Mathematical RECREATIONS. OR, A Collection of many Problems, extracted out of the Ancient and Modern Philosophers, as Secrets and Experiments in Arithmetic, Geometry, cosmography, Horologiographie, Astronomy, Navigation, Music, Optics, Architecture, Stati●k, Mechanics, Chemistry, Water-works, Fireworks, etc. Not vulgarly manifest till now. Written first in Greek 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 General horological Ring: And The Double horizontal Dial. Invented and written by WILLIAM OUGHTRED. LONDON: Printed for William Leake, at the Sign of the Crown in Fleetstreet, between the two Temple Gates, MDCLIII. On the Frontispiece and Book. ALL Recreations do delight the mind, But these are best being of a learned kind: Here Art and Nature strive to give content, In showing many a rare experiment, Which you may read, & on their Schemes here look Both in the Frontispiece, and in the Book. Upon whose table new conceits are set, Like dainty dishes, thereby for to whet And win 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 Arithmetic are wrought, And hidden secrets unto light are brought, The like it in Geometry doth unfold, And some too in cosmography are told: It divers pretty Dial's doth descry, With strange experiments in Astronomy, And Navigation, with each several Picture, In Music, Optics, and in Architecture: In Statick, Machanicks, and Chemistry, In Water-works, and to ascend more high, In Fireworks, like to Jove's Artillery. All this I know thou in this Book shalt find, And here's enough for to content thy mind. 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 W. MATHEMATICAL Recreations Or a Collection of sundry excellent Problems out of ancient & modern Philosophers Both useful and Recreative Printed for William Leake and are to be sold at the Crown in fleet street between the two Temple gates. TO The thrice Noble and most generous Lo. the Lo. Lambert Verreyken, Lo. of Hinden, Wolverthem, etc. My honourable Lo. AMongst the rare and curious Propositions which I have learned out of the studies of the Mathematics in the famous University of Pont a Mousson, I have taken singular pleasure in certain Problems no less ingenious than recreative, which drew me unto the search of demonstrations more difficult and serious; some of which I have amassed and caused to pass the Press, and here dedicate them now unto your Honour; not that I account them worthy of your view, but in part to testify my affectionate desires to serve you, and to satisfy the curious, who delight themselves in these pleasant studies, knowing well that the Nobility, and Gentry rather study the Mathematical Arts, to content and satisfy their affections, in the speculation of such admirable experiments as are extracted from them, than in hope of gain to fill their Purses. All which studies, and others, with my whole endeavours, I shall always dedicate unto your Honour, with an ardent desire to be accounted ever, Your most humble and obedient Nephew and Servant, H. VAN ETTEN. By way of advertisement. Five or six things I have thought worthy to declare before I pass further. FIrst, that I place not the speculative demonstrations with all these Problems, but content myself to show them as at the finger's end: which was my plot and intention, because those which understand the Mathematics can conceive them easily; others for the most part will content themselves only 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 subtlety of the way; for that which doth ravish the spirits is, an admirable effect, whose cause is unknown: which if it were discovered, half the pleasure is l●st; therefore all the fineness consists in the dexterity of the Act, concealing the means, and changing often the stream. Thirdly, great care ought to be had that one deceive not himself, 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 shows it: when the cause is not in the Mathematical principles, but in him that fails in the acting of it. Fourthly, in certain Arithmetical propositions they have only their answers as I found them in sundry Authors, which any one being studious of Mathematical learning, may find their original, and also the way of their operation. Fifthly, because the number of these Problems, and their dependences are many, and intermixed, I thought it convenient to gather them into a Table: that so each one according to his fancy, might make best choice of that which might best please his palate, the matter being not of one nature, nor of like subtlety: But whosoever will have patience to read on, shall find 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 Modern, have sported and delighted themselves upon several 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 show the greatness of their glory in the smallness of the subject: And have amongst most solid and artificial conclusions, composed and produced sundry Inventions both Philosophical and Mathematical, to solace the mind, and recreate the spirits, which the succeeding ages have embraced, and from them gleaned and extracted many admirable, and rare conclusions; judging that borrowed matter oftentimes yields praise to the industry of its author. Hence for thy use (Courteous Reader) I have with great search and labour collected also, and heaped up together in a body of these pleasant and fine experiments to stir up and delight the affectionate, (out of the writings of Socrates, Plato, Aristotle, Demosthenes, Pythagoras, Democrates, Pliny, 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 always in the variety of things, which she produceth to satisfy the mind of curious inquisitors. And though perhaps these labours to some humourous persons may seem vain, 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 employ the mind in useful knowledge, rather than to be busied in vain Pamphlets, Playbooks, fruitless Legends, and prodigious Histories that are invented out of fancy, which abuse many Noble spirits, dull their wits, & alienate their thoughts from laudable and honourable Studies. In this Tractate thou mayst therefore make choice of such Mathematical Problems and Conclusions as may delight thee, which kind of learning doth excellently adorn a man; seeing the usefulness thereof, and the manly accomplishments it doth produce, is profitable and delightful for all sorts of people, who may furnish and adorn themselves with abundance of matter in that kind, 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 soar in an higher Contemplation, having so glittering a Canopy 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 several Arts specified in the Title-page. Experiments of Arithmetic. 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 Geometry. 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 cosmography. 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 Astronomy. Pag. 220, 221, 222, 223, 224. Experiments in Navigation. Pag. 105, 233, 234, 237, 238. Experiments in Music. Pag. 78, 87, 126. Experiments in Optics. 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 Chemistry. Pag. 198, 255, 256, 257, 260, 262, 263, 264. Experiments in Water-works. Pag. 190, 191, 192, 193, 194, 196, 247, 249, 250, 252, 253. Experiments in Fireworks. From page, 265. to the end. FINIS. A Table of the Contents, and chief points contained in this Book. PROBLEM. II. HOw visible objects that are without, and things that pass by, are most lively represented to those that are within. Page 6 Prob. 1 Of finding of numbers conceived in the mind. 1, 2, 3 Prob. 5 Of a Geographical Garden-plot fit for a Prince or some great personage. 14 Prob. 37 Any liquid substance, as water or wine, placed in a Glass, may be made to boil 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 several numbers being taken by two sundry persons, how subtly to discover which of those numbers each of them took. 46 Prob. 4 That a staff may be broken▪ placed upon two Glasses, without hurting of the Glasses. 12 Prob. 7 How to dispose Lots that the 5, 6, 9, etc. of any number of persons may escape. 16 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 air, and move upon the Point of a needle. 27 Prob. 17 Of a deceitful bowl, to bowl withal. 32 Prob. 16 A ponderous or heavy body may be supported in the air without any one touching it. 30 Prob. 18 How a Pear, or Apple, may be parted into any parts, without breaking the rind thereof. 33 Prob. 15 Of a fine kind of door which opens and shuts on both sides. 30 Prob. 9 How the half of a Vessel which contains 8 measures may be taken, being but only two other measures, the one being 3, and the other 8 measures. 22 Prob. 8 Three persons having taken each of them several things, to find which each of them hath taken. 19 Prob. 6 How to dispose three staves which may support each other in the air. 15 Prob. 14 Many things being disposed Circular (or otherwise) to find which of them any one thinks upon. 28 Prob. 19 To find a number thought upon without ask questions. 33 Prob. 11 How a Millstone 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 uniform and inflexible may pass through a hole which is round, square and Triangular; or round, square and ovall-wise, and exactly fill those several 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 find a number thought upon after another manner than those which are formerly delivered. 39 Prob. 23 To find 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 shall 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 may in a right line pass 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 remains after certain operations being ended, without ask any question. 44 Prob. 29 Of the play with two several things. 45 Prob. 31 How to describe a circle that shall touch 3 points placed howsoever upon a plain, if they be not in a right line. 47 Prob. 32 How to change a circle into a square form. 48 Prob. 33 With one and the same compasses, and at one and the same extent or opening, how to describe many circles concentrical, that is, greater or lesser one than another. 49 Prob. 34 Any number under 10. being thought upon, to find 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 Vessel which holds Wine or Water being cast into it at a certain height, but being filled higher it will run all out of its own accord. 56 Prob. 39 Of a Glass very pleasant. 58 Prob. 40. If any one should hold in each hand as many pieces of money as in the other, how to find 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 kind, weighing alike, being privately placed into two like boxes, to find 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 equal weight, being put in a Box as B.G. to find in which end the Gold or Copper is. 65 Prob. 44 How to represent divers sorts of Rainbows here below. 66 Prob. 45 How that if all the powder in the World were enclosed in a bowl of paper or glass, and being fired on all parts, it could not break that bowl. 68 Prob. 46 To find a number which being divided by 2. there will remain 1. being divided by 3. there will remain 1. and so likewise being divided by 4, 5, or 6. there will still remain one, but being divided by 7 will remain nothing. 69 Prob. 47 One had a certain number of Crowns, and counting them by 2 and 2, there rested 1. counting them by 3, and 3, there rested 2. counting them by 4, and 4, there rested 3. counting them by 5, and 5, there rested 4. counting them by 6, and 6, there rested 5. but counting them by 7 and 7, there rested nothing, how many Crowns might he have? 71 Prob. 48 How many sorts of weights in the least manner must there be to weigh all sorts of things between one pound and 121 pound, and so unto 364 pound? 71 Prob. 49 Of a deceitful balance which being empty seems to be just, because it hangs in Aequilibrio, notwithstanding putting 12 pound in one balance, and 11 in the other, it will remain 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, stars, shadow, or compass, to find out the North, or South, or the 4 Cardinal points of the World, East, West, etc. 75 Prob. 52 Three persons having taken Counters, Cards, or other things, to find how much each one hath taken. 7● Prob. 53 How to make a consort of Music of many parts with one voice or one instrument only. 78 Prob. 54 To make or describe an oval form, or that which is near resembled unto it at one turning, with a pair 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 find out the Centre of it? 82 Prob. 56 Any one having taken 3 Cards, to find how many points they contain. 83 Prob. 57 Many Cards placed in divers ranks, to find which of those Cards any one hath thought. 85 Prob. 58 Many Cards being offered to sundry persons to find which of those Cards any one thinketh upon. 86 Prob. 59 How to make an instrument that helps to hear, as Gallileus made to help to see. 87 Prob. 60 Of a fine Lamp which goeth not out, though one carries it in one's pocket, or being rolled on the ground will still burn. 88 Prob. 61 Any one having thought a Card amongst many Cards, how artificially to discover it out. 89 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 itself with Oil, and burns a long time. 95 Prob. 65 Of the play at Keyle or Ninepins. 97 Prob. 66 Of Spectacles of pleasure Of Spectacles which give several colours to the visage. 98 Of Spectacles which make a Town seem 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 pass 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 Sun, the Stars which move about the Planets, and perspicuously things far remote. Of the parts of Gallileus his Glass. 102 Prob. 67 Of the Magnes and Needles touched therewith. How Rings of Iron may hang one by another in the air. 103 Of Mahomet's Tomb which hangs in the air by the touch of the Magnes. 104 How by the Magnes only to find out North and South 105 Of a secrecy in the Magnes, for discovering things far remote. 106 Of finding the Poles by the Magnes 107 Prob. 68 Of the properties of Aeolipiles or Bowls to blow the fire. 108 Prob. 69 Of the Thermometer, or that which measures the degrees of heat and cold by the air. 110 Of the proportion of humane bodies, of statues, of Colossuses, or huge Jmages and monstrous Giants. 113 Of the commensuration of the parts of the body the one to the other in particular, by which the Lion was measured by his claw, the Giant by his thumb, and Hercules by his foot. 115, 116 Of Statues or Colossuses, or huge Images; that mount Athos metamorphosed by Dynocrites into a statue, in whose hand was a Town able to receive ten thousand men. 117 Of the famous Colossus at Rhodes which bade 70 cubits in height, and loaded 900. Camels, which weighed 1080000 l. 118 Of Nero his great Colossus which had a face of 12 foot large. 119 Of monstrous Giants Of the Giant Og and Goliath. 119, 120 Of the Carcase of a man found which was in length 49 foot; and of that monster found in Crect, which had 46. Cubits of height. 120 Of Campesius his relation of a monster of 300 foot found in Sicily, whose face according to the former proportion should be 30 foot in length. 121 Prob. 71 Of the game at the Palm, at Trap, at Bowls, Paile-maile, and others. 122 Prob. 72 Of the game of square forms. 124 Prob. 73 How to make the string of a Viol sensibly shake without any one touching it. 126 Prob. 74 Of a Vessel which contains 3 several kinds of liquor, all put in at one bunghole, and drawn 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 spherical Glasses which burn, the cause and demonstration of burning with Glasses. 131 Of Maginus his way of setting fire to Powder in a Mine by Glasses. 131 Of the examination of burning by Glasses. 133 Prob. 76 Of pleasant questions by way of Arithmetic. Of the Ass and the Mule. 134 Of the number of Soldiers that fought before old Troy. 135 Of the number of Crowns that two men had. 136 About the hour of the day. 137 Of Pythagoras' Scholars. 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 Cupid's Apples. 139 Of a Man's Age. 140 Of the Lion of Bronze placed upon a fountain with his Epigram. ibid. Prob. 77 In Optics, excellent experiments. Principles touching reflections. 141 Experiments upon flat and plain Glasses. 142 How the Images seem to sink into a plain Glass, and always are seen perpendicular to the Glass, an● also inversed. 143 The things which pass by in a street may by help of a plain glass be seen in a Chamber, and the height of a tower or tree observed. 143 How several Candles from one Candle are represented in a plain Glass, and Glasses alternately may be seen one within another, as also the backparts of the body as well as the foreparts are evidently represented. 144 How an Image may be seen to hang in the air by help of a Glass: and writing read or easily understood. 146 Experiments upon Gibbous, or convex Spherical Glasses. How lively to represent a whole City, fortification, or Army, by a Gibbous Glass. 147 How the Images are seen in Concave Glasses. 149 How the Images are transformed by approaching to the centre of the Glass, or point of concourse; and of an exceeding light that a Concave Glass gives by help of a Candle. 151 How the Images, as a man, a sword, or hand, doth come forth out of the Glass. 152, 153 Of strange apparitions of Images in the air, by help of sundry Glasses. 152, 154 Of the wonderful augmentation of the parts of man's body coming near the point of inflammation, or centre of the Glass. 155 How writing may be reverberated from a Glass upon a Wall, and Read. 156 How by help of a Concave Glass to cast light into a Camp, or to give a perspective light to Pyoneers in a Mine, by one Candle only. 156 How excellently by help of a Concave Glass and a 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 four eyes, two Mouths, two Noses, two heads. Of Glasses which give a colour to the visage, and make the face seem fair and foul. 160 Prob. 78 How to show one that is suspicious, what is in another Chamber or Room, notwithstanding the interposition of that wall. 160 corollary, 1. To see the Besiegers of a place, upon the Rampa●●t of a fortification 161 corollary 2. and 3. Notwithstanding the interposition of Walls and Chambers, by help of a Glass things may be seen, which pass by. 162 Prob. 79 How with a Musket to strike a mark not looking towards it, as exactly as one aimed at it. 162 How exactly to shoot out of a Musket to a place which is not seen, being hindered by some obstacle or other interposition. 163 Prob. 80 How to make an Image to be seen hanging in the air, having his head downward. 164 Prob. 81. How to make a company of representative soldiers seem to be as a regiment, or how few in number may be multiplied to seem to be many in number. 165 corollary. Of an excellent delightful Cabinet made of plain Glasses. 165 Prob. 82 Of fine and pleasant Dial's in Horologiographie. Of a Dial of herbs for a Garden. 166 Of the Dial upon the finger and hand, to find what of the Clock it is. 167 Of a Dial which was about an Obelisk at Rome. 168 Of Dial's with Glasses. 168 Of a Dial which hath a Glass in the place of the stile. 169 Of Dial's 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 find how much time the Bullet of a Cannon spends in the Air 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 blank, or shooting downward. 174 Whether 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, etc. Of grains of Mustardseed, and that one grain being sown, with the increase thereof for 20 years will produce a heap greater than all the earth a hundred thousand times. 178 Of Pigs, and that the great Turk with all his Revenne, is not able to maintain for one year, a Sow with all her increase for 12 years. 179 Of grains of Corn, and that 1 grain with all its increase for 12 years, will amount to 244140625000000000000 grains, which exceeds in value all the treasures in the World. 183 Of the wonderful increase of Sheep. 182 Of the increase of Codfish. 182 Of the Progressive Multiplication of souls; that from one of Noah's Sons, from the flood unto Nimrods' Monarchy, should be produced 111350 souls. 183 Of the increase of Numbers in double proportion, and that a pin being doubled as often as there are weeks in the year, the number of pins that should arise is able to load 45930 ships of a thousand Tun apiece, which are worth more than ten 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 musical instruments, transmutation of Places, in Numbers, Letters, Men and such like▪ 185 Of the wonderful interchange of the Letters in the Alphabet: the exceeding number of men, and time to express the words that may be made with these letters, and the number of Books to comprehend them. 187, 188 Of a servant hired upon certain condition, that he might have land lent him to sow one grain of Corn with its increase for 8 years' time, which amounted to more than four hundred thousand Acres of Land. 188 Prob. 85 Of Fountains, Hydriatiques; Stepticks, Machinecks, and other experiments upon water, or other liquor. First, how water at the foot of a Mountain may be made to ascend to the top of it, and so to descend on the other side of it 190 Secondly, to find how much Liquor is in a Vessel, only by using the tap-hole. 191 Thirdly, how is it, that a Vessel is said to hold more water at the foot of a Mountain, then at the top of it 191 4 How to conduct water from the top of one Mountain to the top of another 192 5 Of a fine Fountain 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 Fountain of pleasure. 196 8 Of a fine watering pot for Gardens. 197 9 How easily to take Wine out of a Vessel at the bung hole without piercing a hole in the Vessel. 198 10 How to measure irregular bodies by help of water. 198 11 To find the weight of water. 199 12 To find the charge that a vessel 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 sink down right. 200 14 How a gross body of metal may swim upon the water. 201 15 How to weigh the lightness of the air. 203 16 Being given a body, to mark it about, and show how much of it will sink in the water, or swim above the water. 204 17 To find how much several metals or other bodies do weigh less in the water than in the air. 204 18 How is it that a balance having like weight in each scale, and hanging in Aequilibrio in the air, 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 19 To show 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 Arithmetic, and first of the number of sands calculated by Archimedes and Clavius. 208 2 Divers metals being melted together in one body, to find the mixture of them. 210 3 A subtle 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 form of figures, which Geometricians call Isoperimeter, or equal in circuit, or Compass. 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 equal in circuit but not in quantity. 215 2 Incident: about two sacks each of them ho●ding but a bushel, and yet were able to hold 4 bushels. 217 3 Incident: showeth the deceit of pipes which conveygh water, that a pipe of two inches diameter, doth cast out four times as much water as a pipe of one such diameter. 218 7 Heaps of Corn of 10 foot every way, is not as much as one heap of Corn of 20 foot every way. 218 Prob. 88 Of sundry questions in matter of cosmography, 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 compass of the World, how much. 219 How much the starry Firmament, the Sun, and the Moon are distant from the centre of the earth. 220 How long a Millstone 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 Moon. 221 The Sun moves more in one day than the Moon in 20 days. 221 If a millstone from the orb of the Sun should descend a thousand miles in an hour how long it would be before it come to the earth. 221 Of the Sun's 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 years make such a distance as it moves every hour. 221 To find the Circle of the Sun by the fingers. 223 Prob. 93 Of finding the new and full Moon in each month. 224 Prob. 94 To find the latitude of Country's. 225 Prob. 95 Of the Climates of Country's, and how to find them. 225 Prob. 96 Of longitude and latitude of the places of the earth, and of the Stars of the Heavens. 227 To find the Longitude of a Country. 228 Of the Latitude of a Country. 229 To find the Latitude of a Country. 230 To find the distance of places. 230 Of the Longitude, Latitude, Declination, and distance of the stars. 231 How is it that two Horses or other creatures coming 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 Certain fine Observations. In what places of the World is it that the needle hangs in Aequilibrio, and vertical? 233 In what place of the world is it the sun is East or West but twice in the year? 233 In what place of the World is it that the Sun's Longitude from the Equinoctial paints and Altitude, being equal, the Sun is due East or West? That the sun comes twice to one point of the Compass in the forenoon or afternoon. 233 That in some place of the World there are but two kinds of wind all the year. 233 Two ships may be two leagues asunder under the equinoctial, and sailing North at a certain parallel they will be but just half so much. 233 To what inhabitants, and at what time the sun will touch the north-part of the Horizon at midnight. 234 How a man may know in his Navigation when he is under the Equinoctial. 234 At what day in the year the extremity of the styles shadow in a Dial makes a right line. 234 What height the Sun is of, and how far from the Zenith, or Horizon, when a man's 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 Axiom of Parallels. 236 Prob. 100 To find the variation of the Compass by the Sun shining. 237 Prob. 101 To know which way the wind is in ones Chamber without going abroad. 238 Prob. 102 How to draw a parallel spherical line with great ease. 239 Prob. 103 To measure an height only by help of ones Hat. 240 Prob. 104 To take an height with two straws. 240 In Architecture how statues or other things in high buildings shall bear a proportion to the eye below either equal, double, etc. 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 man alone may place a Cannon upon his Carriage. 245 Prob. 109 How to make a Clock with one wheel 246 Of Water-works. Prob. 110 How a child may draw up a Hogshead of water with ease. 247 Prob. 111 Of a Ladder of Cords to carry in ones pocket, by which he may mount a wall or Tower alone. 248 Prob. 112 Of a marvellous Pump which draws 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 fountain very high. 252 Prob. 115 How to empty the water of a Pit by help of a Cistern. 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 perpetual motion. 255 Prob. 119 Of the admirable invention of making the Philosopher's 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 Pyramidal figure that moves upon a Table 258 Prob. 122 How an Anvil may be cleaved by the blow of a Pistol. 258 Prob. 123 How a Capon may be roasted in a man's travels at his sa●●le-bowe. 259 Prob. 124 How a Candle may be made to burn three times longer than usually it doth 259 Prob. 125 How to draw Wine out of water 260 Prob. 126 Of two Marmouzets, the one of which lights a Candle, and the other blows 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 lasts as marble, resisting air and water. 262 Prob. 129 How to melt metal upon a shell with little fire. 263 Prob. 130 Of the hardening of Iron and steel. 263 Prob. 131 To preserve fire as long as you will, imitating the inextinguible fire of the Vestales. 264 FINIS. 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. MATHEMATICAL RECREATION. PROBLEM I. To find 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 add 6, 8, 10, or any Number at pleasure: and let him take the half of the sum, then ask how much it comes to, for than if you take away half the number from it which you willed him at first to add to it, there shall remain the double of the number thought upon. Example The Number thought upon 5 The Quadruple of it 20 Put 8 unto it, makes 28 The half of it is 14 Take away half the number added from it, viz 4, the rest is 10 The double of the number thought upon, viz. 10 Another way to find what Number was thought upon. BId him which thinketh double his Number, and unto that double add 4, and bid him multiply that same product by 5, and unto that product bid him add 12, and multiply that last number by 10 (which is done easily by setting a cipher 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. Example. The number thought upon 7 For which 700 account only but the number of the hundreds viz. 7. so have you the number thought upon. His double 14 For which 700 account only 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 only but the number of the hundreds viz. 7. so have you the number thought upon. Which multiplied by 5 makes 90 For which 700 account only 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 only but the number of the hundreds viz. 7. so have you the number thought upon. This multiplied by 10 which is only by adding a cipher to it, makes 1020 For which 700 account only but the number of the hundreds viz. 7. so have you the number thought upon. From this subtract 320 For which 700 account only but the number of the hundreds viz. 7. so have you the number thought upon. Rest 700 For which 700 account only but the number of the hundreds viz. 7. so have you the number thought upon. To find 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 odd 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 nine there is in the last half, and for every nine, account four in your memory, for that shall show the number thought upon, if both the triples were even: but if it be odd at the first triple, and ev●n at the second, for the one added unto the least half keep one in memory: if the first triple be even, and the second odd, for the one added unto the least half keep two in memory; lastly, if at both times in tripling, the numbers be odd, for the two added unto the least halves, keep three in memory, these cautions observed, and added unto as many fours as the party says there is nine contained in the last half, shall never fail you to declare or discern truly what number was thought upon. Example. The number thought upon 4 or 7 The triple 12 or 21 The half thereof 6 or 10, one put to it makes 11 The triple of the half 18 or 33 The half 9 or 1●, one put to it makes 17 The number of nine in the last half 1 or 1 The first 1. representeth the 4. number thought upon, and the last 1. with the caution makes 7. the other number thought upon. Note. Order your method so that you be not discovered, which to help, you may with dexterity and industry make Additions▪ Subtractions, Multiplications, Divisions, etc. and instead of ask how many nine there is, you may ask how many eights ten, etc. there is, or subtract 8.10. etc. from the Number which remains, for to find 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 Euclid: upon which 2. Book & 4. proposition this may be 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 add 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. The square of 3 makes 9 the sum of these three numbers 25, the squa●e Root of which is 5, the number thought upon The square of 2 makes 4 the sum of these three numbers 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 numbers 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, add the square of half the difference of the parts, than the Root Quadrat of the aggregate is half the number conceived. EXAMINATION. THe Problems which concern Arithmetic, we examine not, for these are easy to any one which hath read the grounds and principles of Arithmetic, but we especially touch upon that, which tends to the speculations of Physic, 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, It is pleasant to see the beautiful and goodly representation of the heavens intermixed with clouds in the Horizon, upon a woody situation, the motion of Birds in the Air, 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 beautiful paint will so naturally represent itself in such a lively Perspective, that hardly the most accurate Painter can represent the like. But here note, that they may be represented right two manner of ways; first, with a concave glass: secondly, by help of another convex glass, disposed or placed between the paper and the other Glass: as may be seen here by the figure. 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 topographical cards, etc. 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 crystalline humour at the small of the Glass, and the bottom of the eye, for the Wall or leaf of paper. EXAMINATION. THe species being pressed together or contracted doth not perform it upon a wall, for the species of any thing doth represent itself not only in one hole of a window, but in infinite holes, even unto the whole Sphere, or at least unto a Hemisphere (intellectual 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 form. 