THE MOST NOBLE ancient, and learned play, called the Philosopher's game, invented for the honest recreation of students, and other sober persons, in passing the tediousness of time, to the release of their labours, and the exercise of their wits. Set forth with such plain precepts, rules, and tables, that all men with ease may understand it, and most men with pleasure practise it. by Ralph Lever and augmented by W. F. Printed at London by james Rowbothum, and are to be sold at his shop under Bowchurch in cheap side. Vulnere virescit virtus. The Lord Robert Duddelye. The Physnogmie here figured, appears by Painters Art: But valiant are the virtues that, possess the inward part. Which in no wise may painted be, yet plainly do appear. & shine abroad in every place with beams most bright & clear TO THE RIGHT Honourable, the Lord Robert Dudley, Master of the Queen's majesties horse, Knight of the most honourable order of the Garter, and one of the Queen's majesties privy Counsel, JAMES ROUBOTHUM heartily wisheth longelife, with increase of godly honour and eternal felicity. SIth that your honour is full bend, (right honourable lord) To wisdom & to godliness with true faithful accord Sigh that in deed you do delight, in learning and in skill: The show whereof doth well express a perfect godly will. Sigh that also you have in hand, affairs of force and weight: And study do both day and night, to set all things full strait. I thought therefore your honour should not lack some godly game: Whereby you might at vacant times yourself to pastime frame. Whereby I say you might release, such travails from your mind: And in the mean while honest mirth and prudent pastime find. Remembering then this ancient play, where wisdom doth abound: Called the Philosopher's game, me thinketh I have one found. Which may your honour recreate, to read and exercise: And which to you I here submit, in rude and homely wise. Pythagoras did first invent, this play as it is thought: And thereby after studies great, his recreation sought. Yea thereby he would well refresh, his studious weary brain: And still in knowledge further wade and ply it to his gain. Accounting that a wicked play, wherein a man lewdly: Mispendes his time & wit also, and no good gets thereby. But grievously offends the Lord, and so in steed of rest: With trouble and vexation great, on every side is priest. Most games and plays abused are, and few do now remain: In good and godly order as, they aught to be certain. For why? all games should recreate, the heavy mind of man: And eke the body gainsaid: with cares and troubles then. But now in stead of pleasant mirth, great passions do arise: In stead of recreation now, revenging we practise. In stead of love and amity, long discords do appear: In stead of truth and quietness, great oaths and lies we hear. In stead of friendship, falsehood now, mixed with cruel hate: We find to be in plays & games, which daily cause debate. Pythagoras therefore I say, to make redress herein: Invented first this godly game, thereby to fly from sin. Since which time it continued hath, in French & Latin eke: Still exercised with learned men, their comforts so to seek. Whereby without a further proof, all men may be right sure: That this game unto gravity, and wisdom doth allure. Else would not that Philosopher, Pythagoras so wise: Have laboured with diligence, this pastime to devise. Else would not so well learned men, have amplified the same: From time to time with travel great, to bring it into fame. But let us nearer now proceed, and come we to th'effect: And then shall we assuredly, this pastime not neglect. For it with pleasure doth assuage, the heavy troubled heart: And with like comforts drives away, all kind of sourging smart. The mind it maketh circumspect, and heedful for to be: The time that thereon is bestowed, is not in vain truly. The body it doth stir and move, to lightsomeness and joy: The senses and the powers all, it no wise doth annoy. It practiseth Arithmetic, and use of number showth: As he that is cunning therein, assuredly well knoweth. In Geometry it truly wades, and therein hath to do: A learned play it is doubtless, none can say nay thereto. Proportion also musical, it joins with tother twain: So that therein three noble arts, are exercised certain. What game therefore like unto this, may gotten be or had? There is not one that I do know, the rest are all to bad. It causeth no contention this, nor no debate at all, By this no hatred wrath nor guile, in any wise doth fall. It stirreth not such troubles that, our friend becomes our foe: It moveth not to mischief this, as many others do. Let us avoid the worst therefore, and cleave we to the best. So shall we shun all wickedness, and purchase quiet rest. So shall we serve the living Lord, and walk after his will: So shall we do the thing is good, and fly that which is ill. So shall we live right christianlike, and do our duties well: So shall we please both god & prince, none shall us need compel. And then the Lord of his mercy, will prospero us always: And grant us here to have on earth, full many godly days. Yea than the Lord of his goodness, and grace celestial: Will guide and govern our affairs, and bless our doings all. Which Lord grant to your honour here, good days & long to have: with much increase of health & wealth and from all hurt you save. Your honours most humble, james Roubothum. To the Reader. I Doubt not but some man of severe judgement so soon as he hath on's read the title of this book will immediately say, that I had more need to exhort men to work, then to teach them to play, which censure if it proceed not of such a froward morosity that can be content with nothing but that he doth himself, I do not only well admit, but also willingly submit myself thereto. And if I could be persuaded that men at mine exhortation would be more diligent to labour, I would not only writ a treatise twice as long as this, but also think my whole time well bestowed, if I did nothing else, but invent, speak, and writ that which might exhort, move, & persuade them to the furtherance of the same. But if after honest labour and travel recreation be requisite, (and that need no further probation because we favour the cause well enough) I had rather teach men so to play, as both honesty may be reserved, their wits exersised, they them selves refreshed, and some profit also