and upon the point of the intersection, make a circle equal to the horizon of your Instrument, in which Circle you shal from the line through the two pricks measure the Horizontal distance of the upright shadow, or Azumith from the me­ridian, that way toward which the Meridian is: draw [Page] a line out of the center, to the end of that arch measured: and the angle which this last line maketh with the per­pendicular, shall be equall to the declination of the wall.

XIII Vse. How to place the dyall upon a post without any other direction but it selfe.

Set the diall upon the post, with the stile into the North, as neere as you can guesse: then move it this way and that way, till the same houre and minute be shewed, both in the outward and inward dials by the severall sha­dowes, as hath been already taught, for then the diall standeth in its truest situation; wherefore let it be nailed down in that very place.

XIIII Vse. To finde the height of the sun at high noon everyday.

Seeke out the diurnall Arch or parallel of the suns course for that day, (by Vse III.) and with a paire of Compasses, setting one foot in the center, and the other in the point of intersection of that parallel with the Meridi­an, apply that same distance unto the Semidiameter di­vided: for that measure shal therein shew the degree of of the Suns altitude above the the Horizon that day at high noon.

XV Vse. To finde the height of the sun at any houre or time of the day.

Seeke out the diurnal Arch, or parallel of the suns course for that day: and marke what point of it is in the very houre and minute proposed. And with a paire of Compasses, setting one foot in the Center, and the other in that point of the parallel, apply the same distance upon the Semidiameter divided: for that measure shall shew the degree of the suns altitude above the Horizon at that time.

[Page]And by this meanes you may finde the height of the Sun above the Horizon at every houre throughout the whole yeere, for the making of rings and cylinders and other instruments which are used to shew the houre of the day.

XVI Vse. The height of the sun being given, to finde out the houre, or what it is a clocke.

This is the converse of the former: Seeke therefore in the Semidiameter divided, the height of the sun given. And with a paire of Compasses, setting one foot in the center, and the other at that height, apply the same di­stance unto the diurnall arch, or parallel of the Sun for that day: for that point of the diurnall arch, upon which that same distance lights, is the true place of the sun upon the dial; and sheweth among the houre lines, the true time of the day.

XVII Use. Considerations for the use of the instrument in the night.

In such questions as concerne the night▪ or the time before sun rising, and after sun setting, the instrument re­presenteth the lower Hemisphaere wherein the Southerne pole is elevated. And therefore the parallels which are above the Aequinoctiall toward the center shall be for the Southerne, or winter parallels: and those beneath the Aequinoctiall, for the Northerne or Summer pa­ral [...]els; and the East shall be accounted for West, and the West for East; altogether contrary to that which was before, when the Instrument represented the upper Hemisphaere.

XVIII Use. To finde how many degrees the sun is un­der the Horizon at any time of the night.

Seeke the Declination of the sun for the day propo­sed [Page] (by Vse II.) And at the same declination the con­trary side imagine a parallel for the sun that night▪ and mark what point of it is in the very houre and mi­nute proposed: And with a pair of compasses, setting one foot in the center, and the other in that point of the pa­rallel, apply that same distance unto the semidiameter di­vided: for that measure shall shew the degree of the suns depression below the Horizon at that time.

XIX Use. To finde out the length of the C [...]epusculum, or twylight, every day.

Seek the declination of the sun for the day proposed (by Vse II.) And at the same declination on the contrary side imagine a parallel for the sun that night. And with a paire of compasses setting one foot in the center, and the other at 72 degrees upon the semidiameter divided, apply that same distance, unto the suns nocturnall parallel: for that point of the parallel, upon which that same di­stance shall light, sheweth among the houre lines, the be­ginning of the twilight in the morning, or the end of the twilight in the evening.

XX Use. If the day of the moneth be not known, to finde it out by the dyall.

For the working of this question, either the diall must be fixed rightly on a post, or else you must have a true Meridian line drawn in some window where the sun shineth, wherefore supposing the diall to be justly set ei­ther upon the post, or upon the Meridian. Look what a clock it is by the outward diall, and observe what point of the upright shadow falleth upon the very same minute in the inner diall, and through that same point imagine a parallel circle for the suns course; that imagi­nary circle in the Ecliptick shall cut the day of the moneth.

I The description of it.

THis Instrument serveth as a Diall to finde the houre of the day, not in one place onely (as the most part of Dials do) but generally in all Countreys lying North of the Aequinoctiall: and therefore I call it the generall H [...]rologicall [...]ing.

It consisteth of two br [...]zen circles: a Diameter, and a little Ring to hang it by.

The two circles are so made, that though they are to be set at right angles, when you use the Instrument: yet for more convenient carrying, they may be one folded in­to the other.

The lesser of the two circles is for the Aequinoctiall, having in the midst of the inner side or thicknesse, a line round it, which is the true Aequinoctiall circle, divided into twice twelue hours, from the two opposite points in which it is fastened within the greater.

The greater and outer of the two circles is the Meridi­an: One quarter whereof, beginning at one of the points in which the Aequin [...]cti [...]ll is hung, is divided into nine­ty degrees.

The Diameter is fastened to the Meridian in two op­posite points or poles, o [...]e of them being the very end of the Quadrant, and is the North Pole. Wherefore it is perpendicular to the [...]quinoctiall, having his due position. The diameter is broad, and slit in the middle: and about the slit on both sides are the moneths and dayes of the yeer. And within this slit is a litt [...]e sliding plate pierced through with a small hole: which hole in the motion of it, while it is applied to the dayes of the yeer, representeth the Axis of the world.

[Page]The little Ring whereby the Instrument hangeth, is made to slip up and down along the Quadrant: that so by help of a little tooth annexed, the Instrument may be rectified to any elevation of the Pole.

II. The use of it.

IN using this Instrument, First, the tooth of the little Ring must carefully be set to the height of the Pole in the Quadrant, for the place wherein you are.

Secondly, the hole of the sliding plate within the slit, must be brought exactly unto the day of the moneth.

Thirdly, the Aeqinoctiall is to be drawn out, and by means of the two studs in the Meridian staying it, it is to be set perpendicular thereto.

Fourthly, Guesse as neer as you can at the houre, and turn the hole of the little plate toward it.

Lastly, Hold the Instrument up by the little Ring, that it may hang freely with the North Pole thereof toward the North: and move it gently this way and that way, till the beams of the Sun-shining thorow that hole, fall upon that middle line within the Aequinoctiall: for there shall be the houre of the day: And the Meridan of the Instrument shall hang directly North and South.

These Instrument all Dials are made in brasse by Eli­as Allen dwelling over against St. Clements Church without Temple Barre, at the signe of the Horse­shooe neere Essex Gate.


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