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 near to the Axis, for these that are near are better proportioned then these which are farther off. But to have them more lively and true, according to the imaginary conical section, let the hole be no greater than a pins head made upon a piece of thin brass, 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 plain will make two like Cones, and will lively represent the things without in a perfect inversed perspective, which drawn by the Pencil of some artificial 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 glass reflect upon a paper or some white thing. PROBLEM III. To tell how much weighs 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 Maximilian the Emperor did propose upon a day this Question, and promised to give the resolution; notwithstanding scaliger 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 counterpoise it; then charging or laying more weight 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 weigh for as Aristotle saith, The motion that is made in striking adds great weight 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 pound weight upon a stone, nay, though it were exceedingly pressed down by way of a Vice, by Levers, or other Mechanic Engine, it would be nothing to the rigour and violence of a blow. Is it not evident that the edge of a knife laid upon butter, and a hatchet upon a leaf 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 quickness of the motion, which breaks and enters without resistance, if it be extreme quick, as experience shows us in the blows of Arrows, of Cannons, Thunderbolts, and such like. EXAMINATION. THis Problem was extracted from Scaliger, who had it from Aristotle, but somewhat 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 several 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 near at hand, is not so forceable as if it fell farther 〈◊〉 when a bullet flying out of a piece and striking the mark near at hand 〈◊〉 not make such an effect as striking 〈…〉 that Scaliger and 〈…〉 this subject▪ would not be less troubled to resolve this, than they have been in that. PROBLEM IU. How to break a staff 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. In like manner may you do upon two Reeds, held with your hands in the air without breaking them▪ thence Kitchen boys 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. Now in this act, the two ends of the staff in breaking slides away from the Glasses, upon which they were placed; hence it cometh that the Glasses are no wise endangered, no more than the knee upon which a staff is broken, forasmuch as in breaking it presseth not: as Aristotle in his Mechanic Questions observeth. EXAMINATION. IT were necessary here to note, that this thing may be experimented, first, without Glasses, in placing a small slender staff upon two props, and then making trial upon it, by which you may see how the Staff will either break, bow, or depart from his props, and that either directly or obliquely: But why by this violence, that one Staff 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 fair Geographical Card in a Garden Plot, fit for a Prince, or great personage. IT is usual amongst great men to have fair Geographical 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 general knowledge, not only of their own Kingdom's form, situation, longitude, latitude, etc. but of all other places in the whole Universe, with their magnitudes, positions, Climates, 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 contentments: 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 general description might particularly and artificially be prefigured his whole Kingdoms and Dominions, the Mountains and hills being raised like small hillocks with turfs of earth, the valleys somewhat 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 proportional to the platform, the Forests 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 artificial Fountain made in the Garden to pass through channels; then may there be composed walks of pleasure, ascents, places of repose, adorned with all variety of delightful herbs and flowers, both to please the eye or other senses. A Garden thus accommodated shall far exceed that of my Lord of Verulam's 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 air, without being supported by any thing but by themselves. TAke the first staff AB, raise up in the air the end B, and upon him croswise place the staff CB, than lastly, in Triangle wise place the third staff OF▪ in such manner that it may be under AB, and yet upon CD. I say that these staves so disposed cannot fall, and the space CBE is made the stronger, by how much the more it is pressed down, if the staves break not, or sever themselves from the triangular form: so that always the Centre of gravity be in the Centre of the Triangle: for AB is supported by OF, and OF 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 remain these which you would have. ORdinarily the proposition is delivered in this wise: 15 Christians and 15 Turks being at Sea in one Ship, an extreme tempest being risen, the Pilot of the Ship saith, it is necessary to cast over board half of the number of Persons to disburden the Ship, and 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 always be cast into the Sea, until there were half thrown over board; Now the Pilot being a Christian endeavoured to save the Christians, how ought he therefore to dispose the Christians, that the lot might fall always upon the Turks, 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 four, and u five: o the first vowel in the first word showeth that there must be placed 4. Christians; the next vowel u, signifieth that next unto the 4. Christian's must be placed 5 Turks, and so to place both Christians and Turks according to the quantity and value of the vowels in the words of the verse, until 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 Problem is not to be limited, seeing it extends to any number and order whatsoever, and may many ways be useful for Captains, Magistrates, or others which have divers persons to punish, and would chastise chiefly the unruliest of them, in taking the 10, 20, or 100 person, etc. as we read was commonly practised amongst the ancient Romans: herefore to apply a general rule in counting the third, 4, 9, 10, etc. amongst 30, 40, 50, persons, and more or less; 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 always 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 always marking the eighth, until you have marked 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 belief 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 Vespasian, Josephus being Governor of that Town, accompanied with a Troop of forty Soldiers, hid themselves in a Cave, in which they resolved rather to famish than to fall into the hands of Vespasian: and with a bloody resolution in that great distress would have butchered one another for sustenance, had not Josephus persuaded them to die by lot and order, upon which it should fall: Now seeing that Josephus did save himself by this Art, it is thought that his industry was exercised by the help of this Problem, so that of the 40 persons which he had, the third was always killed. Now by putting himself in the 16 or 31 place he was saved, and one with him which he might kill, or easily persuade to yield unto the Romans. PROBLEM. VIII. Three things, and three persons proposed, to find 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 yourself 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 yourself once known by the aforesaid Vowels, which signify for the first vowel 1, for the second vowel 2, for the third vowel 3. Now if the said three persons should by the mutual consent of each other privately change their names, it is most facile by the course and excellency of numbers, distinctly to declare each one's name so interchanged, or if three persons in private, the one should take a Ring, the other a piece of Gold, and the third should take a piece of Silver; it is easy to find 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 conceal 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 vowel; to the second 2. Counters, which represent e, the second vowel; and to the third 3. Counters, which stand for ay, the third vowel: then leaving the other Counters upon the Table, retire apart, and bid him which hath the Ring, take as many Counters as you gave him, and he that hath the Gold, for every one that you gave him, let him take 2, and he that hath the Silver for every one that you gave him, let him take 4. this being done, consider to whom you gave one Counter, to whom two, and to whom three; and mark what number of Counters you had at the first, for there are necessarily but 24. as was said before, the surpluse you may privately reject. And then there will be left either 1.2.3.5.6 or 7. and no other number can remain, which if there be, than they have failed in taking according to the directions delivered: but if either of these numbers do remain, the resolution will be discovered by one of these 6 words following, which ought to be had in memory, viz. Salve, certa, anima, semita, vita, quies·s 1. 2. 3. 5. 6. 7. As suppose 5. did remain, the word belonging unto it is semita, the vowels in the first two syllables are e and i, which showeth according to the former directions, that to whom you gave 2 Counters, he hath the Ring (seeing it is the second vowel represented by two as before) and to whom you gave the 3. Counters, he hath the Gold, for that i represents the third vowel, or 3. in the former direction, and to whom you gave one Counter, he hath the Silver, and so of the rest: the variety of changes, in which exercise, is laid open in the Table following. rest men hid rest men hid 1 1 a 5 1 2 e 2 3 i 3 2 1 e 6 1 2 a 2 3 i 3 3 1 a 7 1 2 i 2 3 e 3 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. without, 2. within and 5. without, etc. the first 1.2.3. which are within with the numbers over them, belongs to the upper semicircle; the other numbers both within and without, to the under semicircle; 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 shows the first, the next is the second, etc. as if 5 remains, it shows to whom he gave 2, he hath the Ring; to whom you gave ●, he hath the Gold, etc. 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, etc. PROBLEM IX. How to part a Vessel which is full of wine containing eight pints into two equal parts, by two other vessels which contain as much as the greater vessel; 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, pour out A into B until it be full, so there will be in B 5 pints, and in A but 3 pints: then pour out of B into C until it be full: so in C shall be 3 pints, in B 2 pints, and in A 3 pints, then pour the wine which is in C into A, so in A will be 6 pints, in B 2 pints, and in C nothing: then pour out the wine which is in B into the pot C, so in C there is now 2 pints, in B nothing, and in A 6 pints,. Lastly, pour out of A into B until 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 pour in unto the pot C until it be full, there will remain in B 4 pints, and if that which is in C, viz. 3 pints be poured into the vessel A, which before had 1 pint, there shall be in the vessel A, but half of its liquor that was in it at the first, viz. 4 pints as was required. Otherwise pour out of A into C until it be full, which pour into B, then pour out of A into C again until it be full, so there is now in A only 2 pints, in B 3, and in C 3, then pour from C into B until it be full, so in C there is now but 1 pint, 5 in B, and 2 in A: pour all that is in B into A, then pour the wine which is in C into B, so there is in C nothing, in B only 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 half 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 vessel which contains 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 two knives or such like of equal poise, at the end of the stick, leaning out somewhat 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, will there rest without 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 gravity is supported by the top of the finger: and seeing that each part by the knives is counterpoised, it cannot fall sidewise, therefore it can fall no wise. In like manner may great pieces of Timber, as Joists, &c be supported, if unto one of the ends be applied convenient proportional counterpoises, yea a Lance or Pike, may stand perpendicular in the Air upon the top of ones finger: or placed in the midst of a Court by help of his Centre of gravity. EXAMINATION. THis Proposition seems doubtful; for to imagine absolutely, that a Pike, or such like, armed with two Knives, or other things, shall stand upright in the Air, 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 Problem 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 superior parts (by reason of the Centrical motion) is made more ponderous, and therefore less in rest. To place therefore this prop really, let the two Knives, or that which is for counterpoise, be longer always then the staff, and so it will hang together as one body: and it will appear admirable if you place the Centre of gravity, near the side of the top of the finger or point; for it will then hang horizontal, and seem to hang only by a touch, yet more strange, if you turn the point or top of the finger upside down. PROBLEM XI. How a millstone 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 centre of gravity 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 counterpoised 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 sink 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 uniform in matter and figure, and that it be erected perpendicular to the Horizon, and lastly, that the Centre 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 Balance, and holding a Needle in your hand, and place the back of that, Knife which lies crosswise to the other two, upon the point of the Needle: as the figure here showeth you; for then in blowing softly upon them, they will easily turn and move upon the point of the Needle with ●ou falling. PROBLEM XIII. To find the weight of Smoke, which is exhaled of any combustible body whatsoever. LEt it be supposed that a great heap of Faggots, or a load of straw weighing 500 pound should be fired, it is evident that this gross substance will be all inverted into smoke and ashes: now it seems that the smoke weighs nothing; seeing it is of a thin substance now dilated in the Air, 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 the rest of the matter is not lost, but is exhaled into smoke, it must necessarily be, that the rest of the weight (to wit) 450 pound, must be the weight of the smoke required. EXAMINATION. 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 smoke therefore certainly is light being in its true medium (the air,) if it should change his medium, then would we change our discourse. PROBLEM XVI. Many things being disposed circular, (or otherwise) to find which of them, any one thinks upon▪ SUppose that having ranked 10 things, as ABCDEFGHIK, Circular (as the figure showeth) and that one had touched or thought upon G, which is the 7: ask the party at what 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 he thought upon, so at E he shall account to himself 7, at D account 8, at C account 9, etc. So the account of 15 will 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, etc. and it will be the same. There are some that use this play at Cards, turned upside down, as the ten simple Cards, with the King and Queen, the King standing for 12, and the Queen for 11, and so knowing the situation of the Cards: and thinking a certain hour 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 show the hour 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 subtlety of this, rests in the artificial disposer of four 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 show how a Ponderosity, or heavy thing, may be supported upon the end of a staff (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 staff or stick which may not roll upon the Table as EC, and place the handle of the Pale upon the staff; then place another staff, or stick, under the staff CE, which may reach from the bottom of the Pale unto the former staff CE, perpendicular wise: which suppose FG, then shall the Pale of water hang without falling, for if it fall it must fall perpendicularly, or plumb wise: and that cannot be seeing the staff CE supports it, it being parallel to the Horizon and sustained by the Table, and it is a thing admirable that if the staff CE were alone from the table, and that end of the staff which is upon the Table were greater and heavier than the other: it would be constrained to hang in that nature. EXAMINATION. NOw without some experience of this Problem, 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 sense without seeing it done. At the first, this proposition seems to be absurd, and impossible. Notwithstanding, being supported with two sticks, as the figure declareth, it is made facile: for the horizontal 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 deceitful Bowl to play withal. MAke a hole in one side of the Bowl, 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 Bowl is cast directly to the play, how it will turn away side-wise: for that on that part of the Bowl 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 Rind. Pass a needle and thread under the kind of the Apple, and then round it with divers turnings, until you come to the place where you began: then draw out the thread 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 always whole. PROBLEM XIX. To find a number thought upon without ask of any question, certain operations being done. BId him add to the number thought (as admit 15) half of it, if it may be, if not the greatest half that exceeds the other but by an unite, which is 8; and it makes 23. Secondly, unto this 23. add the half of it if it may be, if not, the greatest half, viz. 12. makes 35. in the mean 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 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 sum, viz. 35. to subtract the double of the number thought, viz. 30. rest 5. will him to take the half of that if he can, if not, reject 1. and then take the half of the rest, which keep in your memory: then will him to take the half again if he can, if not, take one from it, which reserve in your memory, and so perpetually halveing until 1. remain: for then mark how many halves there were taken, for the first half account 2, for the second 4, for the third 8, etc. and add unto those numbers the ones which you reserved in memory, so there being 5 remaining in this proposition, there were 2 halfing: for which last! account 4, but because it could not exactly be halved without rejecting of 1. I add the 1 therefore to this 4, maketh 5, which half or sum always multiplied by 4, makes 20. from which subtract the first 3 and 2, because the half could not be formerly added, leaves 15, the number thought upon. Other Examples. The number thought upon. The number thought 12 The half of it 6 The sum 18 The half of it 9 The sum of it 27 The double of the number, 24 Which taken away, rests 3 The half of it 1 For which account 2 and 1 put to it because the 3 could not be halfed, makes 3 this multiplied by 4 makes 12 The number thought 79 The greatest half 40 3 The sum 119 The greatest half of which is 60 2 The sum of it is 179 The double of 79 is 158 Which taken from it, rests 21 The lesser half 10. which halve: The half of this is 5 which makes The half of this is 2 which is 10 The half of this is 1, with 10 and 11 is 21. this 21 which is the double of the last half 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 uniform, & an inflexible body, to pass through two small holes of divers forms, as one being circular, and the other square, Quadrangular, and Triangular-wise, yet so that the holes shall be exactly filled. THis Problem is extracted from Geometrical observations, and seems at the first somewhat obscure, yet that which may be extracted in this nature, will appear more difficult and admirable. Now in all Geometrical practices, the lesser or easier Problems do always make way to facilitate the greater: and the aforesaid Problem is thus resolved. Take a Cone or round Pyramid, and make a Circular hole in some board, or other hard material, which may be equal to the bases of the Cone, and also a Triangular hole, one of whose sides may be equal to the Diameter of the circle, and the other two sides equal to the length of the Cone: Now it is most evident, that this conical or Pyramidal 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 Pyramids conjoined, then if a Circular hole be made, whose Diameter is equal to the Diameter of the Cones conjoined, and a Quadrangular hole, whose sloping sides be equal to the length of each side of the Pyramid, and the breadth of the hol equal to the Diameter of the Circle, this conjoined Pyramid shall exactly fill both the Circular hole, and also the Quadrangle hole. PROBLEM. XXI. How with one uniform body or such like to fill three several holes: of which the one is round, the other a just square, and the third an oval form? THis Proposition seems more subtle than the former, yet it may be practised two ways: for the first, take a cylindrical body as great or little as you please: Now it is evident that it will fill a Circular hole, which is made equal to the basis of it, if it be placed down right, and will also fill a long square; whose sides are equal unto the Diameter and length of the Cylinder, and according to Pergeus, Archimedes, etc. in their cylindrical demonstrations, a true Oval is made when a Cylinder is cut slopewise, therefore if the oval have breadth equal unto the Diameter of the Basis of the Cylinder, & any length whatsoever: the Cylinder being put into his own Oval 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 equal to the Diameter of the Circle: and lastly, make a hole Ovalwise, whose breadth may be equal unto the diagonal of the Square; then let a cylindrical body be made, whose Basis may be equal unto the Circle, and the length equal also to the same: Now being placed down right shall fall in the Circle, and flat-wise will fit the Square hole, and being placed sloping-wise will fill the Oval. EXAMINATION. YOu may note upon the last two Problems farther, that if a Cone be cut Ecliptick-wise, it may pass through an Issoc●●● Triangle through many Scalen Triangles, and through an Ellipsis; and if there be a Cone cut scalen-wise, it will pass through all the former, only for the Ellipsis place a Circle: and further, if a solid column be cut Ecliptick-wise it may fill a Circle, a Square, divers Parallelogrammes, and divers Ellipses, which have different Diameters. PROBLEM XXII. To find a number thought upon ●fter another manner, than 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 again 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 seems 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 add the number thought upon to it, and ask him how much it makes: then subtract your Quotient from it, there will remain 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 multiplied 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, maketh 8, which 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 Quotient ● be added unto the number thought, viz. ●, it makes 8, which being known, the number thought upon is also known. PROBLEM XXIII. To find out many numbers that sundry persons, or one man hath thought upon. IF the multitude of numbers thought upon be odd, as three numbers, five numbers, seven, etc. as for example, let 5 numbers thought upon be these ● 2, 3, 4, 5, 6. bid him declare the sum of the first and second, which will be 5, the second and third, which makes 7, the third and fourth, which makes 9, the fourth and fifth, which makes 11, and so always adding the two next together, ask him how much the first and last makes together, which is 8. then take these sums, and place them in order, and add all these together, which were in the odd places: that is the first, third, and fifth, viz. 5, 9, ●, makes 22. In like manner add all these numbets together, which are in the even places, that is in the second and fourth places, viz. 7 and 1● makes 18, subtract this from the former 22, then there will remain the double of the first number thought upon, viz. 4. which known, the rest is easily known: seeing you know the sum 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 party to declare the sum of each two, by antecedent and consequent, and also the sum of the second and last, which will be 5, 7, 9, 11, 13, 10, then add the odd places together, except the first, that is 9, and 13, makes 22, add also the even places together, that is 7, 11, 10, which makes 28, subtract the one from the other, there shall remain the double of the second number thought upon, which known all the rest are known. 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 facile, 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 cosmography, & first dialogue, reported of one that thought he was led by one of the Muses to hell, where he saw Lucifer sitting 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 half on this side of the Centre of the earth, and the other half 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 brag that he may in a right line pass from place to place above 3000 miles. THe opening of this is easy, forasmuch as he that possesseth a Rod of ground possesseth 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 heritage's & possession are as so many Pyramids, whose summets or points meet in the centre of the earth, and the basis of them are nothing else but each man's possession, field, or visible quantity; and therefore if there were made or imagined so to be made, a descent to go to the bottom 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 party be under either of the Poles, for if he be under the North-pole, then looking any way he looketh South, because all the Meridian's concur 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 remains after certain operations being ended, without ask 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 add 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 subtract from this Quotient the number thought upon: In the same time divide apart the number which was added by that which multiplied, so than your Quotient shall be equal to his remainder, wherefore without ask him any thing, you shall tell him what did remain, 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 add 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 several 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 shown in this 〈…〉 the mean time to produce always 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 subtly, and by way of divination, or artificially in which hand the Gold or Silver is; attribute t● the Gold, or suppose it have a certain 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 Crowns, or Shillings, and the Silver at ● Crowns, or 3 Shillings, or any other number, so that one be odd▪ 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 add these two products together, and ask him if it be even or odd; if it be even, than the Gold is in the right hand; if odd, the Gold is in the left hand. PROBLEM. XXX. Two numbers being proposed unto two several 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 odd, as 10. and 9 and let the one of them take one of the numbers, and the other party take the other number, which they place privately to themselves: how artificially, according to the congruity, and excellency of numbers, to find which of them did take 10. and which 9 without ask any qustion: and this seems most subtle, yet delivered howsoever differing little from the former, and is thus performed: Take privately to yourself also two numbers, the one even, and the other odd, 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: add your two products together, & mark if it be even or odd, then bid the two parties put their numbers together, and bid them take the half 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 odd, and such would be the aggregate or sum of the double of Peter's number and John's number, if Peter had taken 10. if otherwise, than they might have taken half, 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. odd: in like manner the double of your number conceived in mind, viz. 4. makes 8. and the triple of the 3. the other number, makes 9 which set together makes 17. odd. Now you cannot take the half 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 half of it may be taken: therefore in such case Peter the took less number: and John the greater, and this being done cleanly carries much grace with it. PROBLEM. XXXI. How to describe a Circle that shall touch 3: Points placed howsoever upon a plain, if they be not in a right line. LEt the three points be A.B.C. put one foot of the Compass upon A. and describe an Arch of a Circle at pleasure: and placed at B. cross that Arch in the two points E. and F. and placed in C. cross the Arch in G. and H. then lay a ruler upon G.H. and draw a line, and 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 pass by the said three points A.B.C. or it may be inverted, having a Circle drawn; to find 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 facile to every practitioner in the principles of Geometry. PROBLEM. XXXII. How to change a Circle into a square form? M●ke a Circle upon pasteboard or other material, 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 always be at the Angle of the square, and so the four quarters of the Circle being placed so, it will make a perfect square, whose side A.A. is equal to the Diameter B.D. Now here is to be noted that the square is greater than the Circle by the vacuity in the middle, viz. M. PROBLEM. XXXIII. With one and the same compasses, and at one and the same extent, or opening, how to describe many Circles concentrical, that is, greater or lesser one than 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 industry of an ingenious Geometrician, who makes it possible, and that most facile, sundry ways; for in the first place if you make a Circle upon a fine plain, and upon the Centre of that Circle, a small peg of wood be placed, to be raised up and put down 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 concentrical, that is, one greater or lesser than another; for the higher the Centre is lifted up, the lesser the Circle will be. Secondly, the compass being at that extent upon a Gibus body, a Circle may be described, which will be less than the former, upon a plain, and more artificially upon a Globe, or round bowl: and this again is most obvious upon a round Pyramid, placing the Compasses upon the top of it, which will be far less than any of the former; and this is demonstrated by the 20. Prop. of the first of Euclids, for the Diameter ●. D. is less than the line AD.A.E. taken together, and the lines AD.AE. being equal to the Diameter BC. because of the same distance or extent of opening the compasses, it follows that the Diameter E.D. and all his Circles together is much less than the Diameter, and the Circle BC. which was to be performed. PROBLEM XXXIV. Any numbers under 10. being thought upon, to find what numbers they were. LEt the first number be doubled, and unto it add 5. and multiply that sum by 5. and unto it add 10. and unto this product add the next number thought upon; multiply this same again by 10. and add unto it the next number, and so proceed: now if he declare the last sum; mark if he thought but upon one figure, for then subtract only 35. from it, and the first figure in the place of ten is the number thought upon: if he thought upon two figures, then subtract also the said ●5. from his last sum, and the two figures which remain 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, etc. so if one thought upon these numbers 5.7.9.6. double the first, makes 1●. to which add 5. makes 15. this multiplied by 5. makes 75. to which add 1●. makes 85. to this add the next number, viz. 7. makes 92. this multiplied by 10. makes 920. to which add the next number, viz. 9 makes 929. which multiplied by 10. makes 9290. to which add 6. makes 9296. from which subtract 3500. resteth 5796. the four numbers thought upon. Now because the two last figures are like the two numbers thought upon: to conceal this, bid him take the half 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 find out in which hand: upon which finger, & joint it is; this will cause great astonishment to ignorant spirits, which will make them believe that he that doth it works by Magic, or Witchcraft: But in effect it is nothing else but a nimble act of Arithmetic, founded upon the precedent Problem: for first it is supposed that the persons stand or sit in order, that one is first, the next second, etc. 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 joints, the one is as 1. the other is as 2. the other as 3. etc. from whence it appears 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 joint, and I would divine and find 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, 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 add the number of the fingers upon which the Ring is, viz. 5. makes 775. this multiplied by 10. makes 7750. to which add the number for the joint upon which the Ring is, viz the third joint, makes 7●53. to which cause him to add 14. or some other number, to conceal it the better: and it makes 7767. which being declared unto you, subtract 3514▪ and there will remain 4.2.5.3. which figures in order declares the whole 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 joint of that finger. PROBLEM. XXXVI. The Play of 34. or more Dice. THat which is said of the two precedent Problems may be applied to this of Dice (and many other particular things) to find what number appeareth upon each Dice being cast by some one, for the points that are upon any side of a Dice are always less 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 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 add 5, then multiply that by 5. and to it add 10, and to the sum bid him add the number of the second Dice: and multiply that by 10: lastly, to this bid him add the number of the last Dice, and then let him declare the whole number: then if from it you subtract ●50. there will remain the number of the three Dice thrown. PROBLEM. XXXVII. How to make water in a Glass seem to boil and sparkle? TAke a Glass near full of water or other liquor; and setting one hand upon the foot of it, to hold it fast: turn slightly one of the fingers of your other hand upon the brim, or edge of the Glass; having before privately wet your finger: and so passing softly on with your finger in pressing a little: for than first, the Glass will begin to make a noise: secondly, the parts of the Glass will sensibly appear to tremble, with notable rarefaction and condensation: thirdly, the water will shake, seem to boil: fourthly, it will cast itself out of the Glass, and leap out by small drops, with great astonishment to the standers by; if they be ignorant of the cause of it, which is only in the Rarefaction of the parts of the Glass, occasioned by the motion and pressure of the finger. EXAMINATION. THe cause of this, is not in the rarefaction of the parts of the Glass, but it is rather in the quick local motion of the finger, for reason showeth us that by how much a Body draweth nearer to a quality, the less is it subject or capable of another which is contrary unto it? now condensation, and rarefaction are contrary qualities, and in this Problem there are three bodies considered, the Glass, the Water, and the Air, now it is evident that the Glass being the most solid, and impenetrable Body, is less subject and capable of rarefaction than the water, the water is less subject than the Air, and if there be any rarefaction, it is rather considerable in the Air then in the Water, which is inscribed by the Glass, and above the Water, and rather in the Water then in the Glass: the agitation, or the trembling of the parts of the Glass to the sense appears not: for it is a continued body; if in part, why then not in the whole? and that the Water turns in the Glass, this appears not, but only the upper contiguous parts of the Water: that at the bottom being less subject to this agitation, and it is most certain that by how much quicker the Circular motion of the finger upon the edge of the Glass is, by so much the more shall the Air be agitated, and so the water shall receive some apparent affection more or less from it, according to that motion: as we see from the quickness of wind 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 vessel which holds wine or water, being cast in●o it at a certain height, but being filled higher, it will run out of its own accord. LEt there be a vessel 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 vessel as at ●▪ are open; let the end ● be somewhat lower than the brim of the Glass: 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 Air enter in thereby, and this Pipe at the bottom may have a small hole to give passage unto the water; then pour in water or wine, and as long as it mounts not above E, it is safe; but if you pour in the water so that it mount above it, farewell all: for it will not cease until it be all gone out; the same may be done in disposing any crooked Pipe in a vessel in the manner of a Faucet or funnel, 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 vessel will be empty incontinent, and the subtlety of this will seem more admirable, if you conceal the Pipe by a Bird, Serpent, or such like, in the middle of the Glass. 