M. BLUNDEVILE His Exercises, containing six Treatises, the titles whereof are set down in the next printed page: which Treatises are very necessary to be read and learned of all young Gentlemen that have not been exercised in such disciplines, and yet are desirous to have knowledge as well in cosmography, Astronomy, and Geography, as also in the Art of Navigation, in which Art it is impossible to profit without the help of these, or such like instructions. To the furtherance of which Art of Navigation, the said M. Blundevile specially wrote the said Treatises and of mere good will doth dedicated the same to all the young Gentlemen of this Realm.  
THOU SHALT LABOUR FOR PEACE PLENTY  
LONDON. Printed by John Windet, dwelling at the sign of the cross Keys, near Paul's wharf, and are there to be sold. 1594.   

The Titles of the Treatises contained in this book.  
FIrst, a very easy Arithmetic so plainly written as any man of a mean capacity may easily learn the same without the help of any teacher.  
Item the first principles of cosmography, and especially a plain treatise of the Sphere, representing the shape of the whole world, together with the chiefest and most necessary uses of the said Sphere.  
Item a plain and full description of both the Globes, aswell Terrestrial as Celestial, and all the chiefest and most necessary uses of the same, in the end whereof are set down the chiefest uses of the Ephemerideses of johannes Stadius, and of certain necessary Tables therein contained for the better finding out of the true place of the Sun and Moon, and of all the rest of the Planets upon the Celestial Globe.  
Item a plain and full description of Petrus Plancius his universal Map, lately set forth in the year of our Lord 1592. containing more places newly found, aswell in the East and West Indieses, as also towards the North Pole, which no other Map made heretofore hath, whereunto is also added how to found out the true distance betwixt any two places on the land or sea, their longitudes and latitudes being first known, and thereby you may correct the scales or Trunks that be not truly set down in any Map or Card.  
Item, A brief and plain description of M. Blagrave his Astrolabe, otherwise called the Mathematical jewel, showing the most necessary uses thereof, and meetest for sea men to know.  
Item the first & chiefest principles of Navigation more plainly and more orderly taught than they have been heretofore by some that have written thereof, lately collected out of the best modern writers, and treaters of that Arte.  
And moreover, I have thought good to add unto mine Arithmetic, as an appendix depending thereon, the use of the Tables of the three right lines belonging to a circle, which lines are called Sines, lines tangent, and lines secant, whereby many profitable and necessary conclusions aswell of Astronomy, as of Geometry are to be wrought only by the help of Arithmetic, which Tables are set down by Clavius the jesuite, a most excellent Mathematician, in his book of demonstrations made upon the Spherickes of Theodosius, more truly printed than those of Monte Regio, which book whilst I read at mine own house, together with a loving friend of mine, I took such delight therein, as I mind (God willing) if God give me life, to translate all those propositions, which Clavius himself hath set down of his own, touching the quantity of Angles, and of their sides, as well in right line triangles, as in Spherical triangles: of which matter, a Monte Regio wrote diffusedlie and at large, so Copernicus wrote of the same briefly, but therewith somewhat obscurely, as Clavius saith. Moreover, in reading the Geometry of Albertus Durcrus, that excellent painter, and finding many of his conclusions very obscurely interpreted by his Latin interpreter (for he himself wrote in high Dutch) I requested a friend of mine, whom I knew to have spent some time in the study of the Mathematicals, not only plainly to translate the foresaid Durerus into English, but also to add thereunto many necessary propositions of his own, which my request he hath (I thank him) very well performed, not only to my satisfaction, but also to the great commodity and profit of all those that desire to be perfect in Architecture, in the Art of Painting, in free Mason's craft, in joiners craft, in Carvers craft, or any such like Art commodious and serviceable in any common Wealth, and I hope that he will put the same in print ere it be long, his name I conceal at his own earnest entreaty, although much against my will, but I hope that he will make himself known in the publishing of his Arithmetic, and the great Art of Algebra, the one being almost finished, and the other to be undertaken at his best leisure, as also in the printing of Durerus, unto whom he hath added many necessary Geometrical conclusions, not heard of heretofore, together with diverse other of his works as well in Geometry as as in other of the Mathematical sciences, if he be not called away from these his studies by other affairs. In the mean time I pray all young Gentlemen and seamen to take these my labours already ended in good part, whereby I seek neither praise nor glory, but only to profit my country.   

To the Reader.  
I Greatly rejoice to see so many of our English Gentlemen, both of the Court and Country in these days so earnestly given to travel aswell by sea as land, into strange and unknown countries, and specially into the East and West Indieses, following therein the good example of divers worthy knights and Gentlemen that have ventured their lives to discover strange countries to the great honour of their country, and to their own immortal fame. And because that to travel by sea requireth skill in the Art of Navigation, in which it is unpossible for any man to be perfect unless he first have his Arithmetic, and also some knowledge in the principles of cosmography, and specially to have the use of the Sphere, of the two Globes, of the Astrolabe, and cross staff, and such like instruments belonging to the Art of Navigation, I thought good therefore to writ the Treatises before mentioned, to serve as an introduction for such young Gentlemen as have not been exercised in such kind of studies, which Treatises if they shall vouchsafe to read with attentive mind, and in such order as they are before set down, I doubt not but that it will 'cause them hereafter to seek for further knowledge therein. And in any wise I wish them to begin with my Arithmetic, the contents whereof are declared in the next chapter following. In the mean time I do earnestly request all young Gentlemen to take these my simple pamphlets no less thankfully than they have done my horse book, and in so doing I shall have just cause to think my labour well bestowed.   