attained, then for lack of exercise to see them either pass the time in idleness, or else to have pleasure in things fruitless and uncomely. And if great Emperors and mighty monarchs of the world have not been ashamed by writing books to teach the art of Dyce-playing, of all good men abhorred, and by all good laws condemned: have I not some colour of defence, to teach the game, which so wise men have invented, so learned men frequented, and no good man hath ever condemned? The invention is ascribed to Pythagoras, it beareth the name of Philosophers, prudent men do practise it & godly men do praise it. But because many herein (as in a play) have challenged much authority, they have filled this game with much diversity. In which as I could perceive the most differens of playing to consist in three kinds, so have I plainly and briefly set them forth in English not as though there might not more diversities be espied, but that I thought these to them whom I have written to be sufficient. yet for that I would be loath, from play & game, to fall to earnest contention, if any man in this doing or any part thereof shall think I have done amiss, and will do better himself, so far am I from envying his good proceeding, that I will be right glad, and give him hearty thanks therefore. All things belonging to this game for reason you may buy: At the book shop under Bochurch, in Chepesyde readily. The books verdict. Wanting I have been long truly, In english language many a day: Lo yet at last now here am I, Your labours great for to delay, And pleasant pastime you to show, minding your wits to move I trow. For though to mirth I do provoke, Unto Wisdom yet move I more: Laying on them a pleasant yoke, Wisdom I mean, which is the door, Of all good things and commendable: Doubt this I think no man is able: CATO. Interpone tuis interdum gaudia curis: ut possis animo quemuis sufferre laborem. The definition THat most ancient and learned play, called the Philosopher's game, being in Greek termed 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, is as much to say in English, as the battle of numbers. Numbers be either even or odd, wherefore the even part is against the odd, either part having a king, which being taken of the adversaries part, and a triumph celebrated within his camp, the game is ended. ¶ Of diverse kinds of playing. Among the diverse kinds of playing this game, we shall set forth three sorts, of which the reader may choose whether of them he liketh best. And of all those three, we shall give such short and easy rules, that no man (although he were altogether ignorant in Arithmetic) shall find the game so hard, but that he may learn to play it. ¶ Of the parts of this Game. HE that will learn this game, any of the three ways, must first be instructed of these six parts. The table as the field .2. the men and the numbers of them as the host .3. the placing of them, as the encamping .4. the order of play and removing the men, as the marching and fighting 5. the manner and laws of conquering and taking .6. and last of all the triumph after the victory, ¶ Of these parts in the first kind of playing. The table must be a plain board containing .128. squares that is .8. in breadth and .16. in length set forth in two diverse colours. Or for a plainer understanding, the table is a double chess board, as it were two chessebordes joined together, the length of two, the breadth of one, whereof this is an example. ¶ Of the men. THe men be in number .48. Whereof .24. be of one side & must be known by one colour, and .24. on the other side, which also must be marked with a contrary colour, as White and Black, Blue and Red, or what colours else you like best. But in the colering these .3. things must be observed, the bottom or lower part of every man (except the two kings) must be marked with his adversaries colour, that when he is taken, he may change his coat and serve him unto whom he is prisoner. The kings because they consist of all three sorts, as it is known by the learned speculation of the numbers, bear the fashion of all three kinds, his foundations are two squares, on which are set, two triangles & upon them rounds. But this difference is between the kings, that the king of the even nubers, hath a pointed top, the king of the odd numbers is not pointed, the cause dependeth upon the consideration of there numbers by which they arise into pyramidal fashion. The third thing considered in the men, is the number that must be written or graven upon them which to learn plainly for practice mark these short rules. There be of each kind of men, two ranks or orders. The first rank or order of rounds be the digites even or odd namely of the even .2.4.6.8. of the odd .3.5.7.9. The second order of rounds are found by multiplying these digites by themselves as .2. times .2. is .4.3. times .3. is .9. Of the even they be .4.16.36.64. of the odd they be .9.25.49, 81. The first order of the triangles are found by adding two of the rounds together one of the first order and another of the second order, as .2. and .4. make six 3. and .9. make twelve, on the even side they are these .6.20.42.72. on the odd side .12.30.56.90. The second order of triangles be made by adding one to every one of the first order of rounds, and then multiplying that number in himself: as .2. is one of the first order of rounds, thereto add one, that is .3. then .3. times .3. is .9. a triangle of the second order, on the even side. Likewise to three a round on the odd side, add .1. so is it .4. then .4. times .4. is .16. On the even part, they be .9.25.49.81. on the odd part .16.36.64.100. The first order of squares (in which are contained the kings) be made by adding two triangles together, one of the first order, and another of the second, as .6. and .9. make .15. likewise 12. and .16. make .28. Among the even they be .15.45. and .91. the King .153. among the odd they be .28.66.120. and .190. the King. The last order of squares be found, by dobling of every one of the first order of rounds, and after adding one, last of all by multipling that number in itself, as twice .2. is .4. and .1. added is .5. so .5. times 5. is .25. likewise twice .3. is .6.1. added is 7. then .7. times .7. is .49. These be on the even side .25.81.169.289. And of the odd side .49.121.225.361. These numbers must be set upon the men both on the upper side, & also on the nether-side. Except