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 until the Pipe be full, then will it run of itself, and it is a fine secret in nature to see, that if the end of the Pipe which is out of the water, be lower than the water, it will run out without ceasing: but if the mouth of the Pipe be higher than the water or level with it, it will not run, 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 always exactly level▪ EXAMINATION. HEre is to be noted, that if the face of the water without be in one and the same plain, with that which is within, though the outtermost Pipe be ten times greater than that which is within; the water naturally will not run, but if the plain of the water without be any part lower than that which is within, it will freely run: 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 Glass very pleasant. SOmetimes there are Glasses which are made of a double fashion, as if one Glass were within another, so that they seem but one, but there is a little space between them. No● pour Wine or other liquor between the two edges by help of a tunnel, into a little hole left to this end, so will there appear two fine delusions or fallacies; for though there be not a drop of Wine within the hollow of the Glass, it will seem to those which behold it that it is an ordinary Glass full of Wine, and that especially to those which are side-wise of it, and if any one move it, it will much confirm it, because of the motion of the Wine; but that which will give most delight, is that, if any one shall take the Glass, and putting it to his mouth shall think to drink the Wine, instead of which he shall sup the Air, and so will cause laughter to those that stand by, who being deceived, will hold the Glass to the light, & thereby considering that the rays or beams of the light are not reflected to the eye, as they would be if there were a liquid substance in the Glass, hence they have an assured proof to conclude, that the hollow of the Glass is totally empty. PROBLEM. XL. If any one should hold in each hand, as many pieces of money as in the other, how to find how much there is? BId him that holds the money that he put out of one hand into the other what 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 take out of it as many as remain in the other hand, and put it into that hand: for then be assured that in the hand which was put the first taking away: there will be found just the double of the number taken away at the first. Example, admit there were in each hand 12 Shillings or Counters, and that out of the right hand you bid him take 7. and put it into the left: and then put into the right hand from the left as many as doth remain 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 easy to find 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 add the points that were upmost together: then putting one of the Dice apart, unto the former sum add the points which are under the other two, then bid him throw these two Dice, and mark how many points a pair are upwards, which add unto the former sum: then put one of these Dice away not changing the side, mark the points which are under the other Dice, and add it to the former sum: lastly, throw that one Dice, and whatsoever appears upward add it unto the former sum; and let the Dice remain thus: this done, coming to the Table, note what points do appear upward upon the three Dice, which add privately together, and unto it add ●1 or 3 times 7: so this Addition or sum shall be equal to the sum which the party privately made of all the operations which he formerly made. As if he should throw three Dice, and there should appear upward 5, 3, 2. the sum of them is 10. and setting one of them apart, (as 5.) unto 10, add 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 appear 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 add 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 appear 2, which I add 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 add secretly 21, (as before was said) so have you 32, the same number which 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 add the points opposite of the Dice; for upon which depends the whole demonstration or secret of the play; for always that which is above and underneath makes 7. but if it make another number, then must you add as often that number. PROBLEM. XLII. Two metals, as Gold and Silver, or of other kin●● weighing alike, being privately placed into two like Boxes, to find which of them the Gold or Silver is in. But because that this experiment in water hath divers accidents, and therefore subject to a caution; and namely, because the matter of the chest, mettle or other things may hinder. Behold here a more subtle and certain invention to find and discover it out without weighing it in the water▪ Now experience and reason showeth us that two like bodies or magnitudes of equal weight, and of divers metals, are not of equal quantity: and seeing that Gold is the heaviest of all metals, it will occupy less room or place; from which will follow that the like weight of Lead in the same form, will occupy or take up more room or place. Now let there be therefore presented two Globes or Chests of wood or other matter alike, & equal 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 it is supposed that the wooden Globes or Chests are of equal weight, form, and magnitude: and to discover in which the Gold or Lead is in, take a broad pair 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 EKE, at the end of which hang a weight G, so that the other end may be counterpoised, and hang in aequilibrio: and do the like to the other Chest or Globe. Now 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 towards the middle, that so the points of the Compass may be against some of the mettle which is enclosed; or just against the extremity of the Gold as in D, and suppose it hang thus in aequilibrio; it is certain that in the other Coffer is the Lead; for the points of the Compasses being advanced as much as before, as at F, which takes up a part of the Lead, (because it occupies a greater place than the Gold) therefore that shall help the weight G. to weigh, and so will not hang in aequilibrio, except G be placed near to F. hence we may conclude, that there is the Lead; and in the other Chest or Globe, there is the Gold. EXAMINATION. IF the two Boxes being of equal magnitude weighed in the air be found to be of equal weight, they shall necessarily take up like place in the water, and therefore weigh also one as much as another: hence there is no possibility to find the inequality of the metals which are enclosed in these Boxes in the water: the intention of Archimedes was not upon contrary metals enclosed 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 divers metals, (as one Gold, and the other Copper) yet of equal weight being put into a box, as BG, to find in which end the Gold or Copper is. THis is discovered by the changing of the places of the two Bowles' or Globes, having the same counterpoise H to be hung at the other side, as in N. and if the Gold which is the lesser Globe, were before the nearest to the handle D●, having now changed his place will be farthest from the handle DE, as in K. 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 always in the middle of the Box, the weight N. must be augmented▪ to keep it in equil●●●● and by this way one may know, that if at the second time, the counterpoise be too light, it is a sign that the Gold is farthest off the handle, as at the first trial it was nearest. PROBLEM. XLIIII. How to represent divers sorts of Rainbows here below? THe Rainbow is a thing admirable in the world, which ravisheth often the eyes and spirits of men in consideration of his rich intermingled colours which are seen under the clouds, seeming as the glistering of the Stars, precious stones, and ornaments of the most beauteous flowers: some part of it as the resplendent stars, or as a Rose, or burning Coal of fire▪ in it one may see Dies of sundry sorts, the Violet, the Blue, the Orange, the Saphir, the Jacinct, and the Emerald colours, as a lively plant placed in a green soil: and as a most rich treasure of nature, it is a high work of the Sun who casteth his rays or beams as a curious Painter draws strokes with his pencil, and placeth his colours in an exquisite situation; and Solomon saith, Eccles. 43. it is a chief and principal work of God. Notwithstanding there is left to industry 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 Oars of a Boat it doth exceeding quickly glide upon the water with a pleasant grace? Aristotle says, that it coloureth the water, and makes a thousand atoms, upon which the beams of the Sun reflecting, make a kind of coloured Rainbow: or may we not see in houses or Gardens of pleasure artificial fountains, which pour forth their droppie streams of water, that being between the Sun and the fountain, there will be presented as a continual Rainbow? But not to go farther, I will show you how you may do it at your door, by a fine and facile experiment. Take water in your mouth, and turn your back to the Sun, 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 atoms vapours in the beams of the Sun to turn into a fair Rainbow, but all the grief is, that it lasteth not, but soon is vanished. But to have one more stable and permanent in his colours: Take a Glass full of water, and expose it to the Sun, so that the rays that pass through strike upon a shadowed place, you will have pleasure to see the fine form of a Rainbow by this reflection. Or take a trigonal Glass or Crystal Glass of divers Angles, and look through it, or let the beams of the Sun pass through it; or with a candle let the appearances be received upon a shadowed place: you will have the same contentment. PROBLEM XLV. How that if all the Powder in the world were in closed within a bowl of paper or glass, and being fired on all parts, it could not break that bowl? IF the bowl and the powder be uniform in all his parts, then by that means the powder would press 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 bowl should be broken in all his parts: for they are infinite. Of like fineness or subtlety may it be that a bowl of Iron falling from a high place upon a plain pavement of thin Glass, it were impossible any wise to break it; if the bowl were perfectly round, and the Glass flat and uniform in all his parts▪ for the bowl would touch the Glass 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 bowl falling upon such a glass will not break it. But this matter is mere Metaphysical, and all the workmen in the world cannot ever with all their industry make a bowl perfectly round, or a Glass uniform. PROBLEM. XLVI. To find a number which being divided by 2, there will remain 1, being divided by 3, there will remain 1; and so likewise being divided by 4, 5, or 6, there would still remain 1; but being didivided by 7, there will remain nothing. IN many Authors of Arithmetic this Problem is thus proposed: A woman carrying Eggs to Market in a basket, met an unruly fellow who broke them: who was by order made to pay for them: and she being demanded what number she had, she could not tell: but she remembered 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 Eggs be discovered? Find a number which may exactly be measured by 7, and being measured by 2, 3, 4, 5, and 6; there will still remain a unite▪ multiply these numbers together, makes 720, to which add 1; so have you the number, viz. 721. in like manner 301 will be measured by 2, 3, 4, 5, 6; so that 1 remains: but being measured by 7, nothing will remain; to which continually add 220, and you have other numbers which will do the same: hence it is doubtful what number she had, therefore not to fail, it must be known whether they did exceed 400, 800, etc. in which it may be conjectured that it could not exceed 4 or 5 hundred, seeing a man or woman could not carry 7 or 8 hundred Eggs, therefore the number was the former ●01. which she had in her Basket: which being counted by 2 and 2, there will remain 1, by 3 and 3, etc. but counted by 7 and 7, there will remain nothing. PROBLEM. XLVII. One had a certain number of crowns, 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 crowns might he have? THis Question hath some affinity to the precedent, and the resolution is almost in the same manner: for here there must be found a number, which multiplied by 7, and then divided by 2, 3, 4, 5, 6; there may always remain a number less by 1 than the Divisor: Now the first number which arrives in this nature is 119, unto which if 420 be added, makes 539, which also will 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 weigh things between 1 and 40, take numbers in triple proportion, so that their sum be equal, 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 balance 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 balance 9 and 1, and in the other balance 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 deceitful balance which being c●●●ty seems i● be just, because it hangs in aequilibrio: notwithstanding putting 12 pound in one balance, and 11 in the other, it will remain in aequilibrio. ARistotle maketh mention of this balance in his mechanic Questions, and saith, that the Merchants of purpose in his time used them to deceive the world: the subtlety or craft of which is thus, that one arm of the balance is longer than another, by the same proportion, that one weight is heavier than another: As if the beam 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 easy to be done: then afterwards put into the balance two unequal weights in such proportion as the parts of the beam have one unto another, which is 12 to 11, but so that the greater be placed in the balance which hangs upon the shorter part of the beam, and the lesser weight in the other balance: it is most certain that the balances will hang in aequilibrio, which will seem most sincere and just; though it be most deceitful, abominable, and false. The reason of this is drawn from the experiments of Archimedes, who shows that two unequal weights will counterpoise one another, when there is like proportion between the parts of the beam (that the handle separates) and the weights themselves: for in one and the same counterpoise, by how much it is farther from the Centre of the handle, by so much it seems heavier, therefore if there be a diversity of distance that the balances hang from the handle, there must necessarily be an ineqality of weight in these balances to make them hang in aequilibrio, and to discover if there be deceit, change the weight into the other balance, for as soon as the greater weight is placed in the balance that hangs on the longer parts of the beam: it will weigh down the other instantly. PROBLEM. L. To heave or lift up a bottle with a straw. TAke a straw that is not bruised, bow it that it make an Angle, and put it into the bottle so that the greatest end be in the neck, than the Reed being put in the bowed part will cast side-wise, and make an Angle as in the figure may be seen: then may you take the end which is out of the bottle in your hand, and heave up the bottle, 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 Sun, Stars, Shadow or Compass, to find out the North or South, or the four Cardinal points of the world, East, West, & c? IT is the opinion of some, that the winds are to be observed in this: if it be hot, the South is found by the winds that blow that way, but this observation is uncertain 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 Sun's motion about the tree: for that the humidity 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 drawn from the widest to the narrowest part of the circles, it shall show the North & South of the world. Another Experiment may be thus: Take a small needle, such as women work with: place it gently down flatwise upon still water, and it will not sink, (which is against the general tenet that Iron will not swim) which needle will by little and little turn 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 find out the Poles of the world or points of North and South in a manner as the magnes doth. EXAMINATION. HEre is observable, that the moisture which aideth to the growth of the tree, is dilated and rarefied by the Meridional heat, and contracted by the Septentrional cold: this rarefaction works upon the part of the humour or moisture that is more thin, which doth easily dissipate and evaporate: which 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 less evaporation and perdition: and so consequently there remains 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 sap and moisture is drawn 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 hardeneth 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 find 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 first take 13, then cause them to put them all together, and declare the sum 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 ask how much one of them hath; (for they will have then all alike,) so half of that number is the number that the third person had at the first: which known all is known. PROBLEM. LIII. How to make a consort of music of many parts with one voice, or one instrument only? THis Problem is resolved, so that a finger or player upon an instrument, be near 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, than a triple, if three times, than an harmony of four 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 means you ●ave a fifth, agreeable consort of music: then 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 music, and so of others, if he will continue his voice with the Echo, and sing alone with two parts. Now experience showeth this to be true, which often comes to pass in many Churches, making one to believe that there are many more parts in the music of a Choir, then in effect truly there are because of the resounding and multiplying of the voice, and redoubling of the Quire. PROBLEM. LIIII. T● make or describe an Oval form, or that which near resembles unto it, at one turning with a pair of common Compasses. THere are many fine ways in Geometrical practices, to make an Oval figure or one near unto it, by several centres: any of which I will not touch upon, but show how it may be done promptly upon one centre only. In which I will say nothing of the Oval form, which appears, when one describeth circles with the points of a common Compasses, somewhat deep upon a skin stretched forth hard: which contracting itself in some parts of the skin maketh an Oval form. But it will more evidently appear upon a Column or Cylinder: if paper be placed upon it, then with a pair of Compasses describe as it were a circle upon it, which paper afterwards being extended, will not be circular but ovall-wise: and a pair of Compasses may be so accommodated, that it may be done also upon a plain thus. As let the length of the Oval be H. K, fasten 2 pins or nails near the end of that line as F. G, and take a thread which is double to the length of G. H, or F. K, then if you take a Compass which may have one foot lower than another, with a spring between his legs: and placing one foot of this Compass in the Centre of the Oval, and guiding the thread by the other foot of the Compasses, and so carrying it about: the spring will help to describe and draw the Oval form. But in stead of the Compasses it may be done with one's hand only, as in the figure may appear. 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. 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 again; or be parted one from another: and so that the ring B, may slide up and down 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 A, 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 ay, therefore bring down B to I, then make the end I pass through the ring B, and also D with his string to pass 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 find out the Centre of it? IT is said that upon a time past, two Mathematicians met, and they would make trial 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 plain Table without other help than the hand, and the mind, is full of admiration; to find the Centre is but to find 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 find the Centre is more difficult, for what attention, vivacity, and subtlety must there be in the spirit, in the eye, in the hand, which will choose the true point amongst a thousand other points? He that makes a circle keeps always the same distance, and is guided by a half distance to finish the rest; but he that must find the Centre, must in the same time take heed to the parts about it, and choose one only point which is equal distant from an infinite of other points which are in the circumference; which is very difficult. Aristotle confirms this amongst his morals, and seems to explain the difficulty which is to be found in the middle of virtue; for it may want a thousand ways, and be far separated from the true Centre of the end of a right mediocrity of a virtuous 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 find the Centre, the round seems to be harder than to find the Centre, because that in finding of it, it is done at once, and hath an equal distance from the whole; But, as before, to draw a round there is a visible point imagined, about which the circle is to be drawn. I esteem that it is as difficult therefore, if not more, to make the circle without a Centre, as to find the middle or Centre of that circle. PROBLEM. LVII. Any one having taken 3 Cards, to find how many points they contain 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, then will him to give you the rest of the Cards, for 4 of them being rejected, the rest show the number of points that his three Cards which he took at the first did contain. 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: last 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 remains 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 less than 52; and that the term be 15, 14, 12, &c, this general rule ensuing may serve: multiply the term by the number of Cards taken at first: to the product add the number of Cards taken, then subtract this sum from the whole number of Cards; the remainder is the number which must be subtracted from the Cards, which remains to make up the game: if there remain nothing after the Subtraction, than the number of Cards remaining doth justly show 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 remain, the sum will be the number of points in the Cards taken, as if the Cards were 7, 10, 5, 8, and the term given were 12; 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 add 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 equal to the points of the four Cards taken, viz. 7, 10, 5, 8. Again, let these five Cards be supposed to be taken, 8, 6, 10, 3, 7; their differences to 15, the terms are 7, 9, 5, 12, 8, which number of Cards taken, there will remain but 6 Cards: then privately multiply 15 by 5, makes 75, to which add 5 makes 80, from this take 52 the number of Cards, rest 28, to which add the remainder of Cards, make 34. the sum with 8, 6, 10, 3, 7. PROBLEM. LVII. Many Cards placed in divers ranks, to find 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 easy to find which of the Cards it is, and it is done thus; ask him in which of the heaps it is, which place in the middle of the other two; then throw down the Cards by 1 and 1 into three several heaps in rank-wise, until all be cast down, then ask him in which of the ranks his Card is, which heap place in the middle of the other two heaps always, and this do four 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 fail. PROBLEM. LVIII. Many Cards being offered to sundry persons, to find which of these Cards any one thinketh upon. ADmit there were 4 persons, then take 4 Cards, and show them to the first, bid him think one of them, and put these 4 away, then take 4 other Cards, and show 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, etc. Then take the 4 Cards of the first person, and dispose them in 4 ranks, 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 show unto eaeh of these parties each of these ranks, and ask him if his Card be in it which he thought, for infallibly that which the first party thought upon will be in the first rank, and at the bottom, the Card of the second person will be in the second rank, the Card of the third thought upon will be in the third rank, and the fourth man's 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 certain 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 Mathematics (which hath furnished us with such admirable helps for seeing) is wanting for that of hearing, it's well known that long trunks or pipes make one hear well far off, and experience shows us that in certain places of the Orcadeses 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 hear him speak at all: And it is a general principle, that pipes do greatly help to strengthen the activity of natural causes: we see that 〈◊〉 contracted in a pipe, burns 4 or 5 foot high, which would scarce heat, being in the open air: the rupture or violence of water issuing out of a fountain, shows us that water being contracted into a pipe, causeth a violence in its passage. The Glasses of Galeileus makes us see how useful pipes or trunks are to make the light and species more visible, and proportionable to our eye. It is said that a Prince of Italy hath a fair hall, in which he can with facility hear distinctly the discourses of those which walk in the adjacent Gardens, which is by certain 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 kind of trunks are used to help the hearing, being made of silver, copper, or other resounding material; 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 ear it may be more uniform and less 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 one's pocket: or being rolled upon the ground will still burn. IT must be observed that the vessel in which the oil is put into, have two pins on the sides of it, one against another, being included within a circle: this circle ought to have two other pins, to enter into another circle of brass, or other solid matter: lastly, this second circle hath two pins, which may hang within some box to contain the whole lamp, in such manner, that there be 6 pins in different position: Now by the aid of these pegs or pins, the lamp that is in the middle will be always well situated according to his Centre of gravity, though it be turned any way: though if you endeavour to turn it upside down, it will lie level▪ which is pleasant and admirable to behold to those which know not the cause: And it is facile from his to make a place to rest quiet in, though there be great agitation in the outward parts. PROBLEM. LXI. Any one having thought a Card amongst many Cards, how artificially to discover it out? TAke any number of Cards as 10, 12, etc. and open some 4 or 5 to the party's sight, and bid him think one of them, but let him note whether it be the first, second, third, etc. then with promptness learn what number of Cards 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, ask him whether his Card were the first, second, etc. then before knowing the number of Cards that were at the bottom, account backwards until 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 mark 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 Problem is easy▪ 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▪ than A said she would not sell her apples so cheap, but would sell them for 3 pence the piece, 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 sums 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 sum of a number, that have equal distance from the half 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 half, the other is less. Secondly, it is difficult to find two numbers whose sum and product is alike, (that is) if the numbers be multiplied one by another, and added together, will be equal, 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 sum and product will be equal one to another. As Clavius shows upon the 36 Pro. of the 9th book of Euclid. Thirdly, the numbers 5 and 6 are called circular numbers, because the circle turns to the point from whence it begins: so these numbers multiplied by themselves, do end always in 5 and 6, as 5 times 5 makes 25, that again by 5 makes 125, so 6 times 6 makes 36, and that by 6 makes 216, etc. Fourthly, the number 6, is the first which Arithmeticians call a perfect number, that is, whose parts are equal unto it, so the 6 part of it is 1, the third part is 2, the half is 3, which are all his parts: now 1, 2, and 3, is equal to 6. It is wonderful 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 privileges carries with it an excellent property: for take what number you will, either in gross or in part, the nine of the whole or in its parts rejected, and taken simply will be the same, as ●7 it makes 3 times 9, so whether the nine be rejected of 27, or of the sum of 2 and 7, it is all one, so if the nine were taken away of 240. it is all one, if the nine were taken away of 2, 4, and 0; for there would remain 6 in either; and so of others. 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, etc. Seventhly, the numbers 220 and 284 being unequal, notwithstanding the parts of the one number do always equalise 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 find in other numbers. I● one be taken from any square number which is odd, the square o● half of it being added to the first square, will make a square number. The square of half 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 odd numbers be continually added from the unity successively, there will be made all square numbers, and if cubick numbers be added successively from the unity, there will be likewise made square numbers. PROBLEM. LXIV. Of an excellent lamp, which serves or furnisheth itself with oil, and burns a long time. I Speak not here of a common lamp which Ca●danus writes upon in his book the subtilita●●, for that's a little vessel in columne-wise, which is full of Oil, and because there is but one little hole at the bottom near the week or match; the oil runs not, for fear that there be emptiness above: when the match is kindled it begins to heat the lamp, and rarefying the oil it issueth by this occasion: and so sends his more airy parts above to avoid vacuity. It is certain that such a lamp the Atheniaus used, which lasted a whole year without being touched: which was placed before the statue of Minerva, for they might put a certain quantity of oil in the lamp CD, and a match to burn without being consumed: such as the naturalists write of, by which the lamp will furnish itself, and so continue in burning: and here may be noted that the oil may be poured in, at the top of th● vessel at a little hole, and then made fast again that the air get not in. PROBLEM. LXV. Of the play at Keyle or nine Pins. YOu will scarce believe that with one bowl and at one blow playing freely, one may strike down all the Keyle at once: yet from Mathematical principles it is easy to be demonstrated, that if the hand of him that plays were so well assured by experience, as reason induceth one thereto; one might at one blow strike down all the Keyle, 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 down the Keyles 3, 6, and 9, and the bowl 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 stricken down at once. PROBLEM. LXIV. Of Spectacles of pleasure. SImple Spectacles of blue, 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 green do somewhat degenerate; instead of showing a lively colour it will represent a pale dead colour, and it is because they are not died green enough, or receive not light enough for green: and colour these images that pass through these Glasses unto the bottom of the eye. EXAMINATION. IT is certain, that not only Glasses died green, but all other Glasses coloured, yield the appearances of objects strong or weak in colour according to the quantity of the dye, more or less, as one being very yellow, another a pale yellow; now all colours are not proper to Glasses to give colour, hence the defect is not that they want faculty to receive light, or resist the penetration of the beams; for in the same Glasses those which are most died, give always the objects more high coloured and obscure, and those which are less died give them more pale and clear: and this is daily made manifest by the painting of Glass, which hinders more the penetration of the light than dying doth, where all the matter by fire is forced into the Glass, leaving it in all parts transparent. Spectacles of Crystal cut with divers Angles diamond-wise do make a marvellous multiplication of the appearances, for looking towards a house it becomes as a Town, a Town becomes like a City, an armed man seems as a whole company caused solely by the diversity of refractions, for as many plains as there are on the outside of the spectacle, so many times will the object be multiplied in the appearance, because of divers Images cast into the eye. These are pleasurable spectacles for avaricious persons that love Gold and silver, for one piece will seem many, or one heap of money will seem as a treasury: but all the mischief is, he will not have his end in the enjoying of it, for endeavouring to take it, it will appear but a deceitful 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 fail often in touching that which you see. Again, there are Spectacles made which do diminish the thing seen very much, and bring it to a fair perspective form, especially if one look upon a fair Garden plat, a greater walk, a stately building, or great Court, the industry of an exquisite Painter cannot come near to express the lively form of it as this Glass 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 visual 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 pass to and fro in the streets, without being seen of any, for their property is to raise up the objects that it looks upon. Now I would not pass this Problem without saying something of Galileus admirable Glass, 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 Sun, which darknet and diminisheth the splendour of that beautiful and shining Luminary: secondly, it shows the new Planets that accompany Saturn and Jupiter: thirdly, in Venus is seen the new, full, and quartile increase; as in the Moon by her separation from the Sun: fourthly, the artificial structure of this instrument helpeth us to see an innumerable number of stars, which otherwise are obscured, by reason of the natural weakness of our sight, yea the stars in via lactea are seen most apparently; where there seem no stars to be, this Instrument makes apparently 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 splendour of some is as the Sun in his most glorious beauty. This Glass hath also a most excellent use in observing the body of the Moon in time of Eclipses, for it augments it manifold, and most manifestly shows the true form of the cloudy substance in the Sun; and by it is seen when the shadow of the earth begins to eclipse the Moon, & when totally she is over shadowed: besides the celestial uses which are made of this Glass, it hath another noble property; it far exceedeth the ordinary perspective Glasses, which are used to see things remote upon the earth, for as this Glass reacheth up to the heavens and excelleth them there in his performance, so on the earth it claimeth preeminency, for the objects which are farthest remote, and most obscure, are seen plainer than those which are near at hand, scorning as it were all small and trivial services, as leaving them to an inferior help: great use may be made of this Glass in discovering of Ships, Armies, etc. Now the apparel or parts of this instrument or Glass, is very mean or simple, which makes it the more admirable (seeing it performs such great service) having but a convex Glass thickest in the middle, to unite and amass the rays, and make the object the greater: to the augmenting the visual Angle, as also a pipe or trunk to amass the Species, and hinder the greatness of the light which is about it: (to see well, the object must be well enlightened, and the eye in obscurity;) then there is adjoined unto it a Glass of a short sight to distinguish the rays, which the other would make more confused if alone. As for the proportion of those Glasses to the Trunk, though there be certain 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 aught to be tried, seeing that exact proportion, in Geometrical calculation cannot serve for diversity of sights in the observation. PROBLEM. LXVII. Of the Adamant or Magnes, and the needles touched therewith. WHo would believe if he saw not with his eyes, that a needle of steel being once touched with the magnes, turns not once, not a year▪ but as long as the World lasteth; his end towards the North and South, yea though one remove it, and turn it from his position, it will come again to his points of North and South. Who would have ever thought that a brute stone black and ill form, touching a ring of Iron, should hang it in the air, and that ring support a second, that to support a third, and so unto 10, 12, or more, according to the strength of the magnes; making as it were a chain without a line, without souldering together, or without any other thing to support them only; but a most occult and hidden virtue, yet most evident in this effect, which penetrateth insensibly from the first to the second, from the second to the third, etc. What is there in the world that is more capable to cast a deeper astonishment in our minds than a great massy substance of Iron to hang in the air in the midst of a building without any thing in the world touching it, only but the air? 