What cause first moved the Author to writ this Arithmetic, and with what order it is here taught, which order the contents of the chapters thereof hereafter following do plainly show.  
I Began this Arithmetic more than seven years since for a virtuous Gentlewoman, and my very dear friend M. Elizabeth Bacon, the daughter of Sir Nicholas Bacon Knight, a man of most excellent wit, and of most deep judgement, and sometime Lord Keeper of the great Seal of England, and lately (as she hath been many years past) the most loving and faithful wife of my worshipful friend M. justice Wyndham, not long since deceased, who for his integrity of life, and for his wisdom and justice daily showed in government, and also for his good hospitality deserved great commendation. And though at her request I had made this Arithmetic so plain and easy as was possible (to my seeming) yet her continual sickness would not suffer her to exercise herself therein. And because that diverse having seen it, and liking my plain order of teaching therein, were desirous to have copies thereof, I thought good therefore to print the same, and to augment it with many necessary rules meet for those that are desirous to study any part of cosmography, Astronomy, or Geography, and specially the Art of Navigation, in which without Arithmetic, as I have said before, they shall hardly profit. But now to return to my matter, the contents of this Arithmetic are these here following. First, having defined what Arithmetic is, and what numeration or numbering is, and of what parts it consisteth, and how to number any great sum written in many figures, I deal with the four special kinds of Arithmetic, that is, Addition, Subtraction, multiplication, and division. Than I show the order of working in whole numbers, called in Latin Integra, by the rule of proportion, otherwise called the golden rule, or the rule of three. Than I treat of broken numbers called Fractions, setting down seven necessary rules belonging thereunto, by the help whereof you shall be the better able to add, subtract, to multiply and divide the same. That done, I show how to use the rule of three aswell in dealing with sole fractions as with fractions annexed to integrums, which rule of three is threefold, that is, the common rule, the rule reverse, and the double rule, the order of all which three kinds I do plainly teach by examples, showing wherein, how, and when, they are to be used. Next to that I set down the rule of fellowship, a necessary rule for those that have to traffic in any trade of merchandise, giving diverse examples thereof. Next to that I treat of Arithmetical, and Geometrical progression, and also of proportion, and of the three kinds thereof, that is, of proportion Arithmetical, Geometrical, and musical. Than I show how to find out the square root of any number, and also the use thereof in setting of Battles, and also how to find out the cubique root of any number. And last of all I treat of Astronomical fractions, showing how to add, to subtract, to multiply and divide the same. And also to take the square root thereof, without the knowledge of which fractions, you can never calculate any thing truly out of the Astronomical Tables.  
Some perhaps do look here that I should speak somewhat of the rules of Algebra, whereby all subtle and intricate questions of Arithmetic are to be unfolded, wherewith I leave indeed deal, partly, for that I have not of long time exercised myself therein, and partly because I know that one hath begun to writ thereof, whose book being once ended, I doubt not but that he will shortly after print the same to the great profit and furtherance of all those that delight in such good exercises. But in the mean time I have thought good to add (as I have said before) unto this Arithmetic a plain description, together with the use of the Tables of Sines, of lines tangent and secant, which Tables will pleasure many that would gladly know how and wherein to use them, and specially such seamen as have some taste of Arithmetic, without the which no good almost is to be done in any science.   

This Treatise of Arithmetic containeth 26. Chapters as followeth.  
WHat Arithmetic is, what numeration is, and of what parts it consisteth, and what signification every digit hath according to his place, and how to express or tell a great number written in many figures. Chapter. 1.  
Of the four special kinds of Arithmetic, and first of Addition, with examples thereof. Chap. 2.  
Of Subtraction, with examples thereof, and how to try the same. Chap. 3.  
Of Multiplication and certain Tables belonging thereunto, together with the use thereof, and what is to be observed therein, with examples and trial thereof. Chap. 4.  
Of Division, and what is to be observed therein, with examples and trial thereof, and of halfing any number. Chap. 5.  
Of the rule of three, called the Golden rule, and what order is to be observed in working thereby, and of the three kinds thereof. Chap. 6.  
Of Fractions what they be, with a Demonstration thereof, together with seven necessary rules belonging to the same, and what every rule teacheth. Chap. 7.  
Of Addition, Subtraction, Multiplication, and Division of Fractions. Chap. 8.  
Of the common rule of three belonging to Fractions with examples. Chap. 9  
Of the rule reverse, called in Latin Regula eversa, and the order of working thereby with examples. Chap. 10.  
Of the double rule called in Latin Regula duplex, and the order of working thereby with examples, and in working thereby how to know when you have to use Regula eversa, or the common rule of three. Chap. 11.  
Of the rule of Fellowship, and the order of working thereby, with diverse examples thereof. Chap. 12.  
Of Progression, what it is, and of the two kinds thereof, that is, Arithmetical and Geometrical. Chap. 13.  
Of Addition belonging to progression Arithmetical, and the order thereof, with examples. Chap. 14.  
Of Addition belonging to progression Geometrical, and the order thereof, with examples. Chap. 15.  
Of proportion, what it is, and of the three kinds thereof, that is, Arithmetical, Geometrical and Musical. Chap. 16.  
Of Arithmetical proportion, what it is, and how it is divided. Chap. 17.  
Of proportion Geometrical, what it is, and how it is divided. chap. 18.  
Of the chief and special kinds of Geometrical proportion, that is, of equality and inequality both greater and lesser. Chap. 19  
Of proportion of the greater inequality, and of the 2. kinds thereof, that is, Simplex, and Multiplex, and of their divers kinds, with certain Tables belonging thereunto Chap. 21.  
Of proportion of the lesser inequality. Chap. 21.  
Of Musical proportion, what it is, and of the two kinds thereof, that is, Simple and compound, of which compound there be also 2. kinds that is proper and unproper. Chap. 22.  
How proportions are to be set down in writing, and how they are to be added, subtracted, multiplied, and divided even like to fractions in all respects. Chap. 23.  
How to find out the square root of any number with examples. Chap. 24.  
The use of the square root in setting of Battles, which according to the Italian use are to be set four manner of ways, the order whereof is here set down with examples. Chap. 25.  
How to find the cubique root of any number, and the order thereof with examples. Chap. 26.  
Of Astronomical Fractions, whereto they serve, and how to add, subtract, multiply, and to divide the same. Chap. 27.  
How to divide such fractions when the divisor is greater than the dividend. Chap. 28.  
How to take the square root of Astronomical Fractions. Chap. 29.  
And immediately after these Chapters do follow the use of the Tables of Sines, lines tangent, and lines secant before mentioned, together with the Tables themselves.   

To the Reader touching the order of finding and correcting the faults escaped in printing.  
FIrst, I pray you vouchsafe with your pen to correct the numbers of the leaves of these books, which numbers are not in some leaves truly set down, that done, for the rest of the faults resort to the Table following, which consisteth of five collums, whereof the first on the left hand showeth the true number of the leaf, the second collum showeth the page both first and second: the third showeth the line: in counting whereof you must begin at the head of every page, and so proceed downward, leaving no printed line untold, though it containeth but one word, omitting notwithstanding such lines as do contain the title of any chapter, so shall you be sure to find any fault mentioned in the said Table, and the correction thereof hard by it on the right hand: for the fourth collum of this Table containeth the faults, and the fift collum the correction thereof: by help of which Table you may correct your own book, where need is, in setting down the correction in the margin right against the fault found.  