one of the kings, which must with the whole number of their pyramid, be marked, only on the bottom. Because the sides must have other numbers, namely the highest point of the even king, must have .1. the round next under him mark with .4. the upermost triangle with .9. the nethermost with .16. The upper most square must have .25. The nethermost square shall have .36. The king of the odd upon his head, which is a round, not pointed hath .16. upon his first triangle .25. on the second triangle .36. upon the first square .49. upon the lowest square .64. ¶ The reason of these numbers and the knowledge of their proportione. FOr them that seek the speculation of these numbers, rather than the practice for playing, and have some sight in the sciens of Arithmetic, some thing must be said of proportion. For this purpose there be three kinds of proportion. Multiplex, superparticuler and superpartiens. ¶ Of multiplex. MULTIPLEX proportion, is when a great number containeth a less number many times, and leaveth nothing, as .8. containeth .2 four times and nothing remaineth .16. containeth .4. etc., this proportion seemeth best to agreed with rounds because the one number containeth the other and nothing remaineeth as the first order 〈◊〉 rounds be. The second order be these. double. quadruple. sextuple. occuple. proportion. triple. quintuple. septupl. noncuple ¶ Of superparticuler proportion. Supper particular proportion is when a greater number containeth a lesser with one part of it, which may measure the whole, as .12. containeth .9. and 3. which is a third part of nine .6. containeth .4. and .2. that is one half to 4. This proportion being the chief, next unto multiplex, is best figured by a trianguler form, which hath fewest lines and angles next unto a circle. For the manner of this proportion consider this figure. sesquialter. sesquiquart. sesqui. sext. sesqu. oct. sesquiter. sesquiquint. sesquisept. sesquinona. ¶ Superpartiens proportion. THE superpartiens proportion is when the greater number containeth the lesser and more parts of it then one as .15. containeth .9. and .6. which is two thirds of .9. like wise .28. coteineth .16. and .12. that is 3/4; of .16. This proportion containeth divers parts beside the whole number therefore is well figured in the square, which also containeth more corners and sides. For the manner of their proportion consider this table. The first order of squares. suꝑparticulres added being the squares. The second order followeth. third fift seventh ninth 5. 9 13. 17. 10. 36. 78. 136. superbipartiens tertias. suꝑquadrupartiens quintas. suꝑsextupartiens septimas. suꝑoctupartiens nonas Fourth sixth eight tenth 7. 11. 15. 19 21. 55. 105. 171. supertripartiens. quartas. suꝑquintupartiens. septimas. suꝑseptupartiens. Octavas. suꝑnonpartiens. decimas. ¶ Of the kings. THE kings contain in them such numbers, as being all added together, make the whole pyramidal number, the lowest square of the even, is 36. which riseth of the multiplying of .6. in itself. The next square that must be less, is .25. arising by the multiplying of five in itself and so followeth .16. of .4 then .9. of .3. last .4. of .2. and single .1. all these added together, make .91. After the same manner consisteth the king of odd. The lowest square is .64. arising of .8. multiplied in himself. The next .49. of 7. times .7. then .36. of .6, 25. of .5. and .16. of 4. these numbers make the whole pyramidal number .190. which because it riseth not to the point of one, ought not to be sharp poyncted, as hath been said before. ¶ Of the placing, encamping or setting in array. TO return again to the plain and easy playing of this game, next to the army & their armour, follow either the order of their battle or encamping. Which because it is more plain and easily seen which the eye, then learned by the ear, I refer thee unto the table where the battle is appointed in such order as this kind of Play requireth. ¶ Of the marching or removing of the men. THE battle being duly placed, it followeth next, to know the manner of marching & removing, for every kind of men, hath their proper kind of motion, and first we must speak of the rounds. ¶ The motion of the rounds. THE rounds must move into the space that is next unto them corner wise, as in the table, from the space A. to any of these. B.C.D. or. E. ¶ Of the triangles. THE triangles pass three spaces counting that in which they stand for one, and that into which they do remove for another, that is leaping over one space. As from the space. A. he may remove into any of these spaces. F.G.H. or. I this is the motion of the triangle in marching or taking. But in flying he may remove the Knights draft of the chess, as from. A. into. X. or. W. etc. ¶ Of the Squares. THe Squares remove into the fourth place from them, that is leaping over two, right forward or sidelong, as from▪ the place of. A. to any of these spaces. L. N. P. R. flying they may remove after the knight's draft, but that they must pass four spaces, as from. P. to. Y. or. T. etc. And this for the marching and removing of the men, where note, that with their flying draught they can take no man, but if need be help to besiege a man. ¶ Of the Kings marching. THe kings because they bear the form of all the three kinds, may remove any of all their draughts when they list, into the next with the round, into the third with the triangle, and into the fourth with the square, and finally in all points like the Queen at the Chess, saving that he can not pass above four spaces at the most. ¶ Of the manner of taking. THe men may be taken six ways, namely by Equality, Obsidion, Addition, Substraction, Multiplication and Division, and also if you will, and soagree, by Proportion Arithmetical. Geometrical. Musical. ¶ Of Equality. BY equality men may be taken, when one man after his motion, seethe his enemy being of the same number that he is, standing in such place as he may remove into, then may he take away his enemy and not remove into his place, as in this example .9. a triangle of the even army, after he hath removed, espieth .9. a round of the odd army, him may he take up and not remove into his place. But if .9. the triangle, espy nine the round, before he remove, standing in his draft, he may take him up and remove into his place. These men may be taken by equality 9.16.25.36.49.64.81. because they are found in both the armies, and