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 Mecca: the sepulchre of that infamous Mah●met rests suspended in the air; and Pliny in his natural History writes that the Architect or Democrates did begin to vault the Temple of Assigned in Alexandria, with store of magnes to produce the like deceit, to hang the sepulchre of that Goddess likewise in the air. I should pass the bounds of my counterpoise, if I should divulge all the secrets of this stone, and should expose myself to the laughter of the world: if I should brag to show others the cause how this appeareth, than in its own natural sympathy, for why is it that a magnes with one end will cast the Iron away, & attract it with the other? from whence cometh it that all the magnes is not proper to give a true touch to the needle, but only in the two Poles of the stone: which is known by hanging the stone by a thread in the air until it be quiet, or placed upon a piece of Cork in a dish of water, or upon some thin board, for the Pole of the stone will then turn towards the Poles of the world, and point out the North and South, and so show 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 pegs and enclosed within two Glasses, showeth the height of the Pole, being elevated as many degrees as the Pole is above the Horizon? What's the cause that fire and Garlic takes away the property 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 remains ignorant of declaring the rrue cause thereof. Some say, that by help of the Magnes persons which are absent may know each others mind, as if one being here at London, and another at Prague in Germany: if each of them had a needle touched with one magnes, than the virtue 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 hour 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 party's needle shall open his intention. The invention is subtle, but I doubt whether in the world there can be found so great a stone▪ or such a Magnes which carries with it such virtue: neither is it expedient, for treasons would be then too frequent and open. EXAMINATION. THe experimental difference of rejection, and attraction proceeds not from the different nature of Stones, but from the quality of the Iron; and the virtue of the stone consisteth only, and especially in his poles, which being hanged in the Air, turns one of his ends always 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 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 near unto them, the North of the one will abhor the North of the other, but the North of the one will always 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 until it be held perpendicular, for that shall be one of the Poles of the stone which you may mark out; in like manner find out the other Pole: Now to find 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 nail of ones finger or such like,) and mark which way the end of the needle that was touched turneth: if to the South, than the point that touched it was the South-Pole, etc. and it is most certain and according to reason and experience: that if it be suspended in aequilibrio in the air, or supported upon the water, it will turn contrary to the needle that toucheth it; for then the pole that was marked for the South shall turn to the North, etc. 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 endure 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 assoon as the heat doth penetrate it, the water begins to rarefie, & issueth forth with a hideous and marvellous force; it is pleasure to see how it blows the fire with great noise. Now touching the form of these vessels, they are not made of one like fashion: some makes them like a bowl, some like a head painted representing the wind, some make them like a Pear: as though one would put it to roast at the fire, when one would have it to blow, for the tail of it is hollow, in form of a funnel, having at the top a very little hole no greater than the head of a pin. Some do accustom to put within the Aeolipile a crooked funnel of many foldings, to the end that the wind that impetuously rolls▪ to and fro within, may imitate the noise of thunder. Others content themselves with a simple funnel placed right upward, somewhat wider at the top than elsewhere like a Cone, whose basis is the mouth of the funnel: and there may be placed a bowl of Iron or Brass, which by the vapours that are cast out will cause it to leap up, and dance over the mouth of the Aeolipile. Lastly, some apply near to the hole small Windmills, or such like, which easily turn by reason of the vapours; or by help of two or more bowed funnels, a bowl may be made to turne● these Aeolipiles are of excellent use for the melting of metals and such like. Now it is cunning and subtlety to fill one of these Aeolipiles with water at so little a hole, and therefore requires the knowledge of a Philosopher to find it out: and the way is thus. Heat the Aeolipiles being empty, and the air which is within it will become extremely rarefied; then being thus hot throw it into water, and the air will begin to be condensed: by which means it will occupy less room, therefore the water will immediately enter in at the hole to avoid vacuity: thus you have some practical speculation upon the Aeolipile. PROBLEM. LXIX. Of the Thermometer: or an instrument to measure the degrees of heat and cold in the air. THis Instrument is like a cylindrical pipe of Glass, which hath a little ball or bowl at the top▪ the small end of which is placed into a vessel of water below, as by the figure may be seen. Then put some coloured liquor into the cylindrical glass, as blue, red, yellow, green, or such like: such as is not thick. This being done the use may be thus. 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 way they may not only distinguish upon what degree the water ascendeth in the morning, at midday, & at any other hour: but also one may know how much one day is hotter or colder than another: by marking how many degrees the water ascendeth or descendeth, one may compare the hottest and coldest days in a whole year together with these of another year: again one may know how much hotter one room is than another, by which also one might keep a chamber, a furnace, a stove, etc. always in an equality of heat, by making the water of the Thermometer rest always upon one & the same degree: in brief, one may judge in some measure the burning of Fevers, and near unto what extension the air can be rarefied by the greatest heat. Many make use of these glasses to judge of the weather: for it is observed that if the water fall in 3 or 4 hours a degree or thereabout, that rain ensueth; and the water will stand at that stay, until the weather change: mark the water at your going to bed, for if in the morning it hath descended rain followeth, but if it be mounted higher, it argueth fair weather: so in very cold weather, if it fall suddenly, it is snow or some sleekey weather that wiil ensue, 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 Maxim 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 greatness from humane bodies. Thirdly, because the symmetry and concordancy of the parts is so admirable, that all works which are well proportionable, as namely the building of Temples, of Ships, of Pillars, and such like pieces of Architecture, are in some measure fashioned and composed after his proportion. And we know that the Ark of Noah built by the commandment of God, was in length 300 Cubits, in breadth 50 Cubits, in height or depth 30 cubits, so that the length contains the breadth 6 times, and 10 times the depth: now a man being measured you will find him to have the same proportion in length, breadth, and depth. Vilalpandus treating of the Temple of Solomon (that chieftain 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 greatness of the work and proportionable symmetry, some dare assure themselves that by knowledge of one only 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 columns, are as the foot; the top, and roof as the head; the rest as the body: those which have been somewhat more curious, have noted that as in humane bodies, the parts are uniform, as the nose, the mouth, etc. 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 writhing of hair, the channels of columns, to the fold of woman's Robes, etc. Now building being a work of the best Artist, there is much reason why man ought to make his imitation from the chief work of nature; which is man. Hence it is that Vitru●ius in his third book, and all the best Architectes, treat of the proportion of man; amongst others Albert Durens hath made a whole book of the measures of man's body, from the foot to the head, let them read it who will, they may have a perfect knowledge thereof: But I will content myself and it may satisfy some with that which followeth. First, the length of a man well made, which commonly is called height, is equal to the distance from one end of his finger to the other: when the arms are extended as wide as they may be. Secondly, if a man have his feet and hands extended or stretched in form of S. Andrews Cross, placing one foot of a pair of Compasses upon his navel, one may describe a circle which will pass by the ends of his hands and feet, and drawing lines by the terms 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 equal to the length of the hand, taken from the small of the arm, unto the extremity of the longest finger. Fiftly, the thickness 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 less. Sixtly, the height of the brow, the length of the nose, the space between the nose and the chin, the length of the ears, the greatness of the thumb, are perfectly equal 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 easy to Painters, Image-makers, etc. 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 divers 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 clear, that by the help of proportion, the body of Hercules was measured by the knowledge of his foot only, 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 only the claw of a Lion, did figure and draw out all the beast according to his true type or form, so the exquisite Painter Timantes, having painted a Pygmy or Dwarf, which he measured with a fathom made with the inch of a Giant, it was sufficient to know the greatness of that Giant- To be short, we may by like method come easily to the knowledge of many fine antiquities touching Statues, Colossus, and monstrous Giants, only 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 special piece of work which he had projected: as to figure out the mount Athos in form of a great Statue, which should hold in his right hand a Town capable to receive ten thousand men: and in his left hand a vessel to receive all the water that floweth from the Mountain, which with an engine should cast into the Sea. This is a pretty project, said Alexander, but because there was not field-roome thereabout to nourish and retain the Citizens of that place, Alexander was wise not to entertain the design. Now let it be required of what greatness this Statue might have been, the Town 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 Mountain itself, and the Mountain was about a mile in height plumb or perpendicular; 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 multiplied by the breadth, makes an hundred twenty five thousand square feet, for the quantity of his hand to make the town in, to lodge the said 10 thousand men, allowing to each man near about 12 foot of square ground: now judge the capacity of the other parts of this Colossus by that which is already delivered. Secondly, Pliny in his 34 book of his natural History, speaks of the famous Colossus that was at Rhodes, between whose legs a Ship might pass with his sails open or displayed, the Statue being of 70 cubits high: and other Histories report that the Saracens having broken it, did load 900 Camels with the metal of it, now what might be the greatness and weight of this Statue? For answer, it is usually allowed for a Camel's burden 1200 pound weight, therefore all the Collosus did weigh 1080000 pound weight, which is ten hundred and fourscore thousand pound weight. Now 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 half, and seeing that the Nose, the brow, and the thumb, are the third part of the face, his Nose was 3 foot and a half long, and so much also was his thumb in length: now the thickness being always the third part of the length, it should seem that his thumb was a foot thick at the least. Thirdly, the said Pliny 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 brass, which was made of 120 foot in height, which Nero caused to be painted in the same height. Now would you know the greatness of the members of this Colossus, the breadth would be 20 foot, his face 12 foot, his thumb and his nose 4 foot, according to the proportion before delivered. Thus I have a fair field or subject to extend myself upon, but it is upon another occasion that it was undertaken, let us speak therefore a word touching the Giants, and then pass away to the matter. Of monstrous Giants. YOu will hardly believe all that which I say touching this, neither will I believe 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 greatness; for the holy writes witness this themselves in Deut, Chap. 3. that there was a certain Giant called Og, of the Town of Rabath, who had a bed of Iron, the length thereof was 9 cubits, and in breadth 4 cubits. So in the first of Kings Chap. 17. there is mention made of Goliath, whose height was a palm and 6 cubits, that is more than 9 foot, he was armed from the head to the foot, and his Curiat only 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 certain, that the rest of his arms 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 bear 200 pound, yet this Giant carries this as a vesture without pain. Solinus reporteth in his 5 Chap. of his History, that during the Grecians war 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 half) therefore according to the proportion delivered, his face should be 5 foot in length, a thing prodigious and monstrous. Pliny in his 7. book and 16 Chap. saith, that in the Isle of Crete or Candie, a mountain being cloven by an Earthquake, there was a body standing upright, which had 46 Cubits of height: some believe that it was the body of Orion or Othus, (but I think rather it was some Ghost or some delusion) whose hand should have been 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 Town, where it is thought that the Giant Antheus was buried, Sertorius could not believe that which was reported of his prodigious greatness, caused his sepulchre to be opened, and found that his body did contain 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 near the capacity of the Colossus at Rhodes. But behold here a fine fable of Symphoris Campesius, in his book entitled Hortus Gallicus, who says that in the Kingdom of Sicily, at the foot of a mountain near Trepane, in opening the foundation of a house, they found a Cave in which was ●aid a Giant, which held in stead of a staff 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 seem that his length was 200 cubits, or 300 foot: if he had said that he had been 300 cubits in length, than he might have made us believe that Noah's 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 certain that there have been men of very great stature, as the holy Scriptures before witness, and many Authors worthy of belief relate: Josephus Acosta in his first book of the Indian History, Chap. 19 a late writer, reporteth, that 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 Palm, at Trap, at Bowls, Paile-maile, and others. THe Mathematics often findeth place in sundry Games to aid and assist the Gamesters, though not unknown unto them, hence by Mathematical 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 Bowl by Geometrical lines and angles. And the first maxim is thus: When a Bowl toucheth another Bowl▪ or when a trapstick striketh the Ball, the moving of the Ball is made in a right line, which is drawn from the Centre of the Bowl by the point of contingency. Secondly, in all kind of such motion; when a Ball or Bowl rebounds, be it either against wood, a wall, upon a Drum, a pavement, or upon a Racket; the incident Angle is always equal to the Angle of reflection. Now following these maxims, it is easy to canclude, first, in what part of the wood or wall, one may make the Bowl or Ball go to reflect or rebound, to such a place as one would. Secondly, how one may cast a Bowl 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 Bowl 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 may be noted that the Maxims of reflections cannot be exactly observed by local motion, as in the beams of light and of other qualities, 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 forms. NVmbers have an admirable secrecy, diversely applied, as before in part is showed, and here I will say something by way of transmutation of numbers. It's answered thus, in the first form the men were as the figure A, than each of these 4 Soldiers 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 Soldiers at the Gates take away each one his Comrade, 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 Viol sensibly shake, without any one touching it? THis is a miracle in music, yet easy to be experimented. Take a Viol 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 losing 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 Viols, 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 sympathy, 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 air, 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. EXAMINATION. IN this Examination we have something else to imagine, than the bare sympathy of the Cords one to another: for first there aught to be considered the different effect that it produceth by extension upon one and the same Cord in capacity: 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 facile to lay open a way to the knowledge of the true and immediate cause of this noble and admirable Phaenomeny. Now this will sensibly appear when the Cords are of equal length and greatness, and set to an unisone; but when the Cords differ from their equality, it will be less 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 proportional 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, etc. & in setting them to one tune: for then in touching the Cord of the one, it will give a sensible motion to the Cord of the other: and not only so but also a harmony. PROBLEM. LXXIIII. Of a vessel which contains three several kinds of liquor, all put in at one bunghole, and drawn out at one tap severally without mixture. THe vessel is thus made, it must be divided into three Cells for to contain the three liquors, which admit to be Sack, Claret, and White-wine: Now in the bunghole there is an Engine with three pipes, each extending to his proper Cell, into which there is put a broach or funnel pierced in three places, in such sort, that placing one of the holes right against the pipe which answereth unto him, the other two pipes are stopped; then when it is full, turn the funnel, and then the former hole will be stopped, and another open, to cast in other wine without mixing it with the other. Now to draw out also without mixture, at the bottom of the vessel there must be placed a pipe or broach, which may have three pipes; and a cock piersed with three holes so artificially done, that turning the cock, the whole which answereth to such of the pipes that is placed at the bottom, may issue forth such wine as belongeth to that pipe, & turning the Cock to another pipe, the former hole will be stopped; and so there will issue forth another kind of wine without any mixtures; but the Cock may be so ordered that there may come out by it two wines together, or all three kinds at once: but it seems best when that in one vessel and at one Cock, a man may draw several kinds of wine, and which he pleaseth to drink. PROBLEM. LXXV. Of burning-Glasses. IN this ensuing discourse I will show the invention of Prom●theus, how to steal fire from Heaven, and bring it down to the Earth; this is done by a little round Glass, or made of steel, by which one may light a Candle, and make it flame, kindle Firebrands to wake them burn, melt Led, ●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. But this is nothing to the burning of those Glasses which are hollow, namely those which are of steel well polished, according to a par●bolicall or oval section. A spherical Glass, or that which is according to the segment of a Sphere, burns very effectually about the fourth part of the Diameter; notwithstanding the Parabolie and Ecliptic sections have a great effect: by which Glasses there are also divers figures represented forth to the eye. The cause of this burning is the uniting of the beams of the Sun, which heat mightily in the point of concourse or inflammation, which is either by transmission or reflection▪ Now it is pleasant to behold when one breatheth in the point of concourse, or throweth small dust there, or sprinkles vapours of hot water in that place; by which the Pyramidal point, or point of inflammation is known. Now some Author's promise to make Glasses which shall burn a great distance off, but yet not seen vulgarly produced, of which if they were made, the Parabolie makes the greatest effect, and is generally held to be the invention of Archimedes or Pro●●us. Maginus in the 5 Chap. of his Treatise of spherical Glasses, shows how one may serve himself with a concave Glass, to light fire in the shadow, or near such a place where the Sun shines not, which is by help of a flat Glass, by which may be made a percussion of the beams of the Sun into the concave Glass, adding unto it that it serves to good use to put fi●e to a Mine, provided that the combustible matter be well applied before the concave Glass; in which he says true: but because all the effect of the practice depends upon the placing of the Glass and the Powder which he speaks not of: I will deliver here a rule more general. How one may place a Burning-glasse with his combustible matter in such sort, that at a convenient hour of the day, the Sun shining, it shall take fire and burn: Now it is certain that the point of inflammation or burning, is changed as the Sun changeth place, and no more nor less, than the shadow turns about the style of a Dial; therefore have regard to the Sun's motion, and ●is height and place: a Bowl of Crystal in the same place that the top of the style is, and the Powder or other combustible matter under the Meridian, or hour of 12, 1, 2, 3, etc. or any other hour, and under the Sun's arch for that day: now the Sun coming to the hour of 12, to ●, 2, ●, etc. the Sun casting his beams through the Crystal Bowl, will fire the material or combustible thing, which meets in the point of burning: the like may be observed of other Burning-glasses. EXAMINATION. IT is certain in the first part of this Problem that conical, concave and spherical Glasses, of what matter soever, being placed to receive the beams of the Sun will excite heat, and that heat is so much the greater, by how much it is near the point of concourse or inflammation. But that Archimedes or Proclus d●d fire or burn Ships with such Glasses, the ancient Histories are silent, yea the selves say nothing: besides the great difficulty that doth oppose it in remoteness, and the matter that the effect is to work upon: Now by a common Glass we fire things near at hand, from which it seems very facile to such which are less read, to do it at a far greater distance, and so by relation 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 show the variety of Antiquity, and truth of History: and as touching to burn at a great distance, as is said of some, it is absolutely impossible; and that the parabolical and Oval Glasses were of Archimedes and ●roclus invention is much uncertain: 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 near at hand; for if it be cast far off, the effect is little, and the heat weak, or otherwise such Glasses must be greatly extended to contract many beams to amass a sufficient quantity of beams in parabolical and conical 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 far remote, as is supposed before, such a Glass cannot be used but at a great inclination of the Sun▪ by which the effect of ●urning is diminished, by reason of the weakness of the Sunbeams. And here may be noted in the last part of this Problem, that by reason of obstacles if one plain Glass be not sufficient, a second Glass may be applied to help it: that so if by one simple reflection it cannot be done, yet by a double reflection the Sunbeams may be ●ast into the said Caverne or Mine, and though the reflected beams in this case be weak▪ yet upon a 〈◊〉 combustible matter it will not fail to do the effect. PROBLEM. LXXVI. Containing m●ny pleasant Questions by way of Arithmetic. I Will not in●ert i● this Problem that which is drawn from the ●reek Epigrams, but proposing the Question immediately will give the answer also, without ●●aying to show the manner how they are answered; in this I will 〈◊〉 be tied to the ●reek terms, w●●ch I 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 satisfied Of the 〈…〉 the Mule. IT 〈◊〉 ●hat ●he Mule and the Ass upon a day 〈◊〉 a voyage each of them carried a Barrel full of Wine: now the las●e Ass f●lt herself overladen, complained and bowed under her burden; which th● Mule seeing said unto her being angry, (for it was in the time when beasts spoke) Thou great Ass, wherefore complainest thou? if I had but only one measure of that which thou carriest, I should be loaden twice as much as thou art, and if I should give a measure of my loading to thee, yet my burden would be as much as thine. Now how many measures did each of them carry? Answer, the Mule did carry 7 measures, and the Ass 5 measures: for if the Mule had one of the measures of the Asses loading, than the Mule would have 8 measures, which is double to 4, and giving one to the Ass, each of them would have equal burdens: to wit, 6 measures apiece. Of the number of Soldiers that fought before old Troy. HOmer being asked by He●iodus how many Grecian Soldiers came against Troy? who answered him thus; The Grecians, said Homer, made 7 fires, or had 7 Kitchens, and before every fire, or in every Kitchen there were 50 broaches turning to roast a great quantity of flesh, and each broach had meat enough to satisfy 900 men: now judge how many men there might be. Answer, 315000. that is, three hundred and fifteen thousand men, which is clear by multiplying 7 by 50, and the product by 900 makes the said 315000. Of the number of Crowns that two men had. JOhn and Peter had certain number of crowns: John said to Peter, If you give me 10 of your crowns, I shall have three times as much as you have: but Peter said to J●hn, If you give me 10 of your crowns I shall have 5 times as much as you have: how much had each of them? Answer, John had 15 crowns and 5 sevenths of a crown, and Peter had 18 crowns, and 4 sevenths of a crown. For if you add 10 of Peter's crowns to those of john's, than should John have 25 crowns and 5 sevenths of a crown, which is triple to that of Peter's, viz. 8▪ and 4 sevenths: and John giving 10 to Peter, Peter should have then 28 crowns, and 4 sevenths of a crown, which is Quintupla, or 5 times as much as John had left, viz. 5 crowns and 5 sevenths. In like manner two Gamesters playing together, A and B▪ after play A said to B, Give me 2 crowns of thy money, and I shall have twice as much as thou hast: and B said to A, Give me 2 crowns 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 hour of the day. SOme one asked a Mathemacian what a clock it was; who answered that the rest of the day is four thirds of that which is past: now judge what a clock it is. Answer, if the day were according to the Jews and ancient Romans, which ma●e it always to be 12 hours, it was then the ● hour, and one seventh of an hou●e, so there remained of the whole day 6, that is, 6 hours, 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 hours, than the hour had been 10 of the clock▪ and two seventhes of an hour, which is found▪ out by dividing 12, or 24 by ●. There might have been added many curious propositions in this kind, but they would be too difficult for the most part of people▪ therefore I have omitted them▪ Of Pythagoras his Scholars. Pythagoras' being asked what number of Scholars he had, answered, that half of them studied Mathematics, the fourth part Physic, the seventh part Rhetoric, and besides he had 3 women: now judge you saith he, how many Scholars I have. Answer, he had in all 28, the half of which is 14, the quarter 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 answer the qeustion, join the number of Graces and Muses together which makes 12, and so many Apples had each Grace: Now may you take the double, triple, etc. of 12 that is 24, 36, etc. conditionally, that if each Grace had but 12, then may there be allotted to each Muse but one only; 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 Crowns amongst his two children; the one being legitimate, and the other a Bastard, conditionally that the fifth part which his legitimate Son should have, should exceed by 10, the fourth part of that which the Bastard should have: what was each 〈◊〉 part? Answer, the legitimate Son had 577 crowns and 7/●, and the Bastard 42● crowns 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 less than ●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 Dams, but each cup was heavier one than another by one Dram: how much did each of them therefore weigh? Answer, the first weighed 102 Dams and a half; the second 101 Dams and a half, the third 100 Dams and ●, the fourth 99 a & half, the fifth 98 & a half; and the sixth Cup weighed 97 Dams and a half▪ which together makes 600 Drams as before. Of Cupid's 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, Erateses the seventh part▪ Terpomene the fourth part, Polyhymnia took away 30, Urania 220, and Calliope 300. so there were left me but 5 Appls, how many had he in all at the first? I answer 3●60. There are an infinite of such like questions amongst the Greek Epigrams: but it would be unpleasant to express them all: I will only add one more, and show a general rule for all the rest. Of a Man's Age. A Man was said to pass the sixth part of his life in childhood, the fourth part in his youth, the ●hird part in Manhood, and 18 years besides in old age: what might his Age be? the answer is, 72 years: which and all others is thus resolved: multiply 1/●▪ ¼ and ⅓▪ together, that is, 6 by 4 makes 24, and that again by 3 makes 72, then take the third part of 72, which is 24, the fourth part of it, which is 18, and the sixth part of it which is 12, these added together make 54, which taken from 72, rests 18 this divided by 18 (spoken in the Question) gives 1, which multiplied by the sum of the parts, viz. 72, makes 72, the Answer as before. Of the Lion of Bronze placed upon a Fountain with this Epigram. Out of my right eye if I let water pass, I can fill the Cistern in 2 days: if I let it pass out of the left eye, it will be filled in 3 days: if it pass out of my feet, the Cistern will be 4 days a●filling; but if I let the water pass out of my mouth, I can fill the Cistern then in 6 hours: in what time should I fill it, if I pour forth the water at all the passages at once? The Greeks (the greatest talkers in the world) variously apply this question to divers statues, and pipes of Fountains: and the solution is by the Rule of ●, by a general Rule, or by algebra. They have also in their anthology many other questions, but because they are more proper to exercise, than to recreate the spirit, I pass them over (as before) with silence. PROBLEM. LXXVII. Divers excellent and admirable experiments upon Glasses. THere is nothing in the world so beautiful as light: and nothing more recreative to the sight, than Glasses which reflect: therefore I will now produce some experiments upon them, not that will dive into their depth (that were to lay open a mysterious thing) but that which may delight and recreate the spirits: Let us suppose therefore these principles, upon which is built the demonstration of the appearances which are made ●n all sort of Glasses. First, that the rays or beams, which reflect upon a Glass, make the Angle of incident equal 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 Glass, 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 Glass. Theo. 17. and 18. And here you are to understand, that there is not meant only those which are simple Glasses or Glasses of steel, but all other bodies, which may represent the visible Image of things by reason of their reflection, as Water, Marble, Metal, or such like. Now take a Glass in your hand and make experiment upon that which followeth. Experiment upon flat and plain 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 equal to the Angle of reflection: therefore when a Glass stands upright, that is, perpendicular to the Horizon, you cannot see that which is above, except the Glass be placed down flat: and to see that on the right hand, you must be on the left hand, etc. Secondly, an image cannot be seen in a Glass if it be not raised above the surface of it; or place a Glass upon a wall, you shall see nothing which is upon the plain of the wall, and place it upon a Table or Horizontal Plain, you shall see nothing of that which is upon the Table. Thirdly, in a plain Glass all that is seen appears or seems to sink behind the Glass, as much as the image is before the Glass, as before is said. Fourthly, (as in water) a Glass lying down flat, or horizontal, Towers, Trees, Men, or any height doth appear, inversed or upside down; and a Glass placed upright, the right hand of the Image 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 Glass must be disposed, that the line upon which the Jmages come on the Glass, make the Angle of incidence equal to that Angle of reflection. Seventhly, present a Candle upon a plain Glass, and look flaunting upon it, so that the Candle and the Glass be near in a right line, you shall see 3, 4, 5, etc. images, from one and the same Candle. Eightly, take two plain Glasses, and hold them one against the other, you shall alternately see them oftentimes one within the other, yea within themselves, again and again. Ninthly, if you hold a plain Glass behind your head, and another before your face, you may see the h●nder part of your head, in that Glass which you hold before your face. Tenthly, you may have a fine experiment if you place two Glasses together, that they make an acute angle, and so the lesser the angle is, the more appearances you shall see, the one direct, the other inversed, the one approaching, and the other retiring. Eleventhly, it is a wonder & astonishment to some, to see within a Glass an Image without knowing from whence it came, and it may be done many ways: 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 upwards. Then place another object, so that it reflect, or cast the Image downward to the eye of the spectator▪ without perceiving it being hid behind something, for then the Glass will represent a quite contrary thing, either that which is before the Glass, or that which is about it, to wit, the other hidden object. Twelfthly, if there be ingraved behind the backside of a Glass, or drawn any Image upon it, it will appear before as an Image, without any appearance: o● portraiture to be perceived. EXAMINATION. THis 12 Article of engraving an Image behind the Glass, 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 usual covering of Glasses behind, and so made up like an ordinary lookingglass 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 subtlest, and that principally if the Glass be in an obscure place, and the light which is given to it be somewhat far off. PLace a Glass near the floor of a Chamber, & make a hole through the place under the Glass, so that those which are below may not perceive it, and dispose a bright Image under the hole so that it may cast his species upon the Glass, 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 adjoining, and so make it to be seen upon the side of the Wall. 14 In these Channel-Images which show one side a death's head, & another side a fair face: and right before some other thing: it is a thing evident, that setting a plain Glass 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 plain Glass writing with such industry, that one may read it in the Glass, and yet out of the Glass there is nothing to be known, which will thus happen, if the writing be writ backward: but that which is more strange, to show a kind of writing to a plain Glass, it shall appear another kind of writing both against sense and form, as if there were presented to the Glass WELL it would show it MET; if it were written thus MIV, and presented to the Glass, it would appear thus VIM; for in the first, if the Glass lie flat, than 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 plain Glasses, either of the Admirable multiplications, or appearances, which is made in a great number of them; for to content the sight in this particular, one must have recourse to the Cabinets of great Personages who enrich themselves with most beautiful ones. Experiments upon Gibbous, or convex Spherical Glasses. IF they be in the form of a Bowl, or part of a great Globe of Glass, there is singular contentment to contemplate on them. First, because they present the objects less and more gracious, and by how much more the Images are separated from the Glass, by so much the more they diminish in Magnitude. Secondly, they that show the Images plaiting, or folding, which is very pleasant, especially when the Glass is placed down, and behold in it some Blanching, feeling, etc. The upper part of a Gallery, the porch of a Hall, etc. 