Errata.  Leaf Page Line The faults. The Correction. 1 2 27 for, 4270570. read 43 20, 70. 2 1 15 for, what, read. Why. 3 2 5 for, calling, read Culling. 5 2 9 for, squarewise, read squirewise. 10 1 32 for, limediate, read immediate. 24 2 33 for, the former 4. read the former doubled 4. 29 1 29 for, 2650. read 2550. 32 1 4 for, the productes, read the particular productes. 32 1 27 for, contain many, read contain so many. 32 2 31 for, over 1. read over 1‴. 32 2 32 for, appointing to his, read appointing to the product his 34 1 1 for, the number must be 24. r. the number 24. must be 37 1 18 for, 53″. read 50″. 42 1 4 for, 10. read 16. 42 2 23 for, 20. read 19 53 1 16 for, 890. read 899. 56 2 13 for, 23. degr. 38. read 23. degr. 28′ 131 1 30 for, precession, read precession. 137 2 13 for, ever the heaven, read every heaven, 139 2 3 for Anges, read Auges. 169 1 6 for returning, read returneth. 173 1 16 for, or, read and 181 1 32 for, 2160. miles, read 2160 b. miles. 193 2 6 for, Island, read Island. 199 2 5 for, Aphiscii, read Amphiscii. 210 1 31 for, before fastened, read before taken. 215 2 2 for, the hour of the north, r. the hour of the night. 216 2 2 for, dawning twilight, read dawning and twilight 217 1 28 for, the Celestial globe, read the terrestrial Globe. 227 2 20 for, mere lately, read more lately. 229 2 7 for, in a great sph are, read in a right sphere. 233 1 32 for, the 7. of Scorpio, read the 8. of Scorpio. 236 2 16 for, 30. deg. 47′. of, read the 3. deg. 47′. of 249 1 22 for, Equinoctionall, read Equinoctial. 300 1  In the figure, for 1. ♉. read ♑. 1.   

Of Arithmetic. Cap. 1.  
WHat is Arithmetic?  
It is the art of counting or numbering by figures.  
What is to number by figures?  
It is to express the value of any number in his proper Characters and figures, which is called by a Latin name Numeration.  
What belongeth to Numeration?  
Two things, to know the shapes of the figures, and the signification of their places.  
How many figures are there?  
These ten. 1. 2. 3. 4. 5. 6. 7. 8. 9 0. Whereof the tenth made like an. 0. as you see here, is called a cipher, which is no number of itself, but serveth only to fill up a number.  
What is number?  
Number is a collection or some of many ones added together.  
How is number divided?  
Into three kinds, that is Digit, Article and compound.  
Which be they?  
The Digit is any of the first 9 figures before set down. Article is any number ending in ten, as ten is one Article, twenty is two Articles, thirty is three Articles, etc. Compound is that which is compounded of Article & Digit, as 13. 14. 17. 24. etc.  
Show what signification every Digit hath according to his place, and in what order such place is to be considered in expressing any number?  
The order as touching the place, is to begin at the right hand, and so to proceed towards the left hand. For any of the 9 Digits whatsoever, standing in the first place, which is on the right hand, signifieth the value of himself only, in the second place ten times himself, in the third place hundredth times himself, in the fourth place a Thousand times himself, in the fift place ten Thousand times himself, in the sixth place a hundredth Thousand times himself, in the seventh place a Myllion, in the eight place ten millions, in the ninth place a hundredth millions in the tenth place a thousand millions, etc.  
Doth the cipher signify nothing?  
Yes it maketh a place wheresoever it standeth, so as it be not the outermost on the left hand: for there it hath no place at all, as here you may see in this number, 04500. whereof the first cipher on the right hand, signifieth the first place, the second cipher the second place, but the last and outermost cipher on the left hand, signifieth no place at all, because it hath no Digit standing before it towards the left, hand, and therefore though in this number there be 5. figures, yet it signifieth no more but four Thousand and five hundredth.  
By what means may a great number written in many figures, be readily expressed or told?  
By dividing the same into diverse parts with stréekes, or pricks made at the end of every third figure, beginning to tell from the right hand towards the left, as in this number, 4/270/570. In which, beginning with the cipher on the right hand, I tell one, two, and three, and there make a Stréeke, and so proceeding forth still towards the left hand, I make a Stréeke at the end of every third figure, by which Stréekes or partitions I make them now severst numbers, and every Stréeke must be named by this word Thousand. Notwithstanding, in expressing this number being thus divided, or any other such like, you must begin at the left hand, and say thus four Thousand thousand, three hundredth twenty Thousand, five hundredth and seventy: for by reason of this Division, the figure 4. standeth here alone, and in the first place, and in deed signifieth 4. millions, and by that means, you may more fitly express the said number in saying thus, four millions, three hundredth and twenty Thousand, five hundredth and seventy: And the better to discern the millions from the rest in great numbers, it shall not be a miss to set an M. signifying millions, right over the head of the Stréek which is drawn betwixt the sixth and seventh figures, as in this example containing a leaven figures, 34/545 ᵐ / 678/694. which is to be uttered thus, thirty four Thousand, five hundredth forty five millions, six hundredth seventy eight Thousand, six hundredth ninety and four Crowns, or Pounds, or whatsoever other denomination or name, it shall please you to give them.   

Of the four special kinds or parts of Arithmetic. Cap. 2.  
WHich are those four kinds?  
These, Addition, Subtraction, Multiplication and Division.  
What, is not Numeration also counted as a part?  
Because Numeration together with the figures, and places whereof it consisteth, are counted rather as first Elements, and principles of Arithmetic, then as parts or special kinds thereof.  

Of Addition.  
WHat is Addition?  
It is that which teacheth to bring many several Sums into one some.  
How is that done?  
First by placing every several number one right under another, under which you must draw a line, that done, you must add together the numbers of the first rank, beginning on the right hand with the lowest figure of the same rank, and so going upward to the highest figure of the same rank, and so from rank to rank, till you come to the last, and if the Somme of any rank do not exceed the number of any of the foresaid 9 Digits, then set down that Digit which comprehendeth that number right under his proper rank, beneath the line, but if the Somme of that rank, exceedeth the number of any one Digit by reason that it consisteth of Articles and Digits, then set down the Digit and keep the Article or Articles in your mind, to be added to the first figure of the next rank on the left hand, but if the Somme be an even Article or Articles: then set down a cipher, keeping the number of Article or Articles in your mind, be it, one, two, or three, to be added to the next rank, all which things you shall better understand by this example here following. As for example, I spent in one year 125.l. in another year 234. l, and in another year 240.l. Now to know the total Somme of all this, I place these several Sums one right under another. 

l  
125.  
234.  
240.  
599.   and then I draw a line under them as here you see.  
Than beginning on the right hand with the lowest figure of the first rank above the line, I say that a cipher and 4. is but 4. Again 4. and 5. maketh 9 which I set down under the line, then proceeding to the second rank towards the left hand, I say that 4. and 3. maketh 7. and 7. and 2. maketh 9 which I also set down, then removing to the third rank, I say that 2. and 2. maketh 4. and 4. and 1. maketh 5. which I also set down as you see in the former example, so as the total Somme under the line is. 599.l.  
Another example having Ciphers mixed with Digits.  