in as much as any man taken being turned with his bottom upward, & that beareth his adversaries colour, may serve his enemy on whose side he is taken, there may yet be taken by equality .4. and 6. ¶ Of taking by obsidion. BY obsidion any man may be taken even the king himself, if he be so compassed with 4. men, that his law full draft be hindered, as for example the round standing in the place of .1. and 4. men of what kind it skilleth not, occupying the places of .2.3.4.5. after you have set your last man in his place may be taken up, also if a triangle be enclosed, as in. a. with any four men standing in. b.c.d.e. he may be taken, even so may a square be taken. Also Triangles and squares may be besieged, it all the four men or any of them, the rest standing nearer, do stand in the third or fourth space from them so that they have no way to remove, as a triangle or square standing in. A. may be besieged by .4. men or any of them (the rest standing nearer) in. F.G.H.I. Also a square standing in. A. may be taken by 〈◊〉, if the four men or some of them (the rest standing nearer) do stand in L.N.P.R. and this is sufficient for Obsidion, by which every man may be taken in manner and form as it hath been taught. Of taking by Addition. WHen two numbers are so brought that they find one of their enemies, which is as much as both they being added together, standing in such place as both they might remove into, they shall take him up, without removing into his place, so soon as the latter of those two is set down, but if the adversaries men be in their danger before they remove, one of them whether the player list, shallbe removed into the place of that man which is taken by Addition. As for example .12. the triangle is in. A. if you can bring six the round, to stand in. B. and .6. the triangle to stand in. G. because .6. and .6. being added make .12. and both may remove to. A. you may take up the triangle .12. by addition. Also .120. the square standing in. P. and .49. the round standing in. B. or else .49. the square standing in. L. which being added together make .69. which standeth in. A. shall take the said square .169. by Addition. ¶ Of taking by Substraction. WHen two men do so stand, that the lesser being substracted out of the greater, the number remaining, is all one with the adversaries man that standeth in both their draughts, so soon as the latter is set in his place, he may take away the adversary, not removing into his place, unless he find him so before he remove: as for as example, 2. the round standing in. B. & .9. the triangle standing in .6. shall take their adversary .7. standing in. A. for .2. out of .9. remaineth .7. Another example. The round .2. standing in. A. may be taken by .30. the Triangle standing in H. and the square .28. standing in. P. for take .28. out of .30. and their remaineth .2. ¶ Of taking by multiplication. WHen two numbers stand so, that being multiplied one by the other, the product is all one with their adversaries man standing in both their draughts, they may take that man so soon as the latter is placed. And if they lie so before they remove, being so left of the adversary, one of them shall succeed in his place that is taken, as in example. The round .3. standeth in. D. and 5. standeth in. E. these two shall take the square 15. standing in. A. because three times five is .15. another example. The round 2. standing in. B. and the triangle .6. standing in. I shall take their enemy the triangle .12. standing in. A. by multiplication for .2. times .6. is .12. ¶ Of taking by division. BY division a man may be taken, when two of his enemies do so stand, that one of them being divided by the other, the product is the same that their enemy is, standing in their draft, immediately after the latter is placed, the enemy may be removed. If he were left in their danger before removing, one of them may remove into his place, an example. The round 4. standing in. D. and the triangle .20. standing in. F. may take the adversary .5. standing in. A. by division, because .4. in .20. is contained .5. times. Another example, the round .5. standing in B. and the triangle 30. standing in. F. may take their enemy .6. standing in A. for .5. in 30. is contained .6. times. ¶ Of the taking of the kings. THe game is never won, until the King be taken. The Kings (as hath been said) may remove any way, so they pass not the fourth space. They can not be taken by equality. But by obsidion the whole king may be taken away. Also his whole number at ones, that is .91. or .190. by Addition, by Subtraction, by Multiplication, or by Division. Also he may be taken by parts, when any of his side numbers may be taken then leseeth he that draught, as when any of his square numbers is go he can not remove the square draft, and so of the rest, till nothing of him be left, then must he be taken away, and the triumph prepared. ¶ The law of prisoners. WHen any is taken captive, he must be turned with his conquerors colour upward & placed in the hindermost space of his victor's camp, and from thence being removed must fight against his conquerors enemies, and serve him also to make his triumph. ❧ A Table to take any of the men, by addition subtraction, multiplication or division. Addition. subtraction Addition. Subtract. 1 4 8 56 64 30 1 2 3 4 4 8 9 30 36 66 1 3 4 4 5 9 9 36 45 30 42 72 1 4 5 4 8 12 9 72 81 30 90 120 1 5 6 4 12 16 9 81 90 30 91 121 1 6 7 4 16 20 9 91 100L 36 1 7 8 4 45 49 12 36 36 72 1 8 9 50 12 16 28 36 45 81 1 15 16 7 12 12 30 42 36 64 100 1 120 121 5 15 20 15 42 2 5 20 25 15 30 45 42 49 91 2 3 5 5 25 30 15 49 64 45 2 4 6 6 15 66 81 45 noth. 2 5 7 6 6 12 16 49 2 6 8 6 9 15 16 20 36 49 72 121 2 7 9 6 30 36 16 56 72 49 120 169 2 28 30 6 36 42 16 153 169 56 2 64 66 6 66 72 20 56 64 120 3 7 20 25 45 56 169 225 3 4 7 7 8 15 20 36 56 64 3 5 8 7 9 16 20. 100 120 64. 225. 289 3 6 9 7 42 49 25 72 3 9 12 7 49 56 25 56 81 72. 81. 153. 3 12 15 8 25 66 91 72. 153. 225. 3 42 45 8 12 20 28 72. 28. 361. 8 20 28 28 36 64 81 nothing 8 28 36 28 72 100L Addition. Multiplication & Division. 90 2 5 9 45 90. 100 190 2 3 6 5 20 100L 91 2 4 8 5 45 125 91 nothing 2 6 12 6 100L 2 8 16 6 6 36 100 noth. 2 15 30 6 7 42 120 2 28 56 6 12 72 120 169. 289 2 36 72 6 15 90 121 2 45 90 6 20 120 121. 169. 