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 seem 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 beautiful contraction of the Images, that appear within the sphericity of these small Glasses: for present the Glass to the lower end of a Gallery, 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 City; all the fair Architecture, and appearances will be seen contracted within the circuit of the Glass with such variety 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 Glass. Exptriments upon hollow, or Concave spherical Glasses. I Have heretofore spoken how they may burn, being made of Glass, or Metal, it remains 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 Glass is, and the Globe from whence it is extracted for it must in proportion as a segment of some be made circle or orb. EXAMINATION. 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 fathom, for it is certain 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 equal or not in section, there will happen an evident difference in one and the same experiment, in the number, situation, quantity, 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 himself that he hath caused many to be polished for sundry great Lords of Italy, and Germany, 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 bear with the cost. EXAMINATION. TOuching Maginus he hath nothing aided 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 himself, 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 seen sometimes upon the surface of the Glasses, sometimes as though they were within it and behind it, deeply sunk into it, sometimes they are seen before, and without the Glass, sometimes between the object and the Glass; sometimes in the place of the Eye, sometimes farther from the Glass than the object is: which comes to pass by reason of the divers concourse of the beams, and change of the place of the Images in the line of reflection. EXAMINATION. THe relation of these appearances pass 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 Glass, and it is a principal point to declare truly in which place the Image is seen in the Glass those that are more learned in optical knowledge affirm the contrary, and nature itself gives it a certain place according to its position being always 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 sufficiently circumspect in experience upon which the principles of this sub●ect aught 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 them for the most part fell into the like errors. As for the Jmages to bid● in the eye▪ it cannot be but is impertinent and absurd; but it followeth that, by how much nearer the ob●ect approacheth to the Glass, 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 near the eye, but passing the point of concourse it goes back again: 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 until the object be within, but contrarily if the eye be between the point of concourse and the Glass, than the Jmages are direct: and if the eye or the object be in the point of concourse, the Glass will be enlightened and the Jmages confused, and if there were but a spark of fire in the said point of concourse, all the Glass would seem a burning firebrand, and we dare say it would occur without chance, and in the night be the most certain and subtlest 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 confirm what we here speak of: & will find new lights with a conveniable position to the Glass, he will have reflection of quantities, of truth, and fine secrets in nature, yet not known, which he may easily comprehend if he have but an indifferent sight, and may assure himself 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 seen in one and the same place immutable; or if the Image and the eye move in their own lines, the representation in the Glass seems to invest itself continually with a different figure. NOw the Image coming thus to the eye, those which know not the secret, draw their sword when they see an Image thus to issue out of the Glass, or a Pistol which some one holds behind: and some Glasses will show a sword wholly drawn out, separated from the Glass, as though it were in the air: and it is daily exercised, that a man may touch the Image of his hand or his face out of the Glass, which comes out the farther, by how much the Glass is great and the Centre remote. EXAMINATION. NOw that a Pistol being presented to a Glass behind a man, should come out of the Glass, and make him afraid that stands before, seeming to shoot at him, this cannot be: for no object whatsoever presented to a concave Glass, if it be not nearer to the Grass than the eye is it comes not out to the sight of the party; therefore he needs not fear that which is said to be behind his back, and comes out of the Glass; for if it doth come out, it must then necessarily be before his face, so in a concave Glass whose Centre is far remote of a sword, stick, or such like be presented to the Glass, it shall totally be seen to come forth of the Glass and all the hand that holds it. And here generally note that if an Image be seen to issue out of the Glass 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 known to all the standers by: so many persons standing before a Glass, if one of the company take a sword, and would make it issue forth towards any o●her that stands there: let him choose his Image in the Glisse and carry the sword right towards it and the effect will follow. In like manner one's hand being presented to the Gloss as it is thrust towards the Centre, s● the representation of it comes towards it, and so the hands will seem to be united, or to touch one another. FRom which may be concluded, if such a Glass be placed at the ceiling or planching of a Hall, so that the face be horizontal 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 fear or scaring: for one should see many men in the air as if they were hanging by the feet. EXAMINATION. TOuching a Glass tied at a ceiling or planching, that one may see a man hang by the feet in the air, and so many Glasses, many men may be seen: without caution this is very absurd for if the Glass or Glasses be not so great that the Centre of the sphere upon which it was made, extend not near to the head of him that is under it, it will not pleasantly appear, and though the Glass should be of that capacity that the Centre did extend so far, yet will not the Images be seen to them which are from the Glass but on●y to those which are under it, or near unto it: and to them it will not ably appear, and it would be most admirable to have a Gallery 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 air, and you could not walk without incountering airy apparitions. SEcondly, in flat or plain Glasses the Image is seen equal to his object, and to represent a whole man, there ought to be a Glass as great as the Image is: In convex Glasses the Images are seen always less, in concave Glasses they may be seen greater or lesser, but not truly proportionable, by reason the divers reflections which contracts or enlargeth the Species: when the eye is between the Centre and the surface of the Glass; the Image appears sometimes very great and deformed, and those which have but the appearance of the beginning of a beard on their chin, may cheer up themselves to see they have a great beard; those that seem to be fair will thrust away the Glass with despite, because it will transform their beauty: those that put their hand to the Glass will seem to have the hand of a Giant, and if one puts his finger to the Glass it will be seen as a great Pyramid of flesh, inversed against his finger. Thirdly, it is a thing admirable that the eye being approached to the point of concourse of the Glass, there will be seen nothing but an intermixture or confusion: but retiring back a little from that point, (because the rays do there meet▪) he shall see his Image inversed, having his head below and his feet above. Fourthly, the divers appearances caused by the motion of objects, either retiring or approaching: whether they turn to the right hand or to the left hand, whether the Glass be hung against a wall, or whether it be placed upon a Pavement, as also what may be represented by the mutual aspect of concave Glasses with plain and convex Glasses but I will with silence pass them over, only say something of two rare experiments more as followeth. 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 big, but inversed upon the surface of the Glass, with some kind of colour, or these letters may be written with wax, (the easier to be taken out again:) for then placing the Glass to the Sun, 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 Moon. In the second place, how a man may sundry ways help himself with such a Glass, with a lighted Torch or Candle, placed in the point of concourse or inflammation, which is near the fourth part of the Diameter: for by this means the light of the Candle will be reverberated into the Glass, and will be cast back again very far by parallel lines, making so great a light that one may clearly see that which is done far off, yea in the camp of an Enemy: and those which shall see the Glass a far off, will think they see a Silver Basin enlightened, or a fire more resplendent than the Torch. It is this way that there are made certain Lanterns which dazzle 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 Glass, so that it may successively be applied to the point of inflammation. In like manner by this reflected light, one may read far 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 peril or suspicion. EXAMINATION. THis will be scarce sensible upon a wall remote from the Glass, and but indifferently seen upon a wall which is near the Glass, and withal it must be in obscurity 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 shown in a concave Glass: for if in the point of inflammation of a parabolical section, a Candle be placed, the light will be reflected by parallel lines, as a column or Cylinder; but in the spherical section it is defective in part, the beams being not united in one point, but somewhat scattering: notwithstanding it casteth a very great beautiful light. Lastly, those which fear to hurt their sight by the approach of Lamps or Candles, may by this artifice place at some corher of a Chamber, a Lamp with a hollow Glass behind it, which will commodiously reflect the light upon a Table, or to a place assigned: so that the Glass be somewhat raised to make the light to streeke upon the Table with sharp Angles, as the Sun doth when it is but a little elevated above the Horizon, for this light shall exceed the light of many Candles placed in the Room, and be more pleasant to the sight of him that useth it. Of other Glasses of pleasure. FIrst, the Columnary and Pyramidal Glasses that are contained under right lines, do represent the Images as plain Glasses do; and if they be bowing, than they represent the Image, as the concave and convex Glasses do. Secondly, those Glasses which are plain, but have ascents of Angels in the middle, will show one to have four Eyes, two Mouths, two Noses, etc. EXAMINATION. TH●se experiments will be found different according to the divers meeting of the Glasses, which commonly are made scuing-wise at the end, 〈◊〉 which there will be two divers superficies in the Glass, making the exterior Angle somewhat raised, at the interior only one superficies, which may be covered according to ordinary Glasses to c●use a reflection, and so it will be but one Glass, which by refraction according to the different thickness of the Glass, and different Angles of the scuing form, do differently present the Images to the eye, as four eyes, two mouths, two noses; sometimes three eyes one mouth, and one nose, the one large and the other long, sometimes two eyes only: with the mouth and the nose deformed, which the Glass (impenetrable) will not show. And if there be an interior 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 four eyes, two noses, and two mouths: and by moving the Glass the Angle will vanish, and so the two superficies will be turned into one, and the duplicity of Images will also vanish and appear but one only: 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 seem pale, red, and coloured in divers manners, which is caused by the dye of the Glass, or the divers refraction of the Species: and those which are made of Silver, Latin, Steel, etc. do give the Images a divers colour also. 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 foul and deformed: and give them a contrary visage: for if a Glass be cut as it may be, or if many pieces of Glass be placed together to make a conveniable reflection: there might be made of a Mole (as it were) a mountain, of one Hair 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 four recreative Problems following. PROBLEM. LXXVIII. 1 How to show to one that is suspicious, what is done in another Chamber or Room: 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 tied to the planching or ceiling, that it may be common to communicate the Species to each Glass by reflection, there being left some hole at the top of the Wall against the Glass to this end: the two other Glasses must be placed against the two Walls at right Angles, as the figure here showeth at B. and C. Then the sight at E by the line of incidence FE, shall fall upon the Glass BASILIUS, and reflect upon the superficies of the Glass 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 Glass, upon which is made the second reflection, 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 Town may be seen upon the Rampart: notwithstanding the Parapet, which the besieged may do by placing a Glass in the hollow of the Ditch, and placing another upon the top of the wall, so that the line of incidence coming to the bottom of the Ditch, make an Angle equal to the Angle of reflection, then by this situation and reflection, 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 reflections 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 Image will be seen easily. corollary 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 equal: that is, placing them in such sort by some Geometrical 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 reflections, 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 plain Glass perpendicular as AB. so the mark C shall be seen in Catheti CA, viz. in D, and the line of reflection is D, now let the Musket FE, upon a rest▪ be moved to and fro until it be seen in the line ODD, which admit to be HG, so giving fire to the Musket, it shall undoubtedly strike the mark. corollaries. 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 seen, admit to be QR, then set up a plain Glass as AB, and let the Musket by GH, placed upon his rest PO. Now because the mark C is seen at D, move the Musket to and fro, until it doth agree with the line of reflection MB, which suppose at LI, so shall it be truly placed, and giving fire to the Musket, it shall not fail to strike the said mark at C. PROBLEM. LXXX. How to make an Image to be seen hanging in the air, having his head downward. 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 seem 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 air, if the Image had wings unto it, and had secretly 〈◊〉 motion: and if the Glass were big enough to receive many reflections, 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 Soldiers seem to be a Regiment, or how few in number may be multiplied 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 plain level Table: between which Glasses let there be ranged in Battalia-wise upon the same Table a number of small men according to the square G, H, I, F, or in any other form or posture: hen may you evidently see how the said battle will be multiplied and seem far bigger in the appearance than it is in effect. Corolarie. BY this invention you may make a little Cabinet of four foot long, and two foot large, (more or less) which being filled with Rocks or such like things, or there being put into it Silver, Gold, Stones of lustre, Jewels, etc. and the walls of the said Cabinet being all covered, or hung with plain glass; these visibles will appear manifoldly increased, by reason of the multiplicity of reflections, 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 Dials'. COuld you choose a more ridiculous one than the natural Dial written amongst the Greek Epigrams, upon which some sound Poet made verses; showing that a man carrieth about him always a Dial in his face by means of the Nose and Teeth? and is not this a jolly Dial? 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 Dial of herbs. 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 hours represented with little bushy herbs, as of Hyssop or such which is proper to be cut in the borders; and at the top of the style to have a Fan to show which way the wind blows? this is very pleasant and useful. Of the Dial upon the fingers and the hand. IS it nor a commodity very agreeable, when one is in the fie●d or in some vil●age without any other Dial, to see only by the hand what of the clock it is? which gives it very near; and may be practised by the left hand, in this manner. Take a straw or like thing of the length of the Index or the second finger, hold this straw very right between the thumb and the forefinger, then stretch forth the hand▪ and turn your back, and the palm of your hand towards the Sun; so that the shadow of the muscle which is under the Thumb, touch the line of life, which is between the middle of the two other great lines, which is seen in the palm of the hand, this done, the end of the shadow will show what 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 joint, it is 9 in the morning, or 3 in the afternoon, 10 & 2 at the second joint, 11 and 1 at the third joint, and midday in the line following, which comes from the end of the Index. Of a Dial which was about an Obeliske at Rome. WAs not this a pretty fetch upon a pavement, to choose an Obeliske for a Dial, having 106 foot in height, without removing the Basis of it? Pliny assures us in his 26 book and 8 Chap. that the Emperor Augustus having accommodated 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 enchased marks of Copper upon the Pavement, and placed also an Apple of Gold upon the top of the said Obeliske, to know the hour and the course of the Sun, with the increase and decrease of days 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 Astronomy. Of Dial's with Glasses. PT●lomie w●ites, as Cardanus reports, that long ago there were Glasses which served for Dial's, and presented the face of the beholder as many times as the hour ought to be, twice if it were 2 of the clock, 9 if it were 9, etc. But this was thought to be done by the help of water, and not by Glasses, which did leak by little and little out of the vessel, discovering anon one Glass, than anon two Glasses, than 3, 4, 5 Glasses, etc. to show so many faces as there were hours, which was only by leaking of water. Of a Dial which hath a Glass in the place of the Style. WHat will you say of the invention of Mathematicians, which find out daily so many fine and curious novelties? they have now a way to make Dial's upon the wainscot or ceiling of a Chamber, and there where the Sun can never shine, or the beams of the Sun cannot directly strike: and this is done in placing of a little Glass in the place of the style which reflecteth the light, with the same condition that the shadow of the style showeth the hour: and it is easy to make experiment upon a common Dial, changing only the disposition of the Dial, and tying to the end of the style a piece of plain Glass. The Almains 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 ceiling of the Chamber. EXAMINATION. IN this there are two experiments considerable, the first is with a very little Glass placed so that it may be open to the beams of the Sun, the other hath respect to a spacious or great Glass placed to a very little hole so that the Sun may shine on it, for then the shadow which is cast upon the Dial is converted into beams of the Sun, 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 pass the beams of the Sun, which represent the extremity of the style, & the Glass representeth the plain of the Dial, upon which the beams being in manner of shadows reflect cast upon a plain opposite: and it is needful that in this second way the Glass may be spacious, as before, to receive the delineaments of the Dial. Otherwise you may draw the lineaments of a Dial upon any plain lookingglass which reflecteth the Sunbeams, for the applying a style or a pearl at the extremity of it: and placed to the Sun, the reflection will be answerable to the delineaments on the Glass: but here note, that the Glass ought to be great, and so the delineaments thereon. But that which is most noble, is to draw houre-lines upon the outside of the Glass 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 certain without any difficulty. Of Dial's with water. Such kind of Dial's were made in ancient times, and also these of sand: before they had skill to make Sun-dyals' or Dial's with wheels; for they used to fill a vessel with water, and having experience by trial that it would run out all in a day, they did mark within the vessel the hours noted by the running of the water; and some did set a piece of light board in the vessel to swim upon the top of the water, carrying a little statue, which with a small stick did point out the hour upon a column or wall, figured with houre-notes, as the vessel was figured within. Now it seems a safer way that the water pass out by drop and drop, and drop into a cylindrical Glass by help of a Pipe: for having marked the exterior part of the Cylinder in the hour notes, the water itself which falls within it, will show what of the clock it is, far better than the running of sand, for by this may you have the parts of the hours most accurate, which commonly by sand is not had: and to which may be added the hours of other Countries with greater ease. And here note, that as soon as the water is out▪ of one of the Glasses, you may turn it over into the same again out of the other, and so let it run anew. PROBLEM. LXXXIII. Of Cannons or great Artillery. Soldiers, and others would willingly see 〈◊〉 Problems, which contain: three or four subtle questions: The first is, how to charge a Cannon without powder? THis may be done with air and water, only having thrown cold water into the Cannon, which might be squirted forceably in by the closure of the mouth of the Piece, that so by this pressure the air might more condense; then having a round piece of wood very just, and oiled well for the better to slide, and thrust the Bullet when it shall be time: This piece of wood may be held fast with some Pole, for fear 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 air and water, and then when one would shoot it, let the pole be quickly loosened, for then the air searching a greater place, and having way now offered, will thrust out the wood and the bullet very quick: the experiment which we have in long trunks shooting out pellats with air only, showeth the verity of this Problem. 2 In the second question it may be demanded, how much time doth the Bull●● of a Cannon spend in the air before i● falls to the ground? THe resolution of this Question depends upon the goodness 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 air before it fell: and the distance was a German 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 pass, that a Cannon shooting upwards, the Bullet flies with more violence than being shot pointblank, or shooting downward? 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 blows which fall Perpendicular upon a wall, are more violent than those which strike byas-wise or glaunsingly. 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 Random, which is near half of the right, conveys her Bullet with a far greater violence than that which is shot at point blank, or mounted parallel to the Horizon. The common reason is, that shooting high, the fire carries the bowl a longer time in the air, and the air moves more facile upwards, than downwards, because that the airy circles that the motion of the bullet makes, are soon broken. Howsoever this be the general tenet, it is curious to find out the inequality of moving of the air; whether the Bullet fly upward, downward, or right forward, to produce a sensible dfference of motion; & some think that the Cannon being mounted, the Bullet pressing the powder maketh a greater resistance, and so causeth all the Powder to be inflamed before the Bullet is thrown out, which makes it to be more violent than otherwise it would be. When the Cannon is otherwise disposed, the contrary arrives, the fire leaves the Bullet, and the Bullet rolling from the Powder resists less: and it is usually seen, that shooting out of a Musket charged only with Powder, to shoot to a mark of Paper placed Point blank, that there are seen many small holes in the paper, which cannot be other than the grains of Powder which did not take fire: but this latter accident may happen from the overcharging 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 form whatsoever: and by some it hath been imagined, that a Bullet shot in this fashion hath been consumed, melted, and lost in the Air, 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 far 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 Trunk, Pipe, or other concavity, is conveyed so much the more violent, or better, by how much it is longer, either in respect of the Sight, Hearing, Water, Fire, etc. & the reason seems to hold in Cannons, because in those that are long, the fire is retained a longer time in the concavity of the Piece, and so throws out the Bullet with more violence; and experience lets us see that taking Cannons of the same boar, but of diversity 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 foot of length. Now the usual Cannon carries 600 Paces, some more, some less, 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 firm ground, about 10 or 12 foot, etc. It hath been seen lately in Germany, where there were made Pieces from 8 foot long to 17 foot of like boar, 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 less his force was: therefore the length ought to be in a mean measure, and it is often seen, 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 mean proportion of length to be taken, otherwise the flame of the fire will be over-pressed with Air: 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, etc. when they are always augmented by certain proportion. HEre we shall show things no less admirable, as recreative, and yet so certain and easy to be demonstrated, that there needs not but Multiplication only, to try each particular: and first, Of grains of Mustardseed. FIrst, therefore it is certain that the increase of one grain of Mustardseed for 20 years' 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 Mustardseed: Now because this seems but words, it must be proved by Art, as may be done in this wise, as suppose one Mustardseed sown to bring forth a tree or branch, in each extendure of which might be a thousand grains: but we will suppose only a thousand in the whole tree, and let us proceed to ●0 years, every seed to bring forth yearly a thousand grains, now multiplying always by a thousand, in less than 17 years you shall have to many grains which will surpass 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 grains of sand will be sufficiently expressed with 49 cyphers, but the number of grains of Mustardseed at the end of 17 years will have 52 cyphers: and moreover, grains of Mustardseed, are far greater than these of the sands: it is therefore evident that at the seventeenth year, all the grains of Mustardseed which shall successively spring from one grain only, cannot be contained within the limits of the whole firmament; what should it be then, if it should be multiplied again by a thousand for the ●8 year: and that again by a thousand for every years increase until you come to the 20 year, it's a thing as clear as the day, that such a heap of Mustardseed would be a hundred thousand times greater than the Earth: and bring only but the increase of one grain in 20 years. Of Pigs. SEcondly, is it not a strange proposition, to say that the great Turk with all his Revenues, is not able to maintain for one years' time, all the Pigs that a Sow may pig with all her race, that is, the increase with the increase unto 12 years: this seems impossible, yet it is most true, for let us suppose and put, the case, that a Sow bring forth but 6, two Males, and 4 Females, and that each Female shall bring forth as many every year, during the space of 12 years, at the end of the time there will be found above 3● millions of Pigs: now allowing a crown for the maintenance of each Pig for a year, (which is as little as may be, being but near a half of a farthing allowance for each day;) there must be at the least so many crowns to maintain them, one a year, viz. 33 millions, which exceeds the Turks revenue by much. Of grains of Corne. THirdly, it will make one astonished to think that a grain of Corn, with his increase successively for the space of 12 years will produce in grains 24414062●000000000000, which is able to load almost all the creatures in the World. To open which, let it be supposed that the first year one grain being sowed brings forth 50, (but sometimes there is seen 70, sometimes 100 fold) which grains sown the next year, every one to produce 50, and so consequently the whole and increase to be sown every year, until 12 years be expired, there will be of increase the aforesaid prodigious sum of grains, viz. 244140625000000000000, which will make a cubical heap of 6258522 grains every way, which is more than a cubical body of 31 miles every way: for allowing 40 grains 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 room to contain the aforesaid quantity of Corn: and suppose a bushel of Corn were equal unto two Cubic feet, which might contain twenty hundred thousand grains then would there be 122070462500000. bushels, and allowing 30 bushels to a Tun, 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 Tun to each ship a: quantity so great that the Sea is scarce able to bear, or the universal world able to find vessels to carry it. And if this Corn should be valued at half 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 sow one grain of Corn; and to continue it in sowing, the increase only for 12 years 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 year, for at the end of 16 years, 100 Sheep 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, which is above 1 million 6 hundred 38 thousand pounds, a fair increase of one Sheep: and a large portion for a Child if it should be allotted. Of the increase of Codfish, Carp, etc. FIfthly, if there be any creatures in the world that do abound with increase or fertility, it may be rightly attributed to fish; for they in their kinds produce such a great multitude of Eggs, and brings forth so many little ones, that if a great part were not destroyed continually, within a ●ittle while they would fill all the Sea, Ponds, and Rivers in the world; and it is easy to show how it would come so to pass, only by supposing them to increase without taking or destroying them for the space of 10 or 12 years: having regard to the solidity of the waters which are allotted for to lodge and contain 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 were saved after the deluge or Noah's flood) should spring such a world of people to begin a Monarchy under Nimrod, being but 200 years after the flood, and that amongst them should be raised an army of two hundred thousand fight men: But it is easily proved if we take but one of the Children of Noah, and suppose that a new generation of people begun at every 30 years, and that it be continued to the seventh generation which is 200 years; for then of one only family there would be produced one hundred and eleven thousand souls, three hundred and five to begin the world: though in that time men lived longer, and were more capable of multiplication and increase: which number springing only from a simple production of one yearly, would be far greater, if one man should have many wives, which in ancient times they had: from which it is also that the Children of Israel, who came into Egypt but only 70 souls, yet after 210 years' captivity, they came forth with their hosts, that there were told six hundred thousand fight 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 adjoin many families together? Of the increase of numbers. SEventhly, what sum of money shall the City of London be worth, if it should be sold, and the money be paid in a year after this manner: the first week to pay a pin, the second week 2 pins, the third week 4 pins, the fourth week 8 pins, the fifth week 16 pins. and so doubling until the 52 weeks, or the year be expired. Here one would think that the value of the pins 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 pins would amount unto the sum of 4519599628681215, and if we should allow unto a quarter a hundred thousand pins, the whole would contain ninety eight millions, four hundred thousand Tun: which is able to load 45930 Ships of a thousand Tun apiece: and if we should allow a thousand pins for a penny, the sum of money would amount unto above eighteen thousand, eight hundred and thirty millions of pounds sterling, an high price to sell a City at, yet certain, according to that first proposed. So if 40 Towns were sold upon condition to give for the first a penny, for the second 2 pence, for the third 4 pence, etc. 