3047.  
4508.  
3049.  
10/604.   
Here I say that 9 and 8. maketh 17. and 17. and 7 maketh 24. wherefore I set down the Digit 4. and keep 2. Articles in mind, which being added to the lowest figure of the second rank, which is 14. maketh 6. then 6. and 4. maketh 10. here I set down a cipher, keeping one Article in mind, which being added to the figure 5. of the third rank maketh 6. which I also set down, than I say that 3. and 4. maketh 7. and 7. and 3. maketh 10. for the which I set down first a cipher, and then because there is no more to be added. I set down on the left hand the one Article which I had in mind, so as the whole Somme cometh to 10/604. as in the former example.  
How are pounds, shillings, pence, half pence, and farthings and all other numbers of diverse Denominations to be added?  
You must divide every several name into diverse Collums or Spaces by themselves, and then beginning with the first on the right hand, you must add every Collum by itself, bringing farthings to half pence, and half pence to pence, pence to shillings, and shillings to pounds, setting the Somme of every Collum under the neither line as you see in this example following.  
l s d ob. que 345 13 1 0 1 234 11 0 1 1 45 14 9 1 0 320 6 8 1 1 946 5 8 0 1  
Here first beginning with the Collum of farthings, I find therein 3. farthings which is one half penny and one farthing. Wherhfore I set down the odd farthing as you see, and keep the half penny in mind: then adding the half penny in mind to the lowest half penny of the second Collum, I say that 1. in mind and 1. maketh 2. and 2. and 1. maketh 3. then 3. and 1. maketh 4. which 4. half pence because they make just two pence, I set down a cipher keeping the two pence in mind, which two pence being added to 8. maketh 10. then 10. and 9 maketh 19 and 19 and 1. maketh 20: How because that 20.d. maketh one shilling and 8.d. I set down the 8.d. keeping the shilling in mind, which on shilling being added to the 6. of the next Collum maketh 7. then 7. and 4. maketh 11. and 11. and 1. maketh 12. then 12. & 3. maketh 15. wherefore I set down 5. keeping the Article in mind, which being added to one of the next Collum maketh 2. and 2. and 1. maketh 3. and 3. and one maketh 4. Articles, which 4. Articles maketh 40. s which is two pound which I keep in mind, and therefore I add the 2.l. to the Collum of pounds, saying that 2. and 5. maketh 7. and 7. and 4. maketh a 11. and 11. and 5. maketh 16. wherefore I set down 6. keeping the one Article in mind, which being added to 2. of the next Collum, maketh 3. then 3. and 4. maketh 7. and 7. and 3. maketh 10. then 10. and 4. maketh 14. wherefore I set down 4. keeping one Article in mind, which being added to 3. of the next Collum maketh 4. then 4. and 2. maketh 6. and 6. and 3. maketh 9 which I also set down, so as the total Somme amounteth to 946.l. 5. s. 8.d. not half penny, one farthing as you see in the former example.  
How shall I know whether these several Sums be truly added or not?  
Some do teach it to be done by calling out all the nine, which way is more tedious than sure: for the surest trial indeed is to be done by Subtracting the several Sums out of the total Somme, of which Subtraction we come now to speak, for all the four special kinds are tried one by an other.    

Of Subtraction. Cap. 3.  
WHat doth Subtraction teach?  
It teacheth to take a lesser number out of a greater and to see what remaineth.  
What is to be observed in this kind?  
First, you must set down your greater number above, and then the lesser number right under the same. As for example, I have lent to one 564.l. and he hath paid me thereof 57l. Here to know what remaineth, I first set down the number lent, and under that the number paid, and then draw a line as you see in this manner.  

Lent. 564.l.  
paid. 57l.   
Here beginning on the right hand, I first say, take 7. out of 4. that cannot be: wherefore I take one Article of the next figure or place of the lent number, which Article being added to 4. maketh 14. then I say take 7. out of 14. and there remaineth 7. which I set down under the 4. then I add that one Article which I borrowed, to the second figure of the paid number which is 5. saying that 5. and 1. in mind maketh 6. then take 6. out of 6. and there remaineth nothing, wherefore I set down a cipher, under the 5. of the paid number, than I proceed to the third figure of the lent number, which is 5. and because I find nothing written under it, nor have nothing in mind to take out of it, I say, take nothing out of 5. and their remaineth still 5. so as the remainder is 507. l as you see in this example following.  

Lent. 564.l.  
Paid. 57l.  
Remain. 507   
How shall I know whether this be right or not?  
By adding the remainder and the number paid together, the Somme whereof (if you have done well) willbe all one with the number lent, as in the former example, I first add 7. and 7. together and that maketh 14. wherefore according to the precepts of Addition before taught, I set down 4. keeping the Article in mind, than I say, one in mind and 5. maketh 6. which I also set down, than I say nothing and 5. is 5. which I set down in the third place, which in all maketh 564. a number equal to the number lent, as you see here following.  

Lent, 564.l.  
Paid. 57:  
Remain. 507.  
Proof. 564.   
You may perceive by this, that if any figure of the paid number be greater than the figure over him, out of the which it is to be Subtracted, you must always borrow one Article of his next fellow, to be added again to him in his proper place. But you have to note, that having to deal with numbers of diverse Denominations, them in borrowing any number, you must always have respect to the Denomination or name of the thing, from whence you borrow, as in borrowing from shillings you borrow 12. and not 10. from pounds you borrow not one Article but 2. Articles which do make 20. s but when the whole number is altogether of one self Denomination, than you must always borrow one Article which is 10. to make up your number that wanteth. As you shall more plainly perceive by this example containing numbers of diverse Denominations, as of pounds, shillings and pence, half pence, and farthings. Suppose therefore that you have lent to one 467. l 13. s. 4.d. ob. q. and he hath paid you again thereof, 89.l. 16. s. 9d. ob. q. Here having set down the Somme lent in several Collums, according to their diverse names, and then the Somme paid, right under the same, draw a line as you see in this example following.  
 l s d ob que Lent. 467 13 4 1 1 paid. 89 16 9 1 0 Remain. 377 16 7 0 1  
Here beginning with the first Collum on the right hand, I say take nothing out of one, and one still remaineth, which I set down, then proceeding to the next, I say take one out of one and nothing remaineth, wherefore I set down a cipher, then proceeding to the next Collum, I say take 9 out of 4. that cannot be, wherefore I borrow a shilling of the next Collum that is 12.d. which being added to 4. maketh 16. pence, than I say take 9 out of 16. and there remaineth 7. which I set down, then proceeding to the next, I add the one shilling which I borrowed, to the 16. which maketh 17. s. than I say take 17. s. out of 13. s. that cannot be. wherefore I borrow one pound which is 2. Articles of the next rank, that is 20. s. which being added to the 13. s. maketh 33. s. than I say take 17. out of 33. s. and there remaineth 16. s. which I set down, than I add the one pound which I borrowed, to 9 and that maketh 10. then I say take 10. out of 7. that cannot be, wherefore I borrow one Article out of the next 6. which being added to the 7. maketh 17. then I say take 10. out of 17. and there remaineth 7. which I set down, than the one, which I borrowed, I add to the 8. of the next rank, and that maketh 9 Again I say take 9 out of 6. that cannot be: wherefore I borrow one Article of the next 4. which being added to 6. maketh 16. then I say take 9 out of 16 and there remaineth 7. which I set down, than I take the one which I borrowed out of the 4. & there remaineth 3. so as the remainder is as you see in the former example. 377.l. 16. s. 7.d. 0. q.  
How shall I try whether this be true or not?  
By adding the remainder and the Somme paid together: as in the former example, and of that Addition, will rise if you have done truly a Some like in every condition to the Somme lent: In making which proof or trial you cannot lightly err, if you remember to reduce pence to shillings, and shillings to pounds, and therefore in the Collum of pence, no particular Somme can be above 11. d, nor in the Collum of shillings no particular some can be above 19, s for if it be 20, s, than it is a pound and must be brought to the Collum of pounds,   