190 3 7 153 3 4 12 7 8 56 169 3 5 15 8 190 noth. 3 12 36 8 9 72 225 3 15 45 8 15 120 289 3 30 90 361 4 9 4 4 16 9 9 81 4 5 20 9 25 225 4 7 28 4 9 36 4 16 64 4 25 100L 4 30 120 5 5 6 30 By this Table any man though he have small or no skill in Arithmetic, may learn to play at this game, and in playing learn some part of Arithmetic. ¶ Of taking by proportion. IF the Gamesters be disposed, they may take men also by proportion, Arithmetical, Geometrical, or Musical. But because it is not necessarily required that they should so do, I will first prosecute the manner of triumph, in which also they may learn to take by proportion, as afterward shallbe seen. For when they can join two or three of their men to one of their adversaries men in such order as the triumph is set, so that those three or four numbers have any of these three proportions they may take their adversaries man. ¶ Of the triumph. WHen the King is taken, the triumph must be prepared to be set in the adversaries camp. The adversaries camp is called all the space, that is between the first front of his men, as they were first placed, unto the neither end of the table, containing .40. spaces or as some will .48▪ When you intend to make a triumph you must proclaim it, admonishing your adversary, that he meddle not with any man to take him, which you have placed for your triumph. Furthermore, you must bring all your men that serve for the triumph in their direct motions, and not in their flying draughts. To triumph therefore, is to place three or four men within the adversaries camp, in proportion Arithmetical, Geometrical or Musical, as well of your own men, as of your enemies men that be taken, standing in a right line, direct or cross, as in. D.A.B. or else 5.1.3. if it consist of three numbers, but if it stand of four numbers, they may be set like a square two against two, as in. E.B.D.C. or .2.3.4.5. and after the same manner must you set them so that your adversaries man make the third or fourth, when you take by proportion. ¶ Of divers kinds of triumphs. THere be three kinds of triumphs a great triumph, a greater triumph, and the greatest and most noble of all. ¶ Of the great triumph. THe great triumph standeth in proportion, either Arithmetical, Geometrical, or Musical only. ¶ Of Arithmetical proportion. Arithmetical proportion, is when the middle number differeth as much from the first, as from the third, that is to say, when the third hath so many more, from the second, as the second hath from the first, as .2.4.6. Here, too, is the distans, for .4. exceedeth 2. by two, & .6. is more than four by .2. ¶ A rule to find out Arithmetical proportion between the first and the last. WHen you have the first and the last if you would find out the middle in proportion. Add the first & the last together, and divide the whole into 2. for the half is the middle in proportion as I would know what is the middle number in proportion between .5. and 25. first I add .5. to .20. that is .30. the half of thirty is .15. which is middle in proportion between .5. and .30. so have I .5.15.35. in Arithmetical proportion. ¶ A table of all the Arithmetical proportions that be in this game. 2 3 4 6 7 8 28 64 100 2 4 6 6 9 12 30 36 42 2 5 8 6 36 66 42 49 56 2 7 12 7 8 9 42 66 90 2 9 16 7 16 25 49 169 289 2 15 28 7 64 121 56 64 71 2 16 30 9 12 15 72 81 90 3 4 5 9 45 81 49. 3 5 7 9 81 153 3 6 9 12 16 20 3 9 15 12 20 28 4 5 6 12 42 72 4 6 8 12 66 120 4 8 12 15 20 25 4 12 20 15 30 45 4 20 36 15 120 225 4 30 56 16 36 56 5 6 7 20 25 30 5 7 9 20 28 36 5 15 25 20 42 64 5 25 45 28 42 56 ¶ Of Geometrical proportion. Geometrical proportion, is when the second hath that proportion to the first, that the third hath to the second, as .2.4.8. as .4. exceedeth .2. by 2. so .8. exceedeth .4. by .4. ¶ A rule to find the middle number in Geometrical proportion. Multiply the first by the third, and of the product find out the root square, for that is the middle, if the numbers have any root square in whole numbers. The root square is a number multiplied in itself, wherefore you must seek such a number, as multiplied in itself, maketh the product of the first and the third number multiplied one by the other. As .20. multiplied by .45. is .900. the root is .30. square, which multiplied in itself is .900. But if you list not to take such pains, here is a Table that may serve your turn for Geometrical proportion to be used in this game. ¶ A table for Geometrical proportion. 2 4 8 16 36 81 2 12 72 20 30 45 3 6 12 25 30 36 4 6 9 25 45 81 4 8 16 36 42 49 4 12 36 36 66 121 4 16 64 36 90 225 4 20 100 49 56 64 5 15 45 49 91 169 9 12 16 64 72 81 9 15 25 64 120 225 9 45 225 81 90 100 16 20 25 81 153 289 16 28 49. 27. ¶ Of Musical proportion. musical proportion is when the differences of the first and last from the mids, are the same, that is between the first and the last, as .3.4.6. between .3. and .4. is .1. between .4. and 6. is .2. the whole difference is .3. which is the difference between .6. and .3. the first and the last. ¶ A rule to find the first, when you have the two last. Multiply the second by the third, divide the product by the distans and the third number, and the quotient is the first, as having .6. and 12. I would find the first, 6. times .12. is 72, the difference between .6. and .12. is 6, which added to .12. is .18, divide .72. by 18. the quotient is .4. so have you .4.6.12. in Musical proportion. ¶ To find the middle between the first and the last. Multiply the first by the last, then double the product, and divide the whole by the first and the last added together, the quotient is then the middle number. As having .6. and .12. I would know the middle in Musical proportion. First I multiply one by the other, the product is .72. that doubled is 144, this divided by .18. which is the addition, of .6. and 12. giveth the quotient 8. so have I .6.8.12. in musical proportion. And thus must you work to find out the third in musical proportion. But if you had rather play than work, this table following shall serve your torn. ¶ A table of Musical proportion. 2 3 6 3 4 6 3 15 16 4 6 12 4 7 28 5 8 20 5 9 45 6 8 12 7 12 42 8 15 120 9 15 45 9 16 72 12 15 20 15 20 30 5 45 225 30 36 45 30 45 49 72 90 120 17. ¶ Of the greater triumph. THe greater victory is, when four numbers be brought together, which agreed in two proportions, either Arithmetical and Geometrical, or else Arithmetical and Musical, or else Geometrical and Musical. Of these three conjunctions the greater triumph consisteth, of the which the table followeth. ❧ A table of Arithmetical, and Geometrical proportion. 