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 four thousand five hundred and fourscore 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 placed in a strait line or right form, 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 half 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 musical 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 Arithmetic, do present unto them a world of confusion and difficulty. 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 own doubts, entering often into a Labyrinth in the search thereof: or if there were 10 voices, how many several notes might there be? These are propositions of such facility, that a child 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 term assigned: so the 6 number that is against 6 in the Table, is 720, and so many (hangs may be made upon 6 Bells, upon 5 there are 120, etc. In like manner against 10 in the Table is 3618800, that is, three millions, six hundred twenty eight thousand, eight hundred: which shows that 10 voices may have so many consorts, each man keeping his own 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 drawn this, or the like proposition. Suppose that 7 Scholars were taken out of a free School to be sent to an University, there to be entertained in some College at commons for a certain sum of money, so that each of them have two meals daily, and no longer to continue there, then that sitting all together upon one bench or form at every meal, 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 entertain them? (who being not skilled in this fetch may answer unadvisedly.) It is most certain that there will be five thousand and forty several 1 a 1 2 b 2 6 c 3 24 d 4 120 e 5 7●0 f 6 5040 g 7 403●0 h 8 362880 i 9 3628800 k 10 39916800 l 11 479001600 m 12 6227020800 n 13 87178291200 o 14 1307674368000 p 15 20922789888000 q 16 355687537996000 r 17 6402375683928000 s 18 121645137994632000 t 19 2432902759892640000 u 20 51090957957745440000 w 21 1124001075070399680000 x 22 25852024726619192640000 y 23 6●0448593438860623360000 z 24 positions or change in the seating, which makes 14 years' time wanting 10 weeks and 3 days. Hence from this mutability of transmutation, it is no marvel 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: which Alphabet of 24 letters may be varied so many times, viz. 620448593438860623360000 which is six hundred twenty thousand, four hundred forty eight millions of millions of millions five hundred ninety three thousand, four hundred thirty eight million of million, & more. Now allowing that a man may read or speak one hundred thousand words in an hour which is twice more words than there are contained in the Psalms of David, (a task too great for any man to do in so short a time) and if there were four thousand six hundred and fifty thousand millions of men, they could not speak these words (according to the hourly proportion aforesaid) in threescore and ten thousand years; which variation & transmutation of letters, if they should be written in books, allowing to each leaf 28000 words, (which is as many as possibly could be inserted,) and to each book a ream or 20 choir 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 contain 250 Galleries of 20 foot broad apiece, it would contain four hundred mill●ons of the said books: so there must be to contain the rest no less 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 certain and infallible in computation. Of a Servant hired upon certain conditions. A Servant said unto his Master, that he would devil with him all his life-time, if he would but only lend him land to sow one grain of Corn with all his increase for 8 years' time; how think you of this bargain? for if he had but a quarter of an inch of ground for each grain, and each grain to bring forth yearly of increase 40 grains, the whole sum would amount unto, at the term aforesaid, 6553600000000 grains: 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 fourteen thousand and four hundred millions of grains, which is 14400000000. thus dividing the aforesaid 6553600000000, the Quotient will be 455, and so many square miles of land must there be to sow the increase of one grain of Corn for 8 years, which makes at the least four 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 year, to be continued for 8 years; a pretty pay for a Master's Servant 8 years' service. PROBLEM. LXXXV. Of Fountains, Hydriatiques, Machinecke, and other experiments upon water, or other liquor. 1. First how to make water at the foot of a mountain 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 fountain A, to the top of the Mountain B; and so to descend on the other side a little lower than the Fountain, as at C. then make a hole in the Pipe at the top of the Mountain, 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 again at B, that no air get in: then unstop the end at A, & at C; then will the water perpetually run 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 vessel without opening the bunghole, and without making any other hole, than that by which it runs out at the top? IN this problem there is nothing but to take a bowed pipe of Glass, and put it into the faucets hole, and stopping it close about: for than you shall see the wine or liquor to ascend in this Pipe, until 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 bunghole. 3. Thirdly, how is it that it is said that a vessel holds more water being placed at the foot of a Mountain, than standing upon the top of it? THis is a thing most certain, because that water and all other liquor disposeth itself 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 vessel is farther from the Centre of the earth, the surface of the water makes a greater sphere, and therefore less gibbous, or swelling over the vessel: from whence it is evident that a vessel 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 bottom of the Mountain holds more water, than being placed on the top of the Mountain. First, therefore one may conclude, that one and the same vessel will always hold more: by how much it is nearer the Centre of the earth. Secondly, if a vessel be very near 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 coming to the Centre will 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 Pail 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 level, but descending it swells and becomes more gibbous. 4. Fourthly, to conduct water from the top of one Mountain, to the top of another. AS admit on the top of a Mountain there is a spring, and at the top of the other Mountain there are Inhabitants which want water: now to make a bridge from one Mountain to another, were difficult and too great a charge; by way of Pipes it is easy and of no great price: for if at the spring on the top of the Mountain be placed a Pipe, to descend into the valley, and ascend to the other Mountlaine, the water will run naturally, and continually, provided that the spring be somewhat higher than the passage of the water at the Inhabitants. 5. Fifthly, of a fine Fountain which spouts water very high, and with great violence by turning of a Cock. LEt there be a vessel as AB, made close in all his parts, in the middle of which let CD be a Pipe open at D near the bottom, and then with a Squirt squirt in the water at C, stopped above by the cock or faucet C, with as great violence as possible you can; and turn the cock immediately. Now there being an indifferent quantity of water and air in the vessel, the water keeps itself in the bottom, and the air which was greatly pressed, seeks for more place, that turning the Cock the water issueth forth at the Pipe, and flies very high, and that especially if the vessel be a little heated: some make use of this for an Ewer to wash hands withal, and therefore putting a movable Pipe above C, such as the figure showeth: which the water will cause to turn 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 turns it, the water descends always 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 Engineer admirable in all Mathematical Arts invented this Instrument to wash King Hieroys great vessels, as some Authors say, also to water the fields of Egypt, as Diodorus witnesseth: and Cardanus reporteth that a Citizen of Milan having made the like Engine, thinking himself to be the first inventor, conceived such exceeding joy, that he be came mad, foll. 2. Again a thing may ascend by descending, if a spiral line be made having many circulations or revolutions; the last being always lesser than the first, yet higher than the Plain supposed it is most certain that then putting a ball into it, and turning the spiral line so, that the first circulation may be perpendicular, or touch always the supposed Plain: the ball shall in descending continually ascend, until at last it come to the highest part of the spiral 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 always by a continual motion and revolution. And this experiment may be more useful, naturally made with a thread of ●ron, or Latin 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 tied some weight at it in such sort, that the water may easily drop if one lift up the said thread: these Helices' or revolutions, notwithstanding will remain 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 equality amongst themselves, and the whirling or turning motion be quick, the sight will be so deceived, that producing the action it will seem to the ignorant no less than a Miracle. 7. Seventhly, of another fine Fountain of pleasure. THis is an Engine that hath two wheels with Cogs, or teeth as AB, which are placed within an Oval CD, in such sort, that the teeth of the one, may enter into the notches of the other; but so just that neither air nor water may enter into the Oval coffer, either by the middle or by the sides, for the wheel must join so near to the sides of the coffer, that there be no vacuity: to this there is an axletree with a handle to each wheel, so that they may be turned, and A being turned, that turneth the other wheel that is opposite: by which motion the air that is in E, & the water that is carried by the hollow of the wheels of each side, by continual motion, is constrained to mount and fly out by the funnel F: now to make the water run what ways one would have it, there may be applied upon the top of the Pipe F, two other movable Pipes inserted one within another; as the figure showeth. But here note, that there may accrue some inconveniency in this Machine seeing that by quick turning the Cogs or teeth of the wheels running one against another, may near break them, and so give way to the air to enter in, which being violently enclosed will escape to occupy the place of the water, whose weight makes it so quick: howsoever, if this Machine be curiously made as an able workman may easily do, it is a most sovereign Engine, to cast water high and far off for to quench fires. And to have it to rain to a place assigned, accommodate a socket having a Pipe at the middle, which may point towards the place being set at the top thereof, and so having great discretion in turning the Axis of the wheel, it may work exceeding well, and continue long. 8. Eightly, of a fine watering pot for gardens. THis may be made in form of a Bottle according to the last figure or such like, having at the bottom many small holes, and at the neck of it another hole somewhat greater than those at the bottom, which hole at the top you must unstop when you would fill this watering pot, for than it is nothing but putting the lower end into a pail of water, for so it will fill itself by degrees: and being full, put your thumb on the hole at the neck to stop it, for than may you carry it from place to place, and it will not sensibly run out, something it will, and all in time (if it were never so close stopped) contrary to the ancient tenet in Philosophy, that air will not penetrate. 9 Ninthly, how easily to take wine out of a vessel at the bunghole, without piercing of a hole in the vessel? IN this there is no need but to have a Cane or Pipe of Glass or such like, one of the ends of which may be closed up almost, leaving some small hole at the end; for than if that end be set into the vessel at the bunghole, 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 full of wine, then opening a little the top above, you may fill a Glass or other Pot with it, for as the Wine issueth out, the air cometh 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 vessel, and keep that which floweth out over, saying it is always equal to the thing cast into the water: let i● is more nea●er this way to pour into a vessel such a quantity of water, which may be thought sufficient to cover the body or magnitude, and make a mark how high the water is in the vessel, then pour out all this water into another vessel, and let the body or magnitude be placed into the first vessel; then pour in water from the second vessel, until it ascend unto the former mark made in the first vessel, so the water which remains in the second vessel is equal 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 find the weight of water. SEeing that 574/1000 part of an ounce weight, makes a cubical inch of water: and every pound weight Haverdepoize makes 27 cubical inches, and 1 9/●; fere, and that ● Gallons and a half wine measure makes a foot cubical, it is easy by inversion, that knowing the quantity of a vessel in Gallons, to find his content in cubical feet or weight: and that late famous Geometrician Master Brigs found a cubical foot of water to weigh near 62 pound weight Haverdepoize But the late learned Simon Stevin found a cubical foot of water to weigh 65 pound, which difference may arise from the inequality of water; for some waters are more ponderous than others, and some difference may be from the weight of a pound, and the measure of a foot: thus the weight and quantity of a solid foot settled, it is easy for Arithmeticians to give the contents of vessels or bodies which contain liquids. 12. To find the charge that a vessel may carry as Ships, Boats, or such like. THis is generally conceived, that a vessel may carry as much weight as that water weigheth, which is equal unto the vessel in bigness, in abating only the weight of the vessel: we see that a barrel of wine or water cast into the water, will not sink to the bottom, 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 vessel were loaden with lead, so much should the water weigh: hence it is that Mariners call Ships of 50 thousand Tons, because they may contain one or two thousand Tun, and so consequently carry as much. 13. How comes it that a Ship having safely sailed 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 kind of water than upon other; now the water of the Sea is thicker and heavier than that of Rivers, Wells, or Fountains; therefore the loading of a vessel 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 vessels more easy to swim, but it is an abuse; for if the loading of a Ship be no heavier than the water that would occupy 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 water be no thicker than a leaf of paper, and weigheth but an ounce under a heavy body, it will support it, as well as if the water under it weighed ten thousand pound weight: hence it is if there be a vessel capable of a little more than a thousand pound weight of water, you may put into this vessel a piece of wood, which shall weigh a thousand pound weight; (but lighter in his kind than the like of magnitude of water:) for then pouring in but a quart of water or a very little quantity of water, the wood will swim on the top of it, (provided that the wood touch not the sides of the vessel:) which is a fine experiment, and seems admirable in the performance. 14. How a gross body of mettle may swim upon the water? THis is done by extending the mettle into a thin Plate, to make it hollow in form of a vessel; so that the greatness of the vessel which the air with it containeth, be equal to the magnitude of the water, which weighs as much as it, for all bodies may swim without sinking, if they occupy the place of water equal in weight unto them, as if it weighed 12 pound it must have the place of 12 pound of water: hence it is that we see floating upon the water great vessels of Copper or Brass, when they are hollow in form of a Cauldron. 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 show how much every kind of metal should be enlarged, to make it swim upon the water: which doth depend upon the proportion that is between the weight of the water and each metal. Now the proportion that is between metals and water of equal magnitude, according to some Authors, is as followeth. A magnitude of 10 pound weight of water will require for the like magnitude of Gold. 187 ½ Lead. 116 ½ Silver. 104 Copper. 91 Iron. 81 Tin. 75 From which is inferred, that to make a piece of Copper of ●0 pound weight to swim, 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 and water of like magnitudes in their ponderosities, are as before, as ●0 to 91. 15. How to weigh the lightness of the air? PLace a Balance of wood turned upside down into the water, that so it may swim, then let water be enclosed within some body, as within a Bladder or such like, and suppose that such a quantity of air should weigh one pound, place it under one of the Balances, and place under the other as much weight of lightness as may counterbalance and keep the other Balance that it rise not out of the water: by which you shall see how much the lightness is. But without any Balance do this; take a cubical hollow vessel, or that which is cylindrical, which may swim on the water, and as it sinketh by placing of weights upon it, mark how much, for than if you would examine the weight of any body, you have nothing to do but to put it into this vessel, and mark how deep it sinks, for so many pound it weighs as the weights put in do make it so to sink. 16. Being given a body, to mark it about, and show how much of it will sink in the water, or swim above the water. THis is done by knowing the weight of the body which is given, and the quantity of water, which weighs as much as that body; for then certainly it will sink so deep, until it occupieth the place of that quantity of water. 17. To find how much several mettle or other bodies do weigh less in the water than in the air: TAke a Balance, & weigh (as for example) 9 pound of Gold, Silver, Led, or Stone in the air, so it hang in aequilibrio; then coming to the water, take the same quantity of Gold Silver, Led, or Stone, and let it softly down into it, and you shall see that you shall need a less counterpoise in the other Balance to counterbalance it: wherefore all solids or bodies weigh less in the water than in the air, and so much the less it will be, by how much the water is gross and thick, because the weight finds a greater resistance, and therefore the water supports more than air; and further, because the water by the ponderosity is displeased, and so strives to be there again, pressing to it, by reason of the other waters that are about it, according to the proportion of his weight. Archimedes demonstrateth, that all bodies weigh less in the water (or in like liquor) by how much they occupy place: and if the water weigh a pound weight, the magnitude in the water shall weigh a pound less than in the air. 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, Led the 12 part, Silver the 10 part, Iron the 8 part, Tin the 7 part and a little more: wherefore in material and absolute weight, Gold in respect of the water that it occupieth weigheth 18, and ¾ times heavier than the like quantity of water, that is, as 18 ¾ to the Quicksilver 15 times, Led 11 and ⅗, Silver 10 and ⅔, Copper 9 and 1/10, Iron 8 and ½, and Tin 8 and 1/●. Contrarily in respect of greatness, 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 balance having like weight in each scale, and hanging in aequilibrio in the air, 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 easy to be resolved by considering different mettles, which though they weigh equal in the air, yet in the water there will be an apparent difference; as suppose so that in the scale of each Balance be placed 18 pound weight of several metals, the one Gold, and the other Copper, which being in aequilibrio in the air, placed in the water, will not hang so, because that the Gold los●eth near the 18 part of his weight, which is about 1 pound, and the Copper loseth but his 9 part, which is 2 pound: wherefore the Gold in the water weigheth but 17 pound, and the Copper 16 pound, which is a difference most sensible to confirm that point. 19 To show what waters are heavier one than another, and how much. PHysicians have an especial respect unto this, judging that water which is lightest is most healthful and medicinal for the body, & Seamen know that the heaviest waters do bear most, and it is known which water is heaviest thus. Take a piece of wax, and fasten Lead unto it, or some such like thing that it may but precisely swim, for than it is equal to the like magnitude of water, than put it into another vessel which hath contrary water, and if it sink, then is that water lighter than the other: but if it sink not so deep, than it argueth the water to be heavier or more grosser than the first water, or one may take a piece of wood, and mark the quantity of sinking of it into several waters, by which you may judge which is lightest or heaviest, for in that which it sinks 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 enclosed in a vessel, being constrained to dilate itself, doth weigh so much as though there were in the concavity of it a solid body of water. There are many ways to experiment this proposition, but to verify it, it may be sufficient to produce two excellent ones only: 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 air; then make a Balance that one of the scales may environ 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 down the hundred pound in the other Balance. PROBLEM. LXXXVI. Of sundry Questions of Arithmetic, 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 was this that the Poets sung, and that which the vulgar believes; nay, that which long ago certain Philosophers to Gelon King of Sicily reported, that the grains of sand were innumerable: But I answer with Archimedes, that not only one may number those which are at the border and about the Sea; but those which are able to fill the whole world, if there were nothing else but sand; and the grains of sands admitted to be so small, that 10 may make but one grain of Poppy: for at the end of the account there need not to express them, but this number 30840979456, and 35 cyphers at the end of it. Clavius and Archimedes make it somewhat more; because they make a greater firmament than Tycho Brahe doth; and if they augment the Universe, it is easy for us to augment the number, and declare assuredly how many grains of sand there are requisite to fill another world, in comparison that our visible world were but as one grain of sand, an atom or a point; for there is nothing to do but to multiply the number by itself, which will amount to ninety places, whereof twenty are these, 95143798134910955936, and 70 cyphers at the end of it: which amounts to a most prodigious number, and is easily supputated: for supposing that a grain of Poppy doth contain 10 grains of sand, there is nothing but to compare that little bowl of a grain of Poppy, with a bowl of an inch or of a foot, & that to be compared with that of the earth, and then that of the earth with that o the firmament; and so of the rest. 2. Divers metals being melted together in one body, to find the mixture of them. THis what a notable invention of Archimedes, related by Vitrivius in his Architecture, where he reporteth that the Goldsmith which King Hiero employed for the making of the Golden Crown, 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 Crown, if there had been made mixture of any other metal with the Gold. The way which he found out was by bathing himself; for as he entered into the vessel of water, (in which he bathed himself) 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 Bowl of pure Gold, Silver, or other metal were cast into a vessel of water, the water proportionally according to the thing cast in would ascend; and so by way of Arithmetic the question lay open to be resolved: who being so intensively taken with the invention, leaps 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 equal to the weight of the Crown, and therefore unequal in magnitude or greatness; and then knowing the several quantities of water which was answerable to the Crown, and the several Masses, he subtly collected, that if the Crown occupied more place within the water than the Mass of Gold did: it appeared that there was Silver or other metal melted with it. Now by the rule of position, suppose that each of the three Masses weighed 18 pound a piece, and that the Mass of Gold did occupy the place of one pound of water, that of Silver a pound and a half▪ and the Crown one pound and a quarter only: then thus he might operate the Mass of Silver which weighed 18 pounds, cast into the water, did cast out half a pound of water more than the Mass of Gold, which weighed 18 pound, and the Crown which weighed also 18 pound, being put into a vessel full of water, threw out more water than the Mass of Gold by a quarter of a pound, (because of mixed metal which was in it:) therefore by the rule of proportion, if half a pound of water (the excess) be answerable to 18 pound of Silver, one quarter of a pound of excess shall be answerable to 9 pound of Silver, and so much was mixed in the Crown. Some judge the way to be more facile by weighing the Crown first in the air, then in the water; in the air 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 we will suppose that Gold and Copper is mixed together, it will weigh less than 17 pound, yet more than 16 pound, and that according to the proportion mixed: let it then be supposed that it weighed in the water 16 pound and 3 quarters, than might one say by proportion, if the difference of one pound of loss, which is between 16 and 17) be answerable to 18 pound, to what shall one quarter of difference be answerable to, which is between 17 and 16 ¾, and it will be 4 pound and a half; and so much Copper was mixed with the Gold. Many men have delivered sundry ways to resolve this proposition since Archimedes invention, and it were tedious to relate the diversities. Baptista Benedictus amongst his Arithmetical Theorems, delivers his way thus: if a Mass of Gold of equal bigness to the Crown did weigh 20 pound, and another of Silver at a capacity or bigness at pleasure, as suppose did weigh 12 pound, the Crown or the mixed body would weigh more than the Silver, and lesser than the Gold, suppose it weighed 16 pound which is 4 pound less than the Gold by 8 pound, then may one say, if 8 pound of difference come from 12 pound of Silver, from whence comes 4 pound which will be 6 pound and so much Silver was mixed in it, etc. 3. Three men bought a quantity of wine, each paid alike, and each was to have alike; it happened at the last partition that there were 21 Barrels, of which 7 were full, 7 half full, and 7 empty, how must they share the wine and vessels, that each have as many vessels one as another, & as much wine one as another? THis may be answered two ways 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 Barrels full, and as many empty ones, and one which is half full; so he shall have 7 vessels and 3 Barrels, and a half of liquor; and one of the other shall in like manner have as much, so there will remain for the third man 1 Barrel full, 5 which are half full, and 1 empty, and so every one shall have alike both in vessels and wine. And generally to answer such questions, divide the number of vessels by the number of persons, and if the Quotient be not an entire number, the question is impossible; but when it is an entire number, there must be made as many parts as there are 3 persons, seeing that each part is less than the half 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 less than half 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 answer is, 2 foot: for by Pythagoras rule the square of DB, the Hypothenusal is equal to the square of DA 6, & AB 10. Now if DA be 6 foot, and AB 10 foot, the squares are 36 and 100, which 36 taken from 100 rests 64, whose Roote-quadrate is 8 so the foot of the Ladder being now at D, the top will be at C, 2 foot lower than it was when it was at B. PROBLEM. LXXXVII. Witty suits or debates between Caius and Sempronius, upon the form of figure's, which Geometricians call Isoperimeter, or equal in circuit or compass. Marvel ●ot at it if I make the Mathematics take place at the Ba●●e, and if I set forth here B●rtoleus, who witnesseth of himself, that being then an ancient Doctor in the Law, he himself took upon him to learn the elements and principles of Geometry, by which he might set forth certain Laws touching the divisions of Fields, Waters, Islands, and other incident places: now this shall be to show in passing by, that these sciences are profitable and behooveful for Judges, Counsellors, or such, to explain many things which fall out in Laws, 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 himself, prayed Caius to change with him for a field which should be equivalent unto his, and the bargain 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 subtlest nor wisest in the world accepted his bargain at the first, but afterwards having conferred with a Land-measures and Mathematician, found that he was overreached in his bargain, and that his field contained 36 square measures, and the other field had but 27 measures, (a thing easy to be known by multiplying the length by the breadth:) Sempronius contested with him in suit of Law, and argued that figures which have equal Perimeter or circuit, are equal amongst themselves: my field, saith he, hath equal circuit with yours, therefore it is equal unto it in quantity. Now this was sufficient to delude a Judge which was ignorant in Geometrical proportions, but a Mathematician will easily declare the deceit, being assured that figures which are Isoperemiter, or equal in circuit, have not always equal capacity or quantity: 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, equal 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 infallible rule of experience, that a square is more capable of quantity 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 equal, otherwise there might be that a regular Triangle, having 24 measures in circuit might have more capacity 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 quantity is but 11. and if it had 6 every way, it gives the same Perimeter, viz. 24. but a quantity of 36 as before. 2. Incident. SEmpronius having borrowed of Caius a sack of Corn, 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 Corn, which had each of them 6 foot high & 1 foot broad: who believed that if the sacks 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 half 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 sacks, is but half of the greater; and so Caius was deceived in half his Corne. 3. Incident. SOme one from a common Fountain of a City hath a Pipe of water of an inch diameter; 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 big, to have just twice as 〈…〉 before: but if the Magistrate of the City understood Geometrical proportions, he would soon cause it to be amended, & show that he hath not only taken twice as much water as he had before, but four times as much: for a Circular hole which is two inches diameter is four times greater than that of one inch, and therefore will cast out four times as much water as that of one inch, and so the deceit is double also in this. Moreover, if there were a heap of Corn of 20 foot every way, which was borrowed to be paid next year▪ the party having his Corn in heaps of 12 foot every way, and of 10 foot every way, proffers him 4 heaps of the greater or 7 heaps of the lesser, for his own heap of 20 every way, which was lent: here it seems that the proffer is fair, nay with advantage, yet the loss would be near 1000 foot. Infinite of such causes do arise from Geometrical figures, which are able to deceive a Judge or Magistrate, which is not somewhat seen in Mathematical Documents. PROBLEM. LXXXVIII. Containing sundry Questions in matter of Cosmography. FIrst, it may be demanded, where is the middle of the world? I speak not here Mathematically, but as the vulgar people, who ask, where is the middle of the world? in this sense to speak absolutely there is no point which may be said to be the middle of the surface, for the middle of a Globe is every where: notwithstanding the Holy Scriptures speak respectively, and make mention of the middle of the earth, and the interpreters apply it to the City of Jerusalem placed in the middle of Palestina, and the habitable world, that in effect taking a map of the world, 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 City 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 compass of the world? FRom the surface of the earth unto the Centre according to ancient traditions, is 3436. miles, so the whole thickness is 6872 miles, of which the whole compass or circuit of the earth is 21600 miles. From the Centre of the earth to the Moon there is near 56 Semidiameters of the earth, which is about 192416 miles. unto the Sun 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 Tycho Brahe. From these measures one may collect by Arithmetical supputations, many pleasant propositions in this manner. First, if you imagine there were a hole through the earth, and that a Millstone should be let fall down into this hole, and to move a mile in each minute of time, it would be more than two days and a half before it would come to the Centre, and being there it would hang in the air. Secondly, if a man should go every day 20 miles, it would be three years wanting but a fortnight, before he could go once about the earth; and if a Bird should fly round about it in two days, then must the motion be 450 miles in an hour. Thirdly, the Moon runs a greater compass each hour, than if in the same time she should run twice the Circumference of the whole earth. Fourthly, admit it be supposed that one should go 20 miles in ascending towards the heavens every day, he should be above 15 years before he could attain to the Orb of the Moon. Fifthly, the Sun makes a greater way in one day than the Moon doth in 20 days, because that the Orb of the Sun's circumference is at the least 20 times greater than the Orb of the Moon. Sixthly, if a Millstone should descend from the p●ace of the Sun a thousand miles every hour, (which is above 15 miles in a minute, far beyond the proportion of motion) it would be above 163 days before it would fall down to the earth. Seventhly, the Sun in his proper sphere moves more than seven thousand five hundred and seventy miles in one minute of time: now there is no Bullet of a Cannon, Arrow, Thunderbolt, or tempest of wind that moves with such quickness. Eightly, it is of a far higher nature to consider the exceeding and unmoveable quickness of the starry firmament, for a star being in the Aequator, (which is just between the Poles of the world) makes 12598666 miles in one hour which is two hundred nine thousand nine hundred and seventy four miles in one minute of time: & if a Horseman should ride every day 40 miles, he could not ride such a compass in a thousand years as the starry firmament moves in one hour, which is more than if one should move about the earth a thousand times in one hour, and quicker than possible thought can be imagined: and if a star should fly in the air about the earth with such a prodigious quickness, it would burn 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 find the Bissextile year, the Dominical letter, and the letters of the month. 