Of Multiplication. Cap. 4.  
WHat is Multiplication?  
It is the producing or bringing forth of a third number, by Multiplying two other numbers the one into the other: And it consisteth of three numbers, that is the multiplycand, the multiplyer, and the product.  
What signify those names?  
The Multiplicand is that number which is to be multiplied, and the multiplyer is that whereby the same is multiplied, & the product is the Somme of such. Multiplication: As for example, if I would multiply 4. by 3. as in saying 3. times 4. maketh 12. here the number of 4. is the Multiplycand, and the number 3. is the multiplyer, and the number 12. is the product of that multiplication.  
What order is to be observed in multiplying, and how are those numbers to be set?  
Before I teach you the true order of multiplying, I think it good to set your down a table of multiplication, which unless you learn perfectly by hart, you shall never multiply readily nor quickly.  

The Table of multiplication.  2 2 4 2 3 6 2 4 8 2 5 10 2 6 12 2 7 14 2 8 16 2 9 18 2 10 20 3 3 9 3 4 12 3 5 15 3 6 18 3 7 21 3 8 24 3 9 27 3 10 30 4 4 16 4 5 20 4 6 24 4 7 28 4 8 32 4 9 36 4 10 40 5 5 25 5 6 30 5 7 35 5 8 40 5 9 45 5 10 50 6 6 36 6 7 42 6 8 48 6 9 54 6 10 60 7 7 49 7 8 56 7 9 63 7 10 70 8 8 64 8 9 72 8 10 80 9 9 81 9 10 90 10 10 100  
How is this Table to be read?  
In this manner, 2. times 2. maketh. 4. and 2. times 3. maketh 6. and 2. times 4. maketh 8. and so forth: multiplying still one Digit by another, until you come to a 100 for this Table serveth only for Digits, which may be made to extend so far as you will, and until you have learned the foresaid Table without book, you may help yourself with this other Table of Digits made squirewise as you see here.  
1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 1  4 6 8 10 12 14 16 18 2   9 12 15 18 21 24 27 3    16 20 24 28 32 36 4     25 30 35 40 45 5      36 42 48 54 6       49 56 63 7        64 72 8         81 9  
In the front of which Table are set down the 9 Digits beginning on the left hand, and so proceeding to the right hand, from 1. to 9 Again on the right side of the said Table are set down the foresaid 9 Digits, beginning above, and so proceeding right down from 1. to 9 the use whereof is thus, first seek the Digit to be Multiplied in the front, and seek the other whereby you have to multiply the same on the right hand, and the square Angle answering to these. 2. Digits will show the product of such multiplication. As for example having to multiply 7. by 6. I first seek the greater Digit which is 7. in the front, and the lesser which is 6. on the right hand, the product whereof I find in the square Angle answering both to 7. and to 6. to be 42. & the like is to be observed in any 2. Digits that are of like value as 7. times 7. the product whereof is 49. But the first Table being perfectly learned without book, whensoever you have to multiply one number by another, you must observe these rules here following.  
First that you set down the first figure of your multiplyer right under the first figure of the number that is to be multiplied, on the right hand, and then orderly to place the rest of the figures of your multiplyer, be they few or many, towards your left hand, directly under the rest of the figures of the number that is to beé multiplied, for if the figures stand not in order one right over another it will breed a confusion in your working. Secondly you must not forget to multiply all the figures of the number that is to be multiplied, by the first figure of your multiplyer before you deal with the next multiplyer, beginning always on your right hand, and so to proceed from one to another, whereby you shall make as many several products as there be figures in your multiplyer. Thirdly you must remember to set down the first figure of every several product, right under the figure of that multiplyer, whereby you do multiply, and having ended your Multiplication, draw a line under the several products: that done, ad the several products together according to the rules of Addition, & the Some thereof shallbe the total or general product of that multiplication: all which rules you shall the better understand by working this example following. Suppose then that you would know how many hours there are in a year, knowing first that a year consisteth of 365. days, here because that every natural day containeth 24. hours comprehending both day & night, you have to multiply 365. by 24. and therefore you must first set down 365. because it is the greater number, and is the multiplycand, which must always stand above, & right under that, your multiplyer which is 24. is to be set down in his due place, according to the Rules before taught thus as you see here. 

365. the multiplicand  
24. the multiplier.   Than say thus 4. times 5. maketh 20. & having set down a cipher right under the 4. keep the 2. Articles in mind, them say 4. times 6. is 24. & 2. in mind is 26. here set down 6. & keep 2. in mind, then say 4. times 3. is 12. which with the 2. in mind maketh 14. here first set down the 4. under the 3. and because you can proceed no further: you must therefore set down hard by the 4. on the left hand the one Article which you had in mind, then having canceled the first figure of the multiplyer, by making a Dash through it with your pen, as you see in the example following: Proceed with the other figure of the multiplyer, saying that 2. times 5. maketh 10. wherefore set down a cipher right under the said 2. keeping the one Article in mind, then say 2. times 6. is 12. and one in mind maketh 13. wherefore set down 3. and keep one in mind, then say 2. times 3. is 6. which with one in mind maketh 7. the which you must set down, and because you have made an end of your multiplication, cancel the 2. and draw another line under the 2. several products, that done, ad together whatsoever is contained betwixt the two lines, and you shall find the general product to be 8/760. hours.  
The multiplycand. 
365.  The multiplyer. 
24.  The several products. 


1460.  
730.    The general product or total Som. 
8760.   
How shall I know whether the last multiplication be right or not?  
By dividing the general product by the Multiplyer, for in so doing your quotient will be like unto the first Somme that was Multiplied, which you cannot do, until such time as you have learned to divide, and therefore having first showed certain compendious ways of multiplication, I will then proceed to Division.  