2 3 4 8 9 12 15 16 2 4 6 8 9 12 15 25 2 4 6 9 9 12 16 20 2 4 5 8 9 45 81 225 2 7 12 72 9 25 45 81 2 9 12 16 9 12 16 20 2 12 42 72 9 15 20 25 3 6 9 12 9 8 153 289 3 4 6 9 12 16 20 25 3 9 15 25 15 16 20 25 4 5 6 9 15 20 30 45 4 6 8 9 16 20 25 30 4 6 9 12 16 36 56 81 4 6 8 16 20 25 30 45 4 12 20 36 30 36 42 49 4 8 12 16 36 42 40 56 4 8 12 36 42 49 56 64 4 8 16 28 49 56 64 72 4 12 20 100 49. 91. 169. 289 4 16 28 49 56 64 72 81 4 16 28 64 64 72 81 90 4 20 36 100 72 81 90 100 5 9 15 25 5 15 25 45 52. 5 25 45 81 6 9 12 16 7 16 20 25 7 49 91 169 8 9 12 16 8 64 120 225 Arithmetical and musical proportion. Geometrical and musical proportion together. 3 4 5 6 2 3 6 12 3 4 5 15 3 4 6 9 3 4 6 9 3 4 6 12 3 5 7 25 3 6 8 12 3 5 9 15 4 6 12 36 3 9 15 45 4 7 28 49 3 4 6 8 5 9 15 45 4 5 6 12 5 9 45 225 4 6 12 15 5 9 45 81 4 6 12 20 9 12 16 72 4 12 15 20 9 15 25 45 5 7 9 45 9 15 45 225 6 7 8 12 9 25 45 225 8 15 120 225 15 20 30 45 9 12 15 45 20 30 36 45 9 12 15 20 25 45 81 225 9 15 30 45 16. 9 15 45 81 12 15 20 25 15 20 25 30 15 20 30 45 15 30 36 45 15 30 45 90 30 36 42 45 72 81 90 120 25. ¶ Of the greatest triumph. THe greatest triumph is of Arithmetical, Geometrical, and Musical proportions all joined together. Arithmetical, Geometrical, and Musical proportions, all together. 2 3 4 6 6 8 12 16 2 3 6 9 6 12 15 20 2 4 6 12 7 12 42 72 2 5 8 20 8 15 64 120 2 7 12 42 8 15 120 225 2 9 16 72 12 15 16 20 3 4 6 8 12 15 20 25 3 4 6 9 15 20 36 45 3 5 9 15 15 30 45 90 3 5 15 25 30 3 9 15 45 4 6 8 12 4 6 9 12 4 7 16 28 4 7 28 49 5 6 25 45 5 9 45 81 5 25 45 225 5 15 25 45 6 8 9 12 And thus is the first kind of playing at an end. And this is sufficient to teach you to play, but if you would learn to play cunningly, you must use to play often, so shall you learn better than by any precepts or rules. ¶ Of the second kind of playing at the Philosopher's game. THere is in this kind of playing to be considered, the table, the men, the marking of them, the setting of them in array, their marching, their laws of taking, and the manner of triumphing. ¶ Of the Table. THe Table is the same that was first described, namely a double chesbord. ¶ Of the men. THe men be as before in number 48.23. on a side, and two contrary kings of even and of odd. They must be of divers colours, as hath been said, the bottom of every one must have his enemy's colour, and his own mark of number, differing in this point from the former playing, that the enemy's men taken, may serve only to celebrated a triumph, but not to fight on his side that taketh them. ¶ Of the marking of the men. THey be marked with the same numbers, that have been showed before and therefore so are to be found out as is taught before. But they be marked beside their numbers, with cossical signs, which be signs used in the rule called regula cossa, or algebra, betokening roots, quadrats, cubes, fouresquared quadrats, sursolides, & quadrates of cubes. All these .6. signs must be contained in this game. The sign of the root.¹ of the quadrate.² of the cube, or solid quadrat.³ of the fouresquared quadrat.⁴ of the sursolide.⁵ of the squared cube.⁶ Every number may be taken for a root, as .2. this number multiplied in itself is a square as .4. The quadrat or square multiplied by the root giveth a cube or solid square, as .4. multiplied by .2. giveth .8. that is a cube. Multiply the cube by the root, so have you a squared quadrat, as .8. by .2. giveth 16. which is a quadrate of a quadrate. Multiply the square or quadrat of quadrat by the root, and the product is the sursolyde, as .2. times .16. is .32. which is a sursolide. Multiply the sursolide by the root, and the product is the quadrate of a cube, as .2. times .32. is .64. which is a quadrat of a cube. So have you the root quadrat, cube, quadrat of quadrat, sursolide, quadrat of cube .2.4.8.16.32.64. So .2. referred to .4. is a root of a square, referred to .8. it is a root of a cube .2. referred to .16. is the root of a four squared quadrate .2. referred to .32. is the root of a sursolide .2. referred to .64. is the root of a quadrate of a cube. These numbers must have the proper cossical signs. Also one number having divers relations, may have divers cossical signs, as 9 referred to .81. being root, hath the sign of a root, but being referred to .3. it hath the sign of a quadrate, for it is a quadrate of .3. and is thus signed.². and so of the rest that have like relation. ¶ The marking of the men. THe first order of rounds in both numbers, must have the sign of the root upon them all after this manner. 8¹ 6¹ 4¹ 2¹ 9¹ 7¹ 5¹ 3¹ THe second order of rounds found out as before, be not all marked with cossical signs, but only .4. and .9. with the root, and .81. with the quadrate. The rest have none, because among their adversaries men there is none that can be cossical root to them in such manner as this game requireth. 64. 36. 16 4¹ 81² 49 25 9¹ THe first order of triangles (having the same numbers that have been taught before) do all lack the cossical signs, except only .6. which is signed with the root. 72. 42. 20 6¹ 81 56 30 12 THe second order of triangles, have all except one (which is the number of .100.) their cossical signs, as 9 both of the root and of the quadrate, 25.36. and .49. have the sign of the quadrate .64. of the quadrate and the cube, and also the quadrat of cube .16. and .81. of the quadrate, and the four squared quadrate. 81².⁴ 49² 25² 9¹ 100 64¹.³ 36² 16² In the first order of squares, only 15. is marked with the root, all the rest do want their cossical signs in this game. 153. 91 45 15¹ 190 120 66 28 THe second order of squares hath .3. numbers marked with cossical signs, that is .25. and .225. with the sign of the quadrate .81. is marked with the sign of the quadrate and the four-squared quadrate. 289. 169. 81².⁴ 25.² 361. 225.² 121. 