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 Leap-year, and that which remains of the division shows the leap-yeare, as if one remain, it shows that it is the first year since the Bissextile or Leap-year, if two, it is the second year etc. and if nothing remain, than it is the Bissextile or Leap-yeare, and the Quotient shows you how many Bissextiles or Leap-yeares there are contained in so many years. To find 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 Sun or whole revolution of the Dominical letters) and that which remains is the number of joints, which is to be accounted upon the fingers by Filius esto Dei, coelum bonus accipe gratis: and where the number ends, that finger it showeth the year which is present, and the words of the verse show the Dominical letter. Example. DIvide 123 by 28 for the year (and so of other years) and the Quotient is 4, and there remaineth 11, for which you must account 11 words: Filius esto Dei, etc. upon the joints beginning from the first joint of the Index, and you shall have the answer. For the present to know the Dominical letter for each month, account from January unto the month required, including January, and if there be 8, 9, 7, or 5, you must begin upon the end of the finger from the thumb and account, Adam degebat, etc. as many words as there are months, for then one shall have the letter which begins the month; then to know what day of the month it is, see how many times 7 is comprehended in the number of days, and take the rest, suppose 4, account upon the first finger within & without by the joints, 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 joint of the first finger or Index: and so you have the present year, the Dominical letter, the letter which begins the Month, and all the days of the Month. PROBLEM. XCIII. To find the New and Full Moon in each Month. Add to t●e Epact for the year, the Month from March, then subtract that surplus from 30, and the rest is the day of the Month that it will be New Moon, and adding unto it 14, you shall have that Full Moon. Note THat the Epact is made always by adding 11 unto 30, and if it pass 30, subtract 30, and add 11 to the remainder, and so ad infinitum: as if the Epact were 12, add 11 to it makes 23 for the Epact next year, to which add 11 makes 34, subtract 30, rests 4 the Epact for the year after, and 15 for the year following that, and 26 for the next, and 7 for the next, etc. PROBLEM. XCIV. To find the Latitude of ● Country. THose that dwell between the North-Pole and the Tropic 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 sun's distance by instrumental observation from the zenith at noon, and add the declination of the sun for that day to it: so the Aggragate showeth such is the Latitude, or Poles height of that Country. Now the declination of the sun 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, than the sun's declination for that day must be taken out of the distance of the sun from the zenith at noon: so shall you have the Latitude, as before. PRBOLEM XCV. Of the Climates of country's, and to find in what Climate any country is under. CLimates as they are taken Geographically signify nothing else but when the length of the longest day of any place, is half an hour longer, or shorter than it is in another place (and so of the shortest day) and this account to begin from the Equinoctial Circle, seeing all Countries under it have the shortest and longest day that can be but 12 hours; But all other Countries that are from the Equinoctial 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 Equinoctial 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 half an hours time: then from each Polar Circle, to each Pole there are reckoned 6. other Climates which differ one from another by a month's time: so the whole earth is divided into 60 Climates, 30 being allotted to the Northern Hemisphere, and 30● to the Southern Hemisphere. And here note, that though these Climates which are between the Equinoctial and the Polar Circles are equal one unto the other in respect of time, to wit, by half an hour; yet the Latitude, breadth, or internal, contained between Climate and Climate, is not equal: and by how much any Climate is farther from the Equinoctial than another Climate, by so much the lesser is the interval between that Climate and the next: so those that are nearest the Equinoctial are largest, and those which are farthest off most contracted: and to find what Climate any Country is under: subtract the length of an Equinoctial day, to wit, 12 hours from the length of the longest day of that Country; the remainder being doubled shows the Climate: So at London the longest day is near 16 hours and a half; 12 taken from it there remains 4 hours and a half, which doubled makes 9 half hours, that is, 9 Climates; so London is in the 9 climate. PROBLEM. XCVI. Of Longitude and Latitude of the Earth and of the Stars. LOngitude of a Country, or place, is an ark of the Aequator contained between the Meridian of the Azores, and the Meridian of the place, and the greatest Longitude that can be is 360 degrees. Note. That the first Meridian may be taken at pleasure upon the Terrestrial Globe or Map, for that some of the ancient Astronomers would have it at Hercules Pillars, which is at the straits at Gibraltar: Ptolemy placed it at the Canary Islands, but now in these latter times it is held to be near the Azores. But why it was first placed by Ptolemy at the Canary Islands, were because that in his time these Islands were the farthest western parts of the world that was then discovered. And why it retains his place now at Saint michael's near the 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 Sun Eastward, or that Ptolemy and others did hold it more convenient to begin from the Western part of the world and so account the Longitude Eastward from Country to Country that was then known; till they came to the Eastern part of Asia, rather than to make a beginning upon that which was unknown: and having made up their account of reckoning the Longitude from the Western part to the Eastern part of the world known, they supposed the rest to be all sea, which since their deaths hath been found almost to be another habitable world. To find the Longitude of a Country. IF it be upon the Globe, bring the Country to the Brazen Meridian, and whatsoever degree that Meridian cuts in the Equinoctial, that degree is the Longitude of that Place: if it be in a Map, then mark what Meridian passeth over it, so have you the Longitude thereof, if no Meridian pass over it, then take a pair of Compasses, and measure the distance between the Place and the next Meridian, and apply it to the divided parallel or Aequator, so have you the Longitude required. Of the Latitude of Country's. LAtitude of a Country is the distance of a Country from the Equinoctial, or it is an Ark of the Meridian contained between the Zenith of the place and the Aequator; which is twofold, viz. either North-Latitude or South-Latitude, either of which extendeth from the Equinoctial to either Pole, so the greatest Latitude that can be is but 90 degrees: If any Northern Country have the Arctic Circle vertical, 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 hours, if the Country have less 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 days, and if the Country be under the Pole, the Sun will make a Circular motion above the Earth, and be visible for a half year: so under the Pole there will be but one day, and one night in the whole year. To find the latitude of Country's. IF it be upon a Globe, bring the place to the Brazen Meridian, and the number of degrees which it meeteth therewith, is the Latitude of the place. Or with a pair of Compasses take the distance between the Country and the Equinoctial, which applied unto the Equinoctial will show the Latitude of that Country; which is equal to the Poles height; if it be upon a Map. Then mark what parallel passeth over the Country and where it crosseth the Meridian, that shall be the Latitude: but if ●o parallel passeth over it, then take the distance between the place and the next parallel, which applied to the divided Meridian from that parallel will show the Latitude of that place. To find the distance of places. IF it be upon a Globe: then with a pair of Compasses take the distance between the two Places, and apply it to the divided Meridian or Aequator, and the number of degrees shall show ●e distance; each degree being 60. miles. ●f it be in a Map (according to wright's pro●ection) take the distance with a pair of Compasses between the two places, and apply this distance to the divided Meridian on the Map right against the two places; so as many degrees as is contained 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 Stars. THe Declination of a star is the nearest distance of a Star from the Aequator; the Latitude of a Star is the nearest distance of a Sarre from the Ecliptic: the Longitude of a Star is an Ark of the Ecliptic contained between the beginning of Aries, and the Circle of the Stars Latitude, which is a circle drawn from the Pole of the Ecliptic unto the star, and so to the Ecliptic. The distance between two Sarres in Heaven is taken by a cross-staff or other Instrument, and upon a Globe it is done by taking between the feet of the Compasses the two Stars, and applying it to the Aequator, so have you the distance between those two starred. 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 certain days travel the one lived more days than the other, notwithstanding they died together in one and the sam● moment also? THis is easy to be answered: let one of them travel toward the West and the other towards the East: then that which goes towards the West followeth the Sun: and therefore shall have the day somewhat longer than if there had been no travel made, and that which goes East by going against the Sun, shall have the day shorter, and so in respect of travel though they die at one and the self same hour 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 travel upon the Saturday, the Christian going West, and the Saracen Eastwards, shall compass the Globe of the earth, thus the Christian at the conclusion shall gain a day, and the Saracen shall lose a day, and so meet with the Jew every one upon his own Sabbath. Certain fine observations. 1 UNder the Equinoctial 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 vertical. 2 In these Countries which are without the Tropical Circles, the Sun comes East and West every day for a half year, but being under the Equinoctial the Sun is never East, nor West▪ but twice in the year, 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 Tropics vertical: then at what time the Sun's Altitude is equal to his distance from any of the Equinoctial points, then t●e Sun is due East or West. 4 If a ship be between the Equinoctial and either of the Tropics, the Sun will come twice to one point of the compass in the forenoon, that is, in one and the same position. 5 Under the Equinoctial near Guinea there is but two sorts of winds all the year, 6 months a Northerly wind, and 6 months a Southerly wind, and the flux of the Sea is accordingly. 6 If two ships under the Equinoctial be 100 leagues asunder, and should sail Northerly until they were come under the Arctic circle, they should then be but 50 leagues asunder. 7 Those which have the Arctic circle, vertical: when the Sun is in the Tropic of Cancer, the Sun setteth not, but toucheth the western part of the Horizon. 8 If the compliment of the Sun's height at noon be found equal to the Sun's Declination for that day, than the equinoctial is vertical: or a ship making such an observation, the Equinoctial is in the Zenith, or direct over them, by which Navigators know when they cross the line, in their travels to the Indies, or other parts. 9 The Sun being in the Equinoctial, the extremity of the stile in any Sunne-dyall upon a plain, maketh a right line, otherwise it is Eclipticall, Hyperbolical, etc. 10 When the shadow of a man, or other thing upon a horizontal 〈◊〉 is equal unto it in length, then is the Sun 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 pleasure: and upon A, describe an Ark BC. then at the same opening, place one of the feet in B, and describe the Ark AC. Lastly, place one of the feet of the Compasses in C. and describe the Ark AB· so shall you have the spherical 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 plain Triangle, whether it be Equilateral, Isocelse, scaleve, Orthogonall, or Opigonall. PROBLEM. XCVIII. To divide a line in as many equal parts as one will, without compasses, or without seeing of it. THis Proposition hath a fallacy in it, & cannot be practised but upon a Maincordion: for the Mathematical 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 until you sound a third in music: if you would have the fourth part of the line, then find the fourth sound, a fifth, etc. so shall you have the answer. PROBLEM. XCIX. To draw a line which shall incline to another line, yet never meet: against the Axiom of Parallels. THis is done by help of a Conoeide line, produced by a right line upon one & the same plain, held in great account amongst the Ancients, and it is drawn 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, IN. transfer 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 IN. 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 plain, so that the Sun may shine on it both before noon and afternoon: in the centre of which Circle place a Gn●●on or wire perpendicular as AB, and an hour before noon mark the extremity of the shadow of AB, which suppose it be at C. describe a Circle at that semidiamiter CDF. then after noon mark when the top of the shadow of AB. toucheth the Circle, which admit in D; divide the distance CD into two equal parts, which suppose at E. draw the line EAF. which is the Meridian line, or line of North & South: now if the Ark of the Circle CD. be divided into degrees. place a Needle GH, upon a plain set up in the Centre, and mark 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 find at any time which way the wind is in ones Chamber, without going abroad? Upon the Plancking or floor of a Chamber, Parlour, or Hall, that you intent to have this device, let there come down 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 show 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, turn this Dart which is within: about which upon the plaster must be described a Circle divided into the 32 points of the Mariner's Compass pointed and distinguished to that end, then may it be marked by placi● to Compass by it; for having noted the North point, the East, &c▪ it is easy to note all the rest of the points: and so at any time coming into this Room, you have nothing to do but to look up to the Dart, which will point you out what way the wind bloweth at that instant. PROBLEM. CII. How to draw a parallel spherical 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 seen nor where the Compasses were placed, will seem very strange how possibly it could be drawn with such exactness, 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 only. THe way of this is easy: for having one's hat upon his head, come near 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 down the brim or edge of your hat until you may but see the other side of the water, then turn about the body in the same posture that it was before towards some plain, and mark where the sight by the brim of the hat glanceth 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 straws or two small sticks. TAke two straws or two sticks 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 until 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 equal to the height thereof above the level of the eye: to which if you add your own height you have the whole height. Otherwise. TAke an ordinary square which Carpenters or other workmen use, as HKL. and placing H. to the eye so that HK. be level, go back or come nearer until that by it you may see the top M. for then the distance from you to the height is equal to the height. PROBLEM. CV. How to make statues, letters, bowls, or other things which are placed in the side of a high building, to be seen below of an equal bigness. LEt BC. be a Pillar 7 yards high, and let it be required that three yards above the level 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 appear 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 Ark KINE. then from K▪ in the perpendicular KX, accounts▪ par●s, viz at L. which shall represent the former three yards, 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 SLIGHTALL, and draw the line SA. cutting the Ark VK. in N. then from K. in the perpendicular account 9 yards, which admit at T. draw TA, cutting YK. in O transfer the Ark MN, from A to P. and draw AP. which will cut the perpendicular in V. so a line drawn from the middle of VF. unto the visual 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 Ark MN, and transfer it from Q to R and draw ARE▪ which will cut the perpendicular in X so the line which passeth by the meddle of XW. perpendicular to the visual line AW, and AX. be the Diameter of the third Globe, to wit 5, 6. which measures transferred in the Pillar BC. which showeth 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 disguise or disfigure an Image, as a head, an arm, a whole body, etc. so that it hath no proportion the ears to become long: the nose as that of a swan, the mouth as a coaches entrance, &c yet the eye placed at a certain point will be seen in a direct & exact proportion. I Will not strive to set a Geometrical figure here, for fear it may seem too difficult to understand, but I will endeavour 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 Books of Paper, where you desire to have the figure disguised in such sort that the height pass athwart the hole of the Picture: then will it carry all the form of the Picture upon the Paper, but with deformity; follow these tracts and mark out the light with a Coals black head or Ink: and you have your desire. To find now the point where the eye must see it in his natural form: it is accustomed according to the order of Perspective, to place this point in the line drawn in height, equal to the largeness 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 wheel L, the axletree PB. upon which the Cannon is placed, at which end towards B, is placed a pillar AE· supported with props D, C, E, F, G▪ about which the Axletree 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 himself behind the wall QR, and so preserves itself from the Enemy's battery, by which means one may avoid many inconveniences which might arise: and moreover, one man may more easily replace it again for another shot by help of poles tied 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 showeth, pierced with holes, alike opposite one unto another as CD, and OF: & let L, and M, be the two bars of Iron which passeth through the holes GH, and F, K, the two supports, or props, AB. the Cannon, OPEN, the Lever, RS, the two notches in the Lever, and Q, the hook where the burden or Cannon is tied to. The rest of the operation is ●cill, that the youngest scholars or learners cannot fail to perform it: to teach Minerva were in vain, and it were to Mathematicians injury in the succeeding Ages. PROBLEM. CIX. How to make a Clock with one only wheel. MAke the body of an ordinary Dial, and divide the hour in the Circle into 12. parts: make a great wheel in height above the Axletree, to the which you shall place the cord of your counterpoise▪ so that it may descend, that in 1● hours of time your Index or Needle may make one revolution, which may be known by a watch which you may have by you: then put a balance which may stop the course of the Wheel, and give it a regular motion, and you shall see an effect as just from this as from a Clock with many wheels. PROBLEM. CX. How by help of two wheels to make a Child to draw up alone a hogshead of water at a time: and being drawn up shall cast out itself into another vessel as one would have it. LEt R be the Pit from whence water is to be drawn, P the hook to throw out the water when it is brought up (this hook must be movable) let AB be the Axis of the wheel SF, which wheel hath divers forks of Iron made at G, equally fastened at the wheel; let I, be a Card, which is drawn by K, to make the wheel S, to turn, which wheel S, bears proportion to the wheel T, as 8 to ●. let N be a Chain of Iron to which is tied the vessel O, and the other which is in the Pit: E● is a piece of wood which hath a mortes in 1, and ●, by which the Cord I, passeth, tied at the brickwall, as KH, and the other piece of timber of the little wheel as M, mortified in likewise for the chain to pass through: draw the Cord I, by K, and the wheel will turn, & so consequently the wheel T, which will cause the vessel O to raise; which being empty, draw the cord again by Y, and the other vessel 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 contain two great vessels to pass up and down one by another▪ PROBLEM. CXI. To make a Ladder of Cords, which may be carried in one's pocket: by which one may easily mount up a Wall, or Tree alone. TAke two Pulleys A, & D, unto that of A, let there be fastened a Cramp of Iron as B; and at D, let there be fastened a staff of a foot and a half long as F, than the Poultry A: place a hand of Iron, as E, to which tie a cord of an half inch thick (which may be of silk because it is for the pocket:) then strive to make fast the Poultry A, by the help of the Cramp of Iron B, to the place that you intent to scale; and the staff F, being tied at the Poultry D, put it between your legs as though you would sit upon it: then holding the Cord C in your hand, you may guide yourself to the place required▪ which may be made more facile by the multiplying of Pulleys. This secret is most excellent in War, and for lovers, its supportablenesse avoids suspicion. PROBLEM. CXII. How to make a Pump whose strength is marvellous 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 Pump which is made, be just for the Case or Pumps head 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, & may be made of wood or brass of 4 inches thick, having a hole at E, which descending 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 tied to the handle TX, which works in the post US. Let A, B, C, D, be a piece of Brass, G the piece which enters into the hole to F, to keep out the Air. H, I, K, L, the piece tied at the funnel or pipe: in which plays the Iron rod or axis G, so that it pass through the other piece MN, which is tied with the end of the pipe of Brass. Note, that the lower end of the Cistern ought to be rested upon a Gridiron or Iron Grate▪ which may be tied in the pit, by which means lifting up and putting down the handle, you may draw ten times more water than otherwise you could. PROBLEM. CXIII. How by means of a Cistern, to make water of a Pit continually to ascend without strength, or the assistance of any other Pump. LEt IL, be the Pit where one would cause water to ascend continually to ●●ach office 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 air 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 Cistern as B, which may be communicative to A, by help of the pipe G, from which Cistern B, may issue the water of pipe D, which may descend to H, which is a little below the level of the water of the pit as much as is GH. to the end of which shall be soldered close a Cock which shall cast out the water by KH. Now to make use of it, let B be filled full of water, and when you would have it run turn the Cock, for then the water in B, will descend by K. and for fear that there should be vacuity, nature which abhors it, will labour to furnish and supply that emptiness out of the spring F, and that the Pit dry not, the Pipe ought to be small of an indifferent capacity according to the greatness or smallness of the spring. PROBLEM. CXIIII. How out of a fountain to cast the water very high: different from a Problem formerly delivered. LEt the fountain be BD, of a round form (seeing it is the most capable and most perfect figure) place into it two pipes conjoined as EA, and HC, so that no Air may enter in at the place of joining: let each of the Pipes have a cock G, & L: the cock at G, being closed, open that at ay▪ & so with a squirt force the water through the hole at H, then close the Cock at A, & draw out the squirt, and open the cock at G. the Air 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 Air approacheth to his natural place, so the force will diminish. PROBLEM. CXV. How to empty the water of a Cistern by a Pipe which shall have a motion of itself. LEt AB, be the vessel; CDE, the Pipe: HG, a little vessel 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 bottom of the vessel at F, then as the vessel filleth so will the Pipe, and when the vessel, shall be full as far as PO, the Pipe will begin to run at E, of his own accord, and never cease until the vessel 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 Brass, Led, or of other matter of equal substance as are the two vessels AB, and BD, & let them be joined together by the two Pillars MN, & OF: then let there be a pipe HG. which may pass through the cover of the vessel CD, and pass through AB, into G, making a little bunch or rising in the cover of the vessel AB, so that the pipe touch it not at the bottom: then let there be soldered fast another Pipe IL, which may be separated from the bottom of the vessel, and may have his bunchie swelling as the former without touching the bottom: as is represented in L, and passing through the bottom of AB, may be continued unto ay, that is to say, to make an opening to the cover of the vessel 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 small Pipe which may pass through the bottom of the vessel AB, as let it be OPEN, and let there be a bunch; or swelling over it as at P, so that it touch not also the bottom: let there be further made to this lesser vessel an edge in form of a Basin to receive the water, which being done pour water into the Pipe IL, until the vessel CD, be full, then turn the whole Machine upside down that the vessel CD, may be uppermost, and AB, undermost; so by help of the pipe GH, the water of the vessel CD, will run 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 boiling fountain, in the form of a thread 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, burn it and take the ashes of it, and let it be calcinated two hours between two Creusets well 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 boiled in some vessel, until it be all consumed: then there will remain a salt at the bottom, which you shall afterwards sow in good Ground well prepared: such as the Theatre of husbandry showeth, and you shall have your desire. PROBLEM. CVIII. How to make an infalliable perpetual motion. Myxe 5. or 6. ounces of ☿ with is equal weight of ♃, grind it together with 10. or 12 ounces of sublimate dissolved in a cellar upon a Marble the space of four days, and it will become like Oil, Olive, which distil with fire of chaff or driving fire, and it will sublime dry substance, than put water upon the earth (in form of Lie) which will be at the bottom of the Limbeck, and dissolve that which you can; filter it, then distil it, and there will be produced very subtle 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 traveled herein without fruit. PROBLEM. CXIX. Of the admirable invention of making the Philosopher's Tree, which one may see with his eye to grow by little and little. TAke two ounces of Aqua fortis, and dissolve in it half an ounce of fine silver refined in a cappel: then take an ounce of Aqua fortis, and two drams of Quicksilver: which put in it, and mix these two dissolved things together, then cast it into a Vial of half 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 hair which is red, or white, without fading until they fall, but here is to be noted that great care ought to be had in anointing the hair, for fear of touching the flesh: for this composition is very Corrosive or searching, that as soon as it toucheth the flesh it raiseth blisters, and bladders very painful. PROBLEM. CXX. How to make the representation of the great world? DRaw salt Niter out of salt Earth▪ which is found along the River's side, and at the foot of Mountains, 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 Glass, well luted, and always in which let there be placed leaves of Gold at the bottom, then put fire under the Limbeck until vapours arise which will cleave unto the Gold; augment your fire until there ascend no more, then take away your receiver, and close it hermetically, and make a Lamp fire under it until you may see presented in it that which nature affords us: as Flowers, Trees, Fruits, Fountains, Sun, Moon, Stars, etc. Behold here the form of the Limbeck, and the receiver: A represents the Limbeck, B stands for the receiver. PROBLEM. CXXI. How to make a Cone, or a Pyramidal body move upon a Table without springs or other Artificial means: so that it shall move by the edge of the Table without falling? THis proposition is not so thorny and subtle as it seems 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 animal will strive always to free herself from the captivity in which she is in by the imprisonment of the Cone: for coming near the edge of the Table she will return to the other side for fear of falling. PROBLEM CXXII. To cleave an Anvil with the blow of a Pistol. THis is proper to a Warrior, and to perform it, let the Anvil be heated red hot as one can possible, in such sort that all the solidity of the body be softened by the fire: then charge the Pistol 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, stuff ●t with Butter; then heat a piece of steel which may be form 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 enclose it in a Box in the Budget, and you shall have your desire: it is said that Count Mansfield served himself with no others, but such as were made ready in this kind, for that it loseth none of its substance, and it is dressed very equally. PROBLEM. CXXIV. How to make a Candle burn and continue three times as long as otherwise it would? Unto the end of a Candle half●burned stick a farthing less or more, to make it hang perpendicular in a vessel of water, so that it swim above the water; then light it, and it will sustain itself & float in this manner; and being placed into a fountain, pond, or lake that runs slowly, where many people assemble, it will cause an extreme fear to those which come therein in the night, knowing not what it is. PROBLEM. CXXV. How out of a quantity of wine to extract that which is most windy, and evil, that it hurt not a sick Person? TAke two vials in such sort that they be of like greatness 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 be uppermost, now because the water is heavier than the wine, it will descend into the other Vial, 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 mutual interchange of liquids, and by this penetration the wine will lose her vapours 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 animal or form, 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 small, 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 will blow out the Candle immediately, 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 immediately be lighted, which will cause an admiration to those which see the action, if it be well done with a secret dexterity. PROBLEM. XXVII. How to keep wine fresh as if it were in a cellar 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 vial of Glass; and place it in a Box made of wood, Leather, or such like: about which vial place Saltpetre, and it will preserve it and keep it very fresh: this experiment is not a little commodious for those which are not near fresh waters, and whose dwellings are much exposed to the Sun. PUOBLEM. CXXVIII. To make a Cement which endureth or lasteth as marble, which resisteth air and water without ever disjoining or uncementing? TAke a quantity of strong and gluing Mortar well beaten, mix with this as much new slaked Lime, and upon it cast Oil of Olive or Linseed-Oile, and it will become hard as Marble being applied in time. PROBLEM. CXXIX. How to melt metal very quickly, yea in a shell upon a little fire. MAke a bed upon a bed of metal with powder of Sulphur, of Saltpetre, and sawdust alike; then put fire to the said powder with a burning Charcoal, and you shall see that the metal will dissolve incontinent and be in a Mass. 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 steel exceeding hard? QVench your Blade or other Instrument seven times in the blood of a male Hog mixed 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 tempering and quenchings of Iron: an experiment of small cost, often proved, and of great consequence for Armoury 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 Chemistry, the fire being kindled of itself, break the Limbeck, and the Irons which are found at the bottom will flame and appear as burning Coals as soon as they feel the air; which if you promptly enclose in a vial of Glass, and that you stop it exactly with some good Lute: or to be more assured it may be closed up with Hermes wax for fear that the Air get not in. Then will it keep more than a thousand years (as a man may say) yea at the bottom of the Sea; and opening it at the end of the time, as soon as it feels the Air 〈◊〉 takes fi●e▪ with which you may light a Match. This secret merits to be traveled after and put in practice, for that it is not common, and full of astonishment, seeing that all kind 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. Artificial fireworks: Or the manner of making of Rockets and Balls of fire, as well for the Water, as for the Air; with the composition of Stars, Golden-rain, Serpent's, Lances, Wheels 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 less sort is very improper to those which are of a more greater form; for the fire being lighted in a great concave, which is filled with a quick composition, burns 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 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. Unto each pound of good musket Powder small beaten, put two ounces of small Cole dust, and with this composition charge the Rocket. For Rockets of 2 or 3 ounces. Unto every four ounces and a half of powder dust, add an ounce of Saltpetre, or to every 4 ounces of powder dust, add an ounce of Cole dust. For Rockets of 4 ounces. Unto every pound of Powder dust add 4 ounces of Salt peter & one ounce of Cole dust: but to have it more slow, unto every 10. ounces of good dust powder add 3 ounces of Saltpetre, and 3 ounces of Cole dust. For Rockets of 5 or 6 ounces. Unto every pound of Powder dust, add 3 ounces and a half of Salt peter, and 2 ounces and a half of Coledust, as also an ounce of Sulphur, and an ounce of file dust. For Rockets of 7 or 8 ounces. Unto every pound of Powder dust add 4 ounces of Salt peter, and 3 ounces of Sulphur. Of Rockets of 10 or 12 ounces. Unto the precedent composition add half an ounce of Sulphur, and it will be sufficient. For Rockets of 14 or 15 ounces. Unto every pound of Powder dust add 4 ounces of Salt peter, or Cole dust 2 ¼ ounces of Sulphur and file dust of 1 ¼ ounce. For Rockets of 1, pound. Unto every pound of Powder dust add 3 ounces of Cole dust, and one ounce of Sulphur. Of Rockets of 2, pound. Unto every pound of Powder dust add 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. Unto every pound of Salt peter add 6 ounces of Cole dust, and of Sulphur 4, ounces. For Rockets of 4, 5, 6, or 7, pound. Unto every pound of Salt peter add 5 ounces of Cole dust, and 2 ½ ounces of Sulphur. For Rockets of 8, 9, or 10 pound. Unto every pound of Salt peter, add 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 greatness of the fire which is lighted at once, which causeth too great a violence, therefore aught to be filled with a more weaker composition. Of the making of Rockets and other Fireworks. FOr the making of Rockets of sundry kinds, divers moulds are to be made, with their Rolling pins, Breathes, Chargers, etc. 