Certain compendious ways of multiplication.  
WHen is any such way to be used?  
When the Multiplyer beginning on the right hand with one cipher or with many, endeth on the left hand with the digit 1. as these numbers following, 10/100 100/1000 / etc.  
Why what is then to be done?  
If you have to Multiply by 10. then you have no more to do, but to set down on the right hand of the number that is to be Multiplied, one cipher, if by a 100 then 2. Ciphers, if by a 1000 then 3. Ciphers, as for example: if you would Multiply 365. by 10. then by setting down on the right hand one cipher as hath been said, the product will be 3/650. if you set down 2. Ciphers the product will be 36/500. if you set down 3. Ciphers, than the product will be 365/000.  
What if the number do end on the left hand with any other Digit, as 2. 3. or more, as 200. 300. 400?  
Than you must Multiply the number of the Multiplycand first by that Digit, and then add to the end of the product on the right hand all the Ciphers annexed to the said Digit, as if you would Multiply 365. by 200. First say 2. times 5. maketh 10. and 2. 6. and one in mind is 13. and so to proceed in Multiplying every figure of the Multiplycand by 2. and you shall find the product to be 730. whereunto if you add on the right hand 2. Ciphers the whole product will be 73/000.    

Of Division. Cap. 5.  
WHat is Division?  
Division is that whereby any number is divided into as many parts as you william.  
How many numbers are incident to Division and how are they called?  
These 4. that is to say, the number which is to be divided which is called the dividend, the second number whereby you do divide which is called the Divisor, the third number is called the quotient, which showeth how many times the Divisor is comprehended in the dividend, and the fourth number is called the remainder, if any be.  
What order is to be observed in Division?  
This here following, first set down your dividend and directly under that, beginning on the left hand, set down your Divisor, that is to say the first figure of your Divisor right under the first figure of your dividend on the left hand, and so consequently one after another proceeding towards the right hand, which is always to be done, so often as your Divisor doth not exceed in quantity, the figure standing right over his head, for if it do, than you must remove your Divisor one figure further towards the right hand. As for example if you would divide 487. by 53. you must not set the first figure of your Divisor which is 5. under the first figure of your dividend, which is but 4. but under the second figure of your dividend which is 8. for you cannot take 53. out of 48. and therefore you must set the first figure of your Divisor, under the second figure of the dividend, and so follow on with the rest and then draw a line as you see here in this example following.  
What is them to be done? 

The dividend. 487.  
The divisor. 53.   (9 the quotient  
Than you must ask how many times 5. is comprehended in 48. and you shall find that 5. is comprehended in 48. 9 times which 9 being the quotient, must be placed on your right hand, behind a crooked line made like a half Moon as you see in the example above, then Multiply the first figure of the Divisor by the quotient 9 and the product thereof shall be 45. which being taken out of 48. there will remain 3. which you must set down over the head of 8. and stréeke out the 48. and also the first figure of your Divisor which is 5. that done Multiply the second figure of your divisor which is 3. by the foresaid quotient which is 9 & the product thereof will be 27. which being taken out of 37. there remaineth 10. which you must set over the head of 37. and cancel the 37. and also the 3. beneath, as you see in this example following.  
Wherein you see that the dividend is 487. the Divisor 53 the quotient 9 and the remainder 10. which if you will apply to any use you may imagine that there is 487.l. to be divided amongst 53. soldiers, and by working as before, you shall found that every Soldier must have 9l. and there shall be remaining 10.l. to be divided amongst the foresaid 53. Soldiers, which 10.l. being reduced to shillings, may be divided amongst the Soldiers as well as the pounds: As for example if you Multiply 10.l. by 20. s. it will make 200. s. which being divided by 53. the first Divisor, the quotient will be 3. s. and the remainder 41. s. and that being reduced into pence, which is done by Multiplying 41. by 12. the quotient, will be 9d. and the remainder 15.d. which if you Multiply by 4. you shall reduce them to farthings, the product whereof will be 60. which product being divided by 53. the first Divisor you shall found in the quotient 1. farthing and the remainder to be 7/53 of a farthing, the value of which Fraction how to found out is taught hereafter when we come to speak of Fractions. So as you see by this means that every Soldier shall have for his share 9l. 3. s. 9d. q. and some what more a thing of no moment.  
How many things are to be remembered in Division?  
These five Rules here following. First that you put no number at one time in the quotient above 9 Secondly to moderate your quotient in such sort, as having Multiplied the first figure of the Deuisor by the quotient, there may remain sufficient for the next figure of the Divisor being Multiplied by the quotient to be deducted out of that number which standeth right over his head. Thirdly that you Multiply every figure contained in the Divisor by the quotient. Fourthly that if at any time in working it happeneth so as your Divisor is not comprehended in the number over his head, then to put a cipher in the quotient and to remove your Divisor one figure further towards the right hand, and that done to work as before. Fiftly to see that the last remainder, if there be any left, do not exceed in quantity the Divisor, all which things you shall better understand by this one example following: Suppose then that you have to divide 819096. by 92. here having set down your dividend and Divisor in such order as is before taught, and as you see here in this example.  
First ask how many times 9 is in 81. and you shall found the quotient to be 8. which being Multiplied into 9 maketh 72. which if you take out of 81. there shall remain 9 which must be set down over the head of the figure 1. and cancel the said 81. together with the first figure of the Divisor, which is 9 then say 2. times 8. is 16. which being Subtracted out of 99 there remaineth 83. which you must set over the head of the 99 and cancel the 99 that is above, and also the 2. beneath, as you see in the former example. That done remove your Divisor towards the right hand, that is to say by setting the last figure of your Divisor which is 2. under the next cipher on the right hand, and place the first figure of your Divisor which is 9 next to that towards the left hand: then ask how many times 9 is comprehended in 83. and you shall found 9 times, which 9 must be set down in the quotient next to the 8. then say 9 times 9 is 81. which being taken out of 83. there remaineth 2. then having set down the 2. over the head of 3. cancel the 83. above & also the 9 beneath, then Multiply the last figure of the Divisor which is 2. by the 9 which is in the quotient, and the product thereof is 18. which being taken out of 20. there remaineth 2. which 2. must be set down over the cipher, and the cipher canceled and also the 2. beneath, as you see in this example following.  
That done remove your Divisor one figure further towards the right hand by setting the 2. under the 9 which is the last figure save one of the dividend, and place the first figure of your Divisor which is 9 next to that on the left hand, then ask how many times 9 is contained in 2. and you shall find none, wherefore you must set down a cipher in the quotient and cancel the Divisor as you see in this example  
That done remove your Divisor again by setting the 2. under the last figure of the diuidend and place the 9 next to that on the left hand, then ask how many times 9 is contained in 29. and you shall found 3. times, which 3. you must set down in the quotient as you see in the example following, then Multiply the first figure of your Divisor which is 9 by 3. and the product thereof will be 27. which being taken out of 29. there will remain 2. which 2. you must set over the head of 9 and cancel the 9 above, together with the 9 beneath, then Multiply the 3. which is in the quotient by the last figure of your Divisor which is 2. and the product thereof will be 6. which being taken out of 26. there remaineth 20. wherefore you must set down a cipher over the 6. which is the last figure of the dividend, and cancel the said 6. above, and also the 2. beneath and then you have done rightly, and in such order as all the former examples and this also next following do plainly show.  
For here you see that the dividend is 819/096. the Divisor 92. the quotient 8903. and the remainder 20. which remainder must be set on the right hand of the quotient over the Divisor, having a line drawn betwixt them as you see in the last example.  
How shall I know whether I have divided truly or not?  
By Multiplying the quotient by your Divisor & by adding to the product thereof the remainder, for so shall you have a number like to the first dividend if you have wrought well, if not, than your Somme will be either more or less than the dividend.  