49. And thus have you all the men that be marked with cossical signs. ¶ The setting in array. THe teachers of this kind of playing, do not so well allow, the former kind of placing or any other, as the natural placing of every man under him of whom he ariseth. So they contain .6. ranks in length, extending to the furthermost edge of the Table after this sort. ¶ The marching or moving. THe men may remove every way, into void places, forward, backward, toward both sides, direct or cornerwyse. So that the round men remove into the next space, the triangles into the third place, and the squares into the fourth place, accounting that place in which they stand for one. Also every man saving the two kings to besiege his enemy, or to fly from the siege himself, may remove the knight's draft in chess, but neither take any man (except it be by siege) nor erect a triumph by such motions. The kings move even as squares, but that they have not the flying draught. It is counted lawful among such as will so agreed, that the Triangles and Squares, may remove into void places, though the spaces between be occupied of other men. ¶ The manner of taking. THe men may be taken seven ways by Obsidion, by Equality, by Addition, by Substraction, by Multiplication, by Division, and by cossical Signs. ¶ Of taking by Obsidion. ALl men may be taken by Obsidion when by four men they be letted of their ordinary draught, as hath been taught before. ¶ Of taking by Equality. BY Equality may these men take or be taken, as hath been said before, 9.16.25.36.49.64.81, as if after you have played your .9. you see your adversaries .9. stand in your man's draft, you may take him up not removing into his place, unless you espy him standing in your draft before you play, then must you take him up and remove into his place. ¶ Of taking by Addition. THe taking by Addition is all one with the first kind of play, in all respects, saving that some require the men that should take by Addition to stand in the next spaces to him that is taken, either directly, or cornerwyse, but the former way is better. Of taking by substraction. THat which was said in the first kind of substraction and that which was last said of Addition may be both referred hither. For this substraction differeth not from the former, but for the opinion of them, that would have the two takers stand only in the next spaces to him that is taken. ¶ Of taking by Multiplication. Taking by multiplication doth differ. For in this kind of playing, it is thus. When your man standeth so, that being lesser than your adversaries man, you may multiply your man by the void spaces between them, and the product is all one with the adversary, you may take him up, not removing into his place, except you espy him so, before you remove your man. ¶ Of taking by Division. Likewise by Division, if your man being greater than the adversary, stand so, that being divided by the void spaces, the quotient is all one with the adversary, you may take him up, not removing into his place, unless you see him so standing before you draw. ¶ Of taking by cossical signs. BY cossical signs any man that hath these signs,². meeting with his root in his ordinary draft that hath this sign. taketh him up, or else is taken of him, without removing into his place, except he may take him before he remove. ¶ Of the Kings, and their taking. THe King of the even must be four square, having six steps, every one lesser than other, on one side he must have on him these roots .1.2.3.4.5.6. on the other side the quadrates arising of these rots, that is 1.4.9.16.25.36. ¶ The King of the odd men, must have but five steps, that is .4.5.6.7.8. lacking the roots that he can not end in .1. The quadrates of his roots be these .16.25.36.49.64. These must be so set on, that the lest must be highest and the greatest lowest. ¶ The Kings be taken by Obsidion, or if their Pyramidal number, be taken by any of the aforesaid means. Also if by such means you can take all his quadrates one after another. ¶ The privilege of the King. IF any of the Kings quadrates be taken, he may redeem it by any of his men having the same number, and must remove into his place, which redeemed him. But if he have none of the same number, he may redeem him for any man of his, that his adversary will choose, and likewise remove into his place by whom he is redeemed. ¶ A table to take the men by Multiplication and Division. even against odd spaces. even. spaces. odd even. spaces. odd 2 8 16 3 5 15 6 2 12 8 8 64 5 5 25 8 2 16 4 9 36 9 5 45 15 2 30 9 9 81 12 6 72 45 2 90 25 9 225 7 7 49 4 3 12 9 10 90 5 9 45 4 4 16 21. 9 9 81 9 4 36 odd against even spaces. 3 12 36 16 4 64 3 14 42 6 5 30 3 2 6 17. 20 5 100 36 2 72 2 6 12 3 3 9 15 6 90 5 3 15 20 6 120 12 3 36 4 7 28 5 4 20 8 7 56 9 4 36 Spaces. 16 4 64 ¶ For Division. even against odd spaces. odd against even spaces. ¶ To take by cossical signs 6 2 3 12 2 6 2 16.⁴ 72 2 36 16 2 8 2 64.⁶ 15 3 5 30 2 15 3 81.⁴ 36 3 12 90 2 45 9 3 3 12 3 4 3 9 20 4 5 16 4 4 4 16.² 36 4 9 36 4 9 4 64.³ 64 4 16 64 4 16 5 25.² 15 5 3 100 4 25 25 5 5 22 5 45 6 36.² 45 5 9 30 5 6 7 49.² 42 6 7 100 5 20 8 64.² 72 6 12 12 6 2 9 81.² 49 7 7 36 6 6 15 225.² 72 8 9 90 6 15 45 9 5 120 6 20 81 9 9 28 7 4 36 12 3 56 7 8 91 13 7 16 8 2 42 14 3 64 8 8 120 8 15 20. 3 9 4 81 9 9 225 9 25 90 10 9 66 11 6 28 14 2 27. ¶ Of the triumph. THe triumph is after the King be clean taken away, to be created in the adversaries camp, as well of your own men as of your adversaries men that be taken, or of both in proportion as hath been showed before, and proclaimed that those men on's placed, may not be taken, as it was declared sufficiently, and no difference between the triumphs, saving that some will not allow a triumph but of four numbers, and two proportions at the least. All three for the greater victory, making but two kinds of triumphs. ¶ Here followeth the third kind of playing at the Philosopher's game. THere must also in this third kind be considered the table, the men, their marking, the order of their battle, the motions, their taking, and last of all their triumphing. The table is the same that hath been twice already described. Yet some will not have it so long, but at the jest it must contain .10. squares in length and always .8. in breadeth. The longest is best. ¶ Of the men. THe men be .48. as it hath been said of two contrary colour, the head and bottom all of one colour, because men on's taken be no more occupied in this kind of playing. ¶ The inscription and fashion. THe fashion is as hath been last declared both of the men, and of the kings, the inscription of numbers the same, but without cossical signs. ¶ Of the order of the battle. THe order of battle is after the first manner, but not so far from the boards end, namely the .4. squares standing in the plats nearest to the boards end the rest accordingly joined to them, as in the first kind of playing. ¶ Of their motions. THe men move froward and