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: now may you load it on the top, with Serpents, Reports, Stars, or Golden Rain: the Serpents are made about the bigness 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 mixed 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 half Powder and half composition, and tying both ends close, they are finished. The best kind of stars are made with this mixture following; unto every 4 ounces of Saltpetre, add 2 ounces of Sulphur, and to it put 1. ounce of Powder-dust, and of this composition make your stars, by putting a little of it within a small quantity of tow; and then tying it up in the form of a ball as great as an Hazelnut or a little Wal-nut, through which there must be drawn a little Primer to make it take fire. Touching the making of the Golden Rain, that is nothing but filling of Quills with the composition of your Rockets somewhat hard. Now if the head of a Rocket be loaded with a thousand of those Quills, it's a goodly sight to see how pleasantly they spread themselves in the Air, and come down like streams of Gold much like the falling down of Snow being agitated by some turbulent wind. Of recreative fires. 1 PHil●strates saith, that if wine in a platter be placed upon a receiver of burning Coals, to exhale the spirit of it, and be enclosed within a Cupboard or such like place, so that the Air may not go in, nor out, and so being shut up for 30 years, he that shall open it, having a wax Candle lighted, and shall put it into the Cubboard there will appear unto him the figure of many clear stars. 2 If Aquavitae have Camphire dissolved in it; and be evaporated in a close Chamber, where there is but a Charcoal fire, the first that enters into the Chamber with a Candle lighted, will be extremely astonished, for all the Chamber will seem to be full of fire very subtle, but it will be of little continuance. 3 Candles which are deceitful are made of half powder, covered over with Tallow, and the other half is made of clean Tallow, or Wax, with an ordinary week; this Candle being lighted, and the upper half 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 small Serpents placed secretly under a Candlestick that is indifferent big, which may have a hole pass through the socket of it to the Candle, through which a piece of primer may be placed, and setting a small Candle in the socket to burn according to a time limited: which Candlestick may be set on a side Table without suspicion 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 fly here and there, intermixing themselves, sometimes in the Air, 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 Pistol; 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 down, forward and backward TAke small Rockets, and place the tail of one to the head of the other upon a Cord according to your fancy, as admit the Cord to be ABCDEFG. give fire to the Rocket at A, which will fly to B, which will come back again to A, and fire another at C, that will fly 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▪ because the Serpents which are in it will variously ●ntermix themselves in the Air, and upon the ground, and every one will extinguish with a report: and here may you note that upon the Rockets may be placed fiery 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● across one the other; upon the crossing of which make a hole, so that it may be placed upon a pin to turn easily, as the figure Q. showeth upon the sides of which hoop or round Circle place your Rockets, to which you may place Lances of fire between each Rocket: let this wheel 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▪ than immediately 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 wheel and casting out of the Serpents. Of night-Combatants. Clubs, Targets, Faulchons, and Maces charged with several fires, do make your night-Combatants, or are used to make place amongst a throng of people. The Clubs at the ends are made like a round Panier with small sticks, filled with little Rockets in a spiral form glued 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 form, 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 form like the figure A, having their backs large to receive many Rockets, the head of one near the neck of another, glued and fastened well together, so that one being spent another may be fired. 〈◊〉 Targets are made of wooden thin boards, which are channeled in spiral lines to contain primer to fire the Rockets one after another, which is all covered with thin covering of wood, or Pasteboard, 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 appear very pleasant to the Spectators: for by the motion of fight, the place will seem to be full of streams of fire: and there may be adjoined to each Target a Sun or a burning Comet with Lances of fire, which will make them more beautiful and resplendent in that action. Of standing Fires. Such as are used for recreation, are Colossus, Statues, Arches, Pyramids, Chariots, Chairs of triumph and such like, which may be accommodated with Rockets of fire, and beautified with sundry other artificial fires, as pots of fire for the Air which may cast forth several figures, Scutcheons, Rockets of divers sorts, Stars, Crowns, 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 Cypress, Lanterns of fire, Candles of divers uses, and colours in burning: and whatsoever the fancy of an ingenious head may allude unto. Of Pots of fire for the Air, which are thrown out of one Case one after another of a long continuance. MAke a long Trunk as AGNOSTUS, and by the side AH, let there be a Channel which may be fired with slow primer or composition; then having charged the Trunk AGNOSTUS, with the Pots of fire for the Air at IGEC, and make the Trunk AGNOSTUS, very fast unto a Post as IK, give fire at the top as at A, which burning downwards will give fire to C, and so throw out that Pot in the Air, which being spent, in the mean time the fire wil-burne from B to D, and so fire E, and throw it out also into the Air, and so all the rest one after another will be thrown out: and if the Pots of fire for the Air which are cast out, be filled with divers Fireworks, they will be so much the more pleasant to the beholders. These Trunks of fire do greatly adorn a Fireworke, and may conveniently be placed at each angle of the whole work. 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 wonderful shout amongst the common people which are standers by; for those Pots being filled with Balls of fire and flying Serpents for the Air, they will so intermix one within another, in flying here and there a little above the ground, and giving such a volley of reports that the Air will rebound with their noise, and the whole place be filled with sundry streams of pleasant fire; which serpents will much occupy these about the place to defend themselves in their upper parts, when they will no less be busied by the balls of fire, which seem to annoy their feet. Of Balls of f●re. THese are very various according to a man's fancy; some of which are made with very small Rockets, the head of one tied 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 roll 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 several fires with reports. In the second kind there may be a channel of Iron placed in divers places in spiral manner, against which may be placed as many small petards of paper as possible may be, the Channel must be full of slow composition, and may be covered a● the former, and made fit with his Rockets in the middle: this Ball may be shot out of a mortar Piece, or charged on the top of a Roc●et: for in its motion it will fly here and there, and give many reports in the Air: 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 works that may be made within it, in which may play 〈◊〉 fires, Petards, etc. and cast out many simple Granades, Balls of fire to burn in the water-Serpents and other things, and often times these boats in their encounters 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 known in respect of the other, and the triumph and victory of the fight gotten. Of Balls of fire which move upon the water. THese may be made in form of a Ball stuffed with other little Balls, glued round about and filled with composition for the water, which fired, 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 Balls full of composition for the water which small Balls must be pierced deep and large, and covered with Pitch, except the hole: in which hole must be first placed a little quantity of grain-Powder; and the rest of the hole filled up with composition; and note further that these Iron Cannons, must be filled with a slow composition; but such which is proper to burn 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, than fire it, and when it begins to blow, throw it into the water, so the fire coming to the holes will fire the grain 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 burn 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 contain sundry Petards, or Rockets, as the figure here showeth, by which is easy to invent others occording to one's fancy. These Lances have wooden handles, that so they may be fastened at some Post, so that they be not overthrown in the flying out of the Rockets or Petards: there are lesser sorts of Lances whose cases are of three or four fold of Paper of a foot long, and about the bigness of ones finger, which are filled with a composition for Lances. But if these Lances be filled with a composition, than (unto every 4 ouncs of powder add● 2 ounces of Salt-Peter, and unto that add 1 ounce of Sulphur) it will make a brick fire red before it be half spent, if the Lance be fired 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 horizontal, or otherwise. Unto the end of the Rocket place an Arrow which may not be too heavy, but in stead of the feathers let that be of thin white tin 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, Granades, etc. and so may be applied to warlike affairs. How a Rocket burning in the water for a certain time, at last shall fly up in the Air with an exceeding quickness. TO do this, take two Rockets, the one equal to the other, and join them one unto another in the middle at C. in such sort that the fire may easily pass from one to another: it being thus done, tie 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 lie upright: then take a pack-thread and tie 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 burn to B. by a small serpent filled there and tied 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 water: and so ascend like a Rocket in the Air, to the admiration of such as know not the secrecy. Of the framing of the parts of a Firework, together, that the several works may fire one after another. 'Cause a frame to be made as ABCD. of two foot square every way, or thereabouts (according to the quantity of your several works) then may you at each angle have a great Lance of fire to stand, which 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 sidewise, and others upright, between these Lances may be placed Pots of fire sloping outwards, 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 behind all the work may be set your Boxes of Rockets, in each of which you may place 6, 9, ●2. or 20 small Rockets: Now give fire at A. (by help of a piece of primer going from one Lance to another) all the Lances will instantly at once be lighted, and as soon as the Lance at A is consumed, it will fire the Channel which is made in the ledge of the frame which runs under the Pots of fire, and as the fire goes along burning, the Pots will be cast forth, and so the rank of Pots upon the sides of the frame AB.BC. and CD. being spent, the soucisons will begin to play being fired also by a Channel which runs under them, upon the ledges AD, HANG, 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 several Fireworks: for the Rockets of the first Box may be loaden with Serpents, the second with Stars, the third with Reports, the fourth with Golden rain, and the fifth with small flying Serpents; these mounting one after another and flying to and fro will much enlighten the Air 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 beautiful forms, some like the branching of Trees, others like fountains of water gliding in the Air, others like flashes of lightning, others like the glittering of stars, giving great contentment, and delight to those which behold them; But if the work be furnished also with Balons (which is the chiefest in recreative Fireworks) then shall you see ascending in the Air but as it were only a quill of fire, but once the Balloon taking fire, the Air will seem 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 Air and Firmament with their rebounding clamour. The making of which with many other rare and excellent Fireworks, and other practices, not only for recreation, but also for service: you may find in a book entitled Artificial Fireworks, made by Mr. Malthas (a master of his knowledge) and are to be sold by William Leake, at the Crown in Fleetstreet, between the two Temple-Gates. Conclusion. In this Book we have nothing omitted what was material in the original, but have abundantly augmented it in sundry experiments: And though the examinations are not so full, and manifold; yet (by way of brevity) we have expressed fully their substance, to avoid prolixity, and so passed by things reiterated. FINIS. Printed or sold by William Leak, at the Crown in Fleetstreet near 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 Readins on the Scat. 21. Hen. 80. Cap 5 of Sewer● Perkins on the Laws of England. Wi●kinsons Office of Sheriff's. Vade Mecum, of a Justice of Peace. The book of Fees. Peasons Law. Mirror of Justice. Topics in the Laws of England. Sken de significatione Verborum. Delaman's use of the Horizontal Quadrant. Wilby's 2d set of Music, 345 and 6 Parts. Corderius in English. D●ctor Fulk's Meteors. Malthus Fireworks. Nyes Gunnery & Fireworks C●to Ma●or with Annotations, by Wil Austin Esquire. Mel Helliconium, by Alex. Ross● Nosce teipsum, by Sir John Davis Animadversions on Lilies Grammar. The History of Vienna, & Paris Lazarillo de Tormes. Hero and L●ander, by G. Chapman and Christoph. Marlowe. Al●ilia or Philotas loving folly. Bishop Andrews Sermons. Adam's on ●eter. Posing of the Accidence. Am●dis de Gaul. Guillieliam's Heraldry. Herbert's Travels. Bacc●s Tales. Man become guilty, by John Francis Sen●●t, and Englished by Henry Earl of Monmouth. The Idiot in 4 books; the first and second of Wisdom; the third of the Mind, the fourth of S●●tick Experiments of the Balance. The life and Reign of Hen. the Eighth, written by the L. Herbet Cornwallis Essays, & Paradoxes. Clenards greek Grammar 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 Solitary Devotions, by Henry Vaughan Silurist With an excellent discourse of Man in glory, written by the Reverend Anselm Arch Bishop of Canterbury. The Fort Royal of Holy Scriptures by I. H. PLAYS. Hen. the Fourth. Philaster. The wedding. The Hollander. Maid's Tragedy. King & no K. The grateful Servant. The strange Discovery. Othello; the Moor of Venice. The Merchant of Venice. THE DESCRIPTION AND USE OF THE DOUBLE horizontal Dial. WHEREBY NOT ONLY THE Hour of the Day is shown; but also the Meridian Line is found: And most ASTRONOMICAL Questions, which may be done by the GLOBE: are resolved. INVENTED AND WRITTEN BY W. O. Whereunto is added, The Description of the general horological RING. LONDON, Printed for WILLIAM LEAKE, and are to be sold at his Shop at the sign of the Crown in Fleetstreet, between the two Temple Gates. 1652. The description, and use of the double horizontal Dial. THere are upon the Plate two several Dial's. That which is outermost, is an ordinary dial, divided into hours and quarters, and every quarter into three parts which are five minutes a piece: so that the whole hour is understood to contain 60 minutes. And for this dial the shadow of the upper oblique, or slanting edge of the style, or cock, doth serve. The other dial, which is within, is the projection of the upper Hemisphere, upon the plain of the Horizon: the Horizon itself is understood to be the innermost circle of the limb: and is divided on both sides from the points of East and West into degrees, noted with 10.20.30, etc. As far as need requireth: And the centre of the Instrument is the Zenith, or Vertical point. Within the Horizon the middle strait 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 hour lines, distinguished accordingly by their figures: and are divided into quarters by the smaller lines drawn between them: every quarter containing 15 minutes. The two arches which cross the hour lines, meeting on both sides in the points of intersection of the six a clock lines with the Horizon, are the two semicircles of the Ecliptic or annual circle of the sun: the upper of which arches serveth for the Summer half year; and the lower for the Winter half year: and therefore divided into 365 days: which are also distinguished into twelve months with longer lines, having their names set down: and into tenths and fifts with shorter lines: and the rest of the days with pricks as may plainly be seen in the dial. And this is for the ready finding out of the place of the Sun every day: and also for the showing of the Sun's yearly motion, because by this motion the Sun goeth round about the heavens in the compass of a year, making the four parts, or seasons thereof▪ namely, the Spring in that quarter of the Ecliptic which begins at the intersection on the East side of the dial▪ and is therefore called the Vernal intersection. Then the Summer in that quarter of the Ecliptic which begin at the intersection with the Meridian in the highest point next the Zenith. After that, Autumn in that quarter of the Ecliptic which beginneth at the intersection on the West side of the dial, and is therefore called the Autumnal 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 yearly motte●, the Sun hath a diurnal, or daily motion, whereby it maketh day and night, with all the diversities and inaequalities thereof: which is expressed by those other circle's drawn cross the hour lines; the middlemost whereof, being grosser than the rest, meeting with the Ecliptic in the points of the Vernal, and Autumnal intersections▪ is the Equinoctial: and the rest on both sides of it are called the parallels, or diurnal arch of the Sun, the two outermost whereof are the Tropics, because in them the sun hath his furthest digression or Declination from the Equinoctial, which is degrees 23 1/●▪ and thence beginneth again to return towards the Equinoctial. The upper of the two Tropics in this nor Northern Hemisphere is the Tropic of Cancer, and the sun being in it, is highest into the North, making the longest day of Summer: And the lower next the Horizon is the Tropic of Capricorn; and the sun being in it, is lowest into the South, making the shortest day of winter. Between the two Tropics and the Equinoctial, infinite such parallel circles are understood to be contained: for the sun, in what point soever of the Ecliptic it is carried▪ describeth by his Lation a circle parallel to the Equinoctial: 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 year in the Ecliptic, a proper circle which may be the diurnal 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 mind thereof, every fifth parallel is herein made a little g●osser than the rest. For this inner dial serveth the shadow of the upright edge of the style; which I therefore call the upright shadow. And thus by the eye and view only to behold and comprehend the course of the sun▪ throughout the whole year both for his annual and diurnal motion, may be the first use of this instrument. TWO Use. To find the declination of the sun every day. Look the day of the month proposed in the Ecliptic, and mark how many degrees the prick showing that day, is distant from the Equinoctial, either on the Summer or Winter side, viz. North or South. 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 find it in the sixth circle above the equinoctial: 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 find it between the second and third northern parallels, as it were an half and one fifth part of that distance from the second: Reckon therefore four degrees for the two circles, and one de●ree for the half space: So shall the Sun's declination be five degrees, and about one fifth 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 find it below the Equinoctial ten parallels, and about one quarter which is 20 degrees and an half southward. So much is the declination. And according to these examples judge of all the rest. III. Use. To find the diurnal arch, or circle of the sun's course every day. The sun every day by his motion (as hath been said) describeth a circle parallel to the equinoctial, which is either one of the circles in the dial, or somewhere between two of them. First, therefore se●k the day of the month; and if it fall upon one of those parallels; that is the circle of the sun's course that same day: But if it fall between any two of the parallels, imagine in your mind●, and estimate with your eye, another parallel through that point between those two parallels keeping still the same distance from each of them. As in the first of the three former examples, The circle of the Sun's course upon 11 of August▪ shall be the very sixth circle above the Equinoctial toward the centre. In example 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 sun's course upon the 13 of November: shall be an imaginary circle between the tenth and eleventh parallels below the Equinoctial, still keeping one quarter of that space from the tenth. IIII Use. To find the r●sing and setting of the sun everyday. 〈…〉 (as was last showed) the imaginary circle or parallel of the sun's course for that day, and mark 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 hour lines which are on both sides of it, by proportioning the distance reasonably, according to 15 minutes for the quarter of the hour, will show the hour 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 nights●hroughout ●hroughout the whole year. When the Sun is in the Equinoctial, it riseth and setteth at 6 a clock, for in the instrument the intersection of the Equinoctial, and the Ecliptic with the Horizon is in the six a clock 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 before 6 in the morning, and after 6 in the evening: and the Diurnal arch greater than 12 hours; and so much more great, the greater the Northern Declination is. Again, 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 Diurnal arch lesser than 12 hours; and by so much lesser, the greater the Southern Declination is. And in those places of the Ecliptic in which the sun most speedily changeth his declination, the length also of the day is most altered: and where the Ecliptic goeth most parallel to the Equinoctial changing the declination, but little altered. As for example, when the sun is near unto the Equinoctial on both sides, the day's increase and also decrease suddenly and apace; because in those places the Ecliptic inclineth to the Equinoctial in a manner like a straight line, making sensible declination. Again, when the sun is near his greatest declination, as in the height of Summer, and the depth of Winter, the days keep for a good time, as it were, at one stay, because in these places the Ecliptic is in a manner parallel to the Equinoctial, the length o● the day also is but little, scarce altering the declination: And because in those two times of the year, 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 nearer every place is to the Equinoctial, the greater is the diversity of days. Wherefore, we may hereby plainly see that the common received opinion, that in every month the days do equally increase, is erroneous. Also we may see that in parallels equally distant from the Equinoctial, the day on the one side is equal to the night on the other side. VI Use. To find how far the sun riseth, and setteth from the true east and west points, which is called the sun's Amplitude ortive, and occasive. Seek out (as was showed in III Use) the imaginary circle, or parallel of the sun's 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 find the length of every day and night. Double the hour of the sun's setting, and you shall have the length of the day; & double the hour of the sun's rising, and you shall have the length of the right. VIII Use. To find the true place of the sun upon the dial, 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 dial be fixed upon a post: Look what a clock it is by the outward dial, that is, look what hour and part of hour the shadow of the slanting edge of the style showeth in the outward limb. Then behold the shadow of the upright edge, and mark what point thereof is upon that very hour and part in the inner dial among the parallels, that point is the true place of the Sun at the same instant. If the dyal be not fixed, and you have a Meridian line no●ed in any window where the Sun 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 find the place of the Sun upon it. If the dyal be not fixed, neither you have a Meridian line, but you know the true hour of the day exactly: hold the dyal even and parallel to the Horizon, moving it till the slanting edge of the stile cast his shadow justly upon the time or hour given; for then the dyal is truly placed, as upon a post. Seek therefore what point of the upright shadow falleth upon that very hour, 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 month in the Ecliptic, find the su●s parallel, or diurnal arch for that day▪ than holding the dyal level to the horizon, move it every way until the slanting shadow of the style in the outward limb, and the upright shadow in the Sun's diurnal arch, both show the very same hour and minute, for that very point of the Sun's 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 thickness of the style, and the bluntness of the angle of the upright edge, the Sun cannot come unto that edge for some space before and after noon. And so during the time that the Sun shineth not on that upright edge, the place of the Sun in the dyal cannot be found. Wherefore they that make this kind of double dyal, are to be careful to file the upright edge of the style as thin and sharp as possible may be. That which hath here been taught concerning the finding out the Sun's 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 Use. To find the hour of the day. If the dyal be fastened upon a post, the hour by the outward dyal, or limb, is known of every one, and the upright shadow in the Sun's parallel, or diurnal arch will also show the very same hour. But if the dial 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 until the hour showed in both dials exactly agreeth, or which is all one, find out the true place of the Sun upon the dial, as was taught in the former question, for that point among the hour lines showeth the hour of the day. X Use. To find out the Meridian, and other points of the Compass. First, you must seek the tru● hour of the day (by the last question) for in that situation the Meridian of the dial standeth directly 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 compass. XI Use. To find out the Azumith of the sun, that is, the distance of the Vertical circle, in which the sun is at that present, from the Meridian. Set your dial upon any plain or flat which is parallel to the horizon, with the Meridian pointing directly north or south, as was last showed: then follow with your eye the upright shadow in a straight line, till it cutteth the horizon: for the degree in which the point of intersection is, shall show how far the suns Azumith is distant from the east and west points, and the compliment thereof unto 90; shall give the distance thereof from the meridian. XII Use. To find 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. Take aboard having one straight edge▪ & a line stricken perpendicular upon it; apply the straight edge 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, until the same hour and minute be showed in both dials: 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 find 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 applied to the wall, as was showed, hang up a thread with a plummet, so that the shadow of the thread may upon the board cross 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 shall 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 a line out of the centre, to the end of that arch measured: and the angle which this last line maketh with the perpendicular, shall be equal to the declination of the wall. XIII Use. How to place the dial upon a post without any other direction but itself. Set the dial upon the post, with the stile into the North, as near as you can guess: then move it this way and that way, till the same hour and minute be showed, both in the outward and inward dials by the several shadows, as hath been already taught, for then the dial standeth in its truest situation; wherefore let it be nailed down in that very place. XIIII Use. To find the height of the sun at high noon everyday. Seek out the diurnal Arch or parallel of the sun's course for that day, (by Use III) and with a pair of Compasses, setting one foot in the centre, 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 shall therein show the degree of of the Sun's altitude above the the Horizon that day at high noon. XV Use. To find the height of the sun at any hour or time of the day. Seek out the diurnal Arch, or parallel of the sun's course for that day: and mark what point of it is in the very hour and minute proposed. And with a pair of Compasses, setting one foot in the Centre, and the other in that point of the parallel, apply the same distance upon the Semidiameter divided: for that measure shall show the degree of the sun's altitude above the Horizon at that time. And by this means you may find the height of the Sun above the Horizon at every hour throughout the whole year, for the making of rings and cylinders and other instruments which are used to show the hour of the day. XVI Use. The height of the sun being given, to find out the hour, or what it is a clock. This is the converse of the former: Seek therefore in the Semidiameter divided, the height of the sun given. And with a pair of Compasses, setting one foot in the centre, and the other at that height, apply the same distance unto the diurnal arch, or parallel of the Sun for that day: for that point of the diurnal arch, upon which that same distance lights, is the true place of the sun upon the dial; and showeth among the hour lines, the true time of the day. XVII Use. Considerations for the use of the instrument in the night. In such questions as concern the night▪ or the time before sun rising, and after sun setting, the instrument representeth the lower Hemisphere wherein the Southern pole is elevated. And therefore the parallels which are above the Equinoctial toward the centre shall be for the Southern, or winter parallels: and those beneath the Equinoctial, for the Northern or Summer parallels; 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 Hemisphere. XVIII Use. To find how many degrees the sun is under the Horizon at any time of the night. Seek the Declination of the sun for the day proposed (by Use 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 hour and minute proposed: And with a pair of compasses, setting one foot in the centre, and the other in that point of the parallel, apply that same distance unto the semidiameter divided: for that measure shall show the degree of the sun's depression below the Horizon at that time. XIX Use. To find out the length of the C●epusculum, or twilight, every day. Seek the declination of the sun for the day proposed (by Use II.) And at the same declination on the contrary side imagine a parallel for the sun that night. And with a pair of compasses setting one foot in the centre, and the other at 72 degrees upon the semidiameter divided, apply that same distance, unto the suns nocturnal parallel: for that point of the parallel, upon which that same distance shall light, sheweth among the hour 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 month be not known, to find it out by the dial. For the working of this question, either the dial 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 dial to be justly set either upon the post, or upon the Meridian. Look what a clock it is by the outward dial, and observe what point of the upright shadow falleth upon the very same minute in the inner dial, and through that same point imagine a parallel circle for the sun's course; that imaginary circle in the Ecliptic shall cut the day of the month. I The description of it. THis Instrument serveth as a Dial to find the hour of the day, not in one place only (as the most part of Dial's do) but generally in all Country's lying North of the Equinoctial: and therefore I call it the general H●rologicall ●ing. It consisteth of two brazen 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 Equinoctial, having in the midst of the inner side or thickness, a line round it, which is the true Equinoctial circle, divided into twice twelve 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 Equinoctial 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 equinoctial, having his due position. The diameter is broad, and slit in the middle: and about the slit on both sides are the months and days of the year. And within this slit is a little sliding plate pierced through with a small hole: which hole in the motion of it, while it is applied to the days of the year, representeth the Axis of the world. 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 month. 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, Guess as near as you can at the hour, 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 Sunshining thorough that hole, fall upon that middle line within the Equinoctial: for there shall be the hour of the day: And the Meridan of the Instrument shall hang directly North and South. These Instrument all Dial's are made in brass by Elias Allen dwelling over against St. Clement's Church without Temple Bar, at the sign of the Horse-shoe near Essex Gate. FINIS