Certain compendious ways of Division.  
IS there no shorter way of Division?  
Not, unless the Divisor hath one cipher or more on the right hand, for than you may use a bréefer way by cutting off the last figure, or figures of the dividend: As for example if you had to divide 3708. by 10. here cut off the last figure 8. with a dash of your pen in this wise 370/8. and the quotient will be 370. and the remainder shall be 8. Again if you had to divide the foresaid number by 100 then cut off the 2. last figures of the dividend, that is the 8. & the cipher standing next it, and then the quotient will be but 37. and the remainder 8. as before. but if you have to divide any number by 1000 then you must cut off 3. figures of the dividend, and so forth, remembering always for every cipher in the Divisor, to cut off one figure or cipher of the dividend.   

Of halfing any number.  
HOw is that done?  
By dividing the number by 2. As for example, if you would know the half of 3708. divide the same by 2. and you shall find the quotient to be 1854. which is the just one half of the dividend.    

Of the Rule of three otherwise called the golden Rule whereof there be three kinds, that is, the common Rule, the rule Reverse, and the Double rule. Cap. 6.  
WHerefore is it called the Rule of three?  
Because that by 3. known numbers, it teacheth you to find out a fourth number unknown.  
How are such 3. numbers to be placed?  
The first is to be placed on the left hand, the third on the right hand, and the second in the midst betwixt both.  
How shall I know in what place each one is to be set?  
By marking to what number the question is annexed, for that must always be the third number, and having the third number, you shall quickly have the first, because the first and third must always be of one self Denomination or name, betwixt which two, the second number is to be placed: As for example let this be the question. If I pay 35. s. for 13. yards of Linen cloth, how much shall I pay for 3. yards, here because the question is annexed to the number 3. yards, that must be the third number and is to be placed on the right hand, then because that the number 13. yards is of the self same Denomination, that must be the first number, and is to be placed on the left hand, & the number 35. s. (which is the price) is to be placed in the midst betwixt the other two, as you see here following. yards, s. yards: 13. 35. 3.  
Now must you make your question in this manner, if 13. yards did cost me 35. s. what shall 3. yards cost me, and so the first and last shall be of one Denomination or name, that is to sayyards.  
What order is to be observed in working by this rule?  
You must Multiply the third and second numbers the one into the other, and divide the product there of by the first, and the quotient will show you what the fourth number should be, which fourth number is always of like Denomination to the second or middle number, as in the former example, first Multiply 35. by 3. and the product will be 105. which if you divide by 13. the quotient will be 8. s. and one third of a shilling which is 4.d. and that is the price of 3. yards which is the number that you seek to know. And this is the common kind of working by the Rule of 3. whereof it is called the common Rule of three.  
What if there be diverse Denominations either of Coin as of pounds, shillings, and pence, or parts of yards, as quarters, and half quarters, and such like?  
Than you must reduce them all to the smallest Denomination which belongeth to Fractions, whereof we come now to treat of.   

Of Fractions. Cap. 7.  
WHat are Fractions?  
They are broken parts of some whole thing called in Latin Integrum.  
What is Integrum?  
Any thing that is whole, and not broken, or divided into parts: As one whole yard, a pound, a shilling.  
Of how many numbers doth a Fraction consist?  
Of two, that is the Numerator, and the denominator, whereof the Numerator is always set above, and the denominator beneath, having a little line drawn betwixt them thus ½ which signifieth one second or one half, again two thirds and three fourth's are written thus, 23/34 whereof the first signifieth two thirds, and the other three fourth's and such like.  
What is meant by these two words Numerator, and Denominator, and whereto serve they?  
The denominator is as much to say as the namer of the parts, which showeth into how many parts the Integrum is to be divided, and the Numerator is as much to say as the Numberer, and showeth how many of those parts are to be taken: As for example ⅔ of a shilling, here the nethermost called the denominator, showeth that the shilling is to be divided into three parts, as into three groats, and the upper number called the Numerator, showeth that you must take 2. out of those 3. parts, so as ⅔ of a shilling is as much to say as 2. groats or 8.d. Again Fractions may be divided into two kinds, that is, into simple Fractions, and into Fractions of Fractions.  
Define those two kinds?  
Simple Fractions are such as I speak of before which are the first & limediate parts of any Integrum that is divided into parts, but if those Fractions be divided into more Fractions, then are they called Fractions of Fractions, as when I say three fourthes of two thirds, of six seventhes and such like.  
How are such Fractions to be set down in writing?  
In manner of simple Fractions thus ¾ of ⅔ of 6/7.  
Make some demonstration of this example that I may the better understand it?  
Imagine that there is a whole piece of Gold of seven. s. as our Angel in times past hath been, which seven. s. be the seven. parts of that piece of Gold, for trial whereof lay down before you seven. twelve penny pieces of Silver, or in steed thereof seven counters, to signify those seven. parts, of which seven. parts you must first take away vi. rejecting or laying a side the odd seventh part, then divide those 6. parts into 3. parts, and every such part will be just 2. s. of which 3. parts you must take away 2. parts that is 4. s. rejecting the other third part which remaineth, that done, divide those 2. parts which you have taken away into 4. parts, which is 4. s. and take 3. of them rejecting the fourth part, so shall you find that ¾ of ⅔ of 6/7 of the foresaid piece of Gold is just 3. s. and there remaineth still of the parts rejected 4. s. which being added to 3. s. that was taken away, do make up again the whole Integrum of 7. s. Notwithstanding Fractions of Fractions do seldom chance in the Division of any number, but if they do, you must reduce them into simple Fractions, before you can deal with them any manner of way, and because there are diverse rules belonging to Fractions, without the knowledge whereof you can neither add, nor subtract them, nor yet join them with any Integrum, I will briefly set down here seven necessary rules for the same.  