backward, to the right hand, and to the left hand, but not cornerwise, except the gamesters so agreed, the rounds into the next space, the triangles into the third, and the squares into the fourth, the kings move as squares. And these be their ordinary draughts in marching. ¶ Of their taking. THey are taken by encountering, by eruption, by laying weight, and by Obsidion. ¶ Of taking by encountering. TO take by encountering is to take by Equality, as hath been twice before declared. ¶ Of taking by eruption. TO take by eruption is when a less number being multiplied by the spaces that are between him & his adversary, the product is as much as his adversary, he may take his enemy away whether he stand directly from him or cornerwise. For men that may be taken by eruption look in the table of taking by multiplication in the second kind of playing. ¶ Of taking by deceit or lying weight. TO take by deceit or lying weight, is to take by addition, not as before when the adversary standeth within the draft of two men which being added make the just number of the adversary, but when the .2. numbers that are to be added, stand in the next spaces to the adversary. For to take by deceit, look in the table that was set forth for taking by addition in the first kind of playing. ¶ Of taking by Obsidion. BY Obsidion all men may be taken, when four men besiege the adversary, standing in the four next spaces about him directly, or cornerwise, the man so besieged can not escape, because he can not remove cornerwyse, therefore may be taken up, so soon as the last of the four is set in his place. In all three kinds of playing no Obsidion can be of any man with some of his fellows, but all four must be his adversaries. In this third kind, these men can be none otherwise taken but by Obsidion. Namely among the even .2.4.4.135. among the odd .3.5.7.190. In all manner of taking this is to be noted, that we must not place the man which taketh in place of him that is taken, but when he may be taken before we draw, then shall we remove our man into his place. ¶ The privilege of the king. THe king standeth for so many men as he hath steps, that is the even for .6. the odd for .5. if any of these (except the lowest and greatest) be taken the king may redeem him, by any man of his that is of the same number. If he have none of the same number, he may redeem him by any of his men that his adversary will choose. But if his lowest square be taken, no ransom will deliver him. Also if the whole king at on's that is the whole number of Pyramid be taken, he can not be redeemed. ¶ Of the triumph. TO take away the tediousness of long play from them that be young beginners, writers of this game have invented divers kinds of short victories, wherefore they divide victory into proper and common. Of the proper victory need nothing here be spoken, for all things thereto belonging are sufficiently set forth in the first kind of playing. ¶ Of the common victory. THe common victory (they say) is after five manners, for men contend either for bodies, goods, quarrels, honour, or else for both quarrels & honour. ¶ Victory of bodies, VIctory of bodies is only to take a certain number of men, as if the gamesters agreed, that he which first taketh 4.02.5.02.6.02.10. men etc., shall win the game. ¶ Uictorye of goods. Victory of goods, is to take a certain number without respect of the men. As if it be covenanted, that he which first taketh men amounting to the number of .100. or .200. shall have the victory. ¶ Victory of quarrel. Victory of quarrel is when neither the men, nor the number, but the characters of the number be considered. As if it be determined that he which first taketh .100. in .8. characters not regarding in how many men they stand, shall win. As .2.4.6.8.24.64. so you have .100. in .8. characters it skilleth not, although there be more than .100. as in this example there is more than .100. by .4. ¶ victory of honour. Victory of honour, is when a determined number is made in a determined number of men, as if it be determined that he which first cometh to .100. in .8. men, shall win the game. As in these .2.4.6.8.4.16.45.15. And though there were somewhat more than 100 so it be in .8. men, it skilleth not. ¶ Of victory of honour and quarrel. THe victory of honour and quarrel, is when one obtaineth the decreed number, in the decreed number of men and the decreed number of characters: as let .100. be the decreed number 8. the determined number of men, and 9 the determined number of characters, He that obtaineth .2.4.6.8.4.6.9.64. obtaineth the victory of honour and quarrel. It shallbe no hindrance though .8. men and .9. characters contain somewhat more than .100. so that there be not .100. upon one man, as in the victory before. ¶ victory of standers. THey have invented another victory, that is of standards, by counterfeiting two armies, one of the Christians, another of the Turks. The white men, that is the even host, containeth .1312. footmen (not counting the roots of squares expressed in the kings) let the first and last be captains and let them divide the whole army into .10. standards so every standard shall have .130. men, beside the two captains and the ten standard bearers. The black men, that is the odd army (except the kings roots) be .1752. The two captains and ten standard bearers taken out, there remaineth .1740. soldiers, to every standard .174. He that winneth more standers hath the victory. If the even host wine .348. men he hath obtained two standards if he win .522 he hath gotten three standards and forth of the rest. If the odd army win .260. they win two standards .390. three standards and so of the rest. ¶ A Table of the victory of standards. One standard of the even, containeth. 130. Two standards. 260. Three standards. 390. Four standards. 520. five standards. 650. Six standards. 780. Seven standards. 910. Eight standards. 1040. Nine standards. 1170. Ten standards. 1300. One standard of the odd, containeth. 174. Two standers. 348. Three standards. 522. Four standards. 696. five standards. 870. Six standards. 1044. Seven standards. 1218. Eight standards. 1392. Nine standards. 1566. Ten standards. 1740. YOu may use any of these six kinds of common victory, in every one of the three kinds of playing. FINIS. printed at London by Roland Hall, for james Rowbothum, and are to be sold at his shop in chepeside under Bow church. 1563.