Seven necessary Rules belonging to Fractions.  
WHat doth the first rule teach?  
To bring Fractions of Fractions into simple Fractions.  
How is that done?  
Thus, Multiply the first Numerator into the second, and if there be any more Fractions, then Multiply also the said product of the first two Numerators into the third Numerator, and that shall be the Numerator of the simple Fraction, then Multiply the denominators in like manner, and the Somme thereof shall be the denominator of the simple Fraction, as in the former example ¾ of ⅔ of 6/7 here I Multiply the first 2. Numerators together, in saying 2. times 3. maketh 6. now because I have a third Numerator, I multiply the product of the first 2. Numerators, which is 6. into the third Numerator which is also 6. saying, 6. times 6. maketh 36. and that shall be the Numerator of the simple Fraction, which I set down. Than by Multiplying the 3. denominators in like manner, I find the denominator of the simple Fraction to be 84. the which I set down under the 36. and draw a line betwixt them so as now I find that ¾ of ⅔ of 6/7 do make 36/84 which indeed is no more but 3/7 as you shall learn hereafter by the sixth rule.  
What doth the second rule teach?  
To bring Fractions being more in value then Integrums into integrums.  
When are Fractions said to be more than integrums?  
When the Numerator is a greater number than the denominator, for if they be both of like value as 34/34 and such like, than such Fraction is an even integrum.  
How shall I know how many Integrums such Fractions as be more than Integrums do contain?  
By dividing the Numerator by the denominator, and if any thing remain, writ that above the denominator, as in this example 806/7 here if you divide 806. by 7. which is the denominator, you shall find the quotient to contain 115. Integrums, and the remainder to be 1. which is 1/7.  
What doth the third rule teach?  
To bring Integrums into parts by Multiplying the number of the Integrums by the denominator of those parts, as if you would bring 64. yards into quarters, Multiply 64. by 4. and there will arise thereof 256. quarters.  
What doth the fourth rule teach?  
It teacheth to bring Integrums having Fractions annexed to them, into one Fraction.  
How is that done?  
Multiply the number of the Integrums by the denominator of the Fraction, and then add to the product the Numerator of the said Fraction, and that some shall be the Numerator, under which writ the denominator aforesaid, and so you shall find that 23. Integrums having ⅔ annexed thereunto shall make ●1/3.  
What doth the fift rule teach?  
It teacheth to express a Fraction written with many figures, in so few as may be.  
How shall I do that?  
Thus, find out some number that may first divide the Numerator, and then the denominator severally by themselves, without leaving in either of them any remainder, and the quotient of the first Division will show the Numerator, and the quotient of the second Division will show the denominator, but if you cannot readily find out a number that will divide them both without leaving some remainder, then leave not to Subtract the lesser number out of the greater, until you find two like numbers, by one of the which two like numbers divide both the Numerator, and also the denominator severally as before, and the quotient will show that which you seek: but if such 2. like numbers cannot be found, than you may assure yourself that the Fraction cannot be written in lesser figures than they already be.  
Give examples of these two ways?  
For example of the first way, suppose that you would express 9/12 in lesser numbers, here seek out some number that may divide evenly both the Numerator, and also the denominator, without leaving in either of them any remainder, which by dividing each of them by the number of 3. you may do: For by ask how many times 3. is contained in 9 the quotient will be 3. and in ask how many times 3 is contained in 12. the quotient shall be 4. so shall you found 9/12 to be no more than ¾. Now for example of the second way, let 27/81 be the Fraction which you would set down in lesser figures, here because you cannot readily find out a number that will evenly divide both the Numerator and the denominator, Subtract the lesser out of the greater, that is to say 27. out of 81. and there will still remain 54. from which Subtract 27. and there will remain 27. which 2. numbers because they are both like, divide the Numerator and denominator of the foresaid Fraction, each of them by one of these numbers, that is to say, by 27. and the quotient of the first Division will show you the Numerator which is 1. and the quotient of the second Division will show you the denominator which is 3. so as you shall found ⅓ to be as much in value as 27/81 and note that when there be Ciphers both in the Numerator and in the denominator standing in such sort as they may be evenly cut off, the remainder will show the fewest figures, where in the Fraction may be written as 200/500 here by cutting off the two Ciphers as well beneath as above the line, with a dash of your pen in this manner ⅖ 00/00 the remainder shall be ⅖ which is as much in value as 200/500.  
What doth the sixth rule teach?  
It teacheth to found out the value of the Fraction of any integrum.  
How is that done?  
Thus, Multiply the Numerator by the known parts of the Integrum, and divide the product thereof by the denominator, so shall you have the value of the Fraction: As for example, if you would know the value of ¾ of an Angel, consider first what parts an Angel hath, and you shall found the parts thereof to be 10. s. or 30. groats, here if you Multiply 3. by 10. s. it will make 30. s. which being divided by 4. you shall find in the quotient 7. s. and the remainder to be 2/4 or one half of a shilling, which is 6.d. Again if you Multiply 3. by 30. groats it will make 90. groats, which being divided by 4. you shall find in the quotient 22. groats, and the remainder to be 2/4 or one half of a groat which is 2. pence, making in all 7. s. 6.d. like unto the first Somme, so as you see that ¾ of an Angel is 7. s. 6.d. and thus you may deal with the Fractions of any other Integrum that hath known parts.  
What doth the seventh rule teach?  
It teacheth to bring Fractions of diverse denominations to one self denomination, without the which neither Addition, nor Subtraction of Fractions can be made: As for example if you would add ⅔ and ⅘ together, you must first bring them to one self denomination thus, Multiply the denominators the one into the other, and the product thereof shall be a common denominator to both the Fractions, wherefore say 3. times 5. do make 15. which must be set down in 2. several places by themselves thus, 15. and 15. then Multiply the Numerator of the first Fraction into the denominator of the second, as 2. into 5. maketh 10. which set down over the first common denominator thus 10/15 then Multiply the Numerator of the second Fraction into the denominator of the first Fraction, as 4. into 3. which maketh 12. and that must be likewise set down over the second common denominator thus 12/15 so shall you found that ⅔ and ⅘ are all one in value with 10/15 and 12/15.    

Addition of Fractions. Cap. 8.  
HOw are Fractions to be added together?  
Fractions being first brought to one denomination are easily added, for than you have no more to do but to add the Numerators together, and to writ the common denominator under the Somme of such Addition, as in the former example, 10/15 being added to 12/15 maketh 22/15 but if the denominators be not like, than you must make them like by the seventh rule before taught.  
What if there be three Fractions of diverse denominations to be added together as ⅔ ¾ ⅘?  
Than having reduced the 2. first to one self denomination add the 2. Numerators together and writ under the Somme thereof the common denominator, that done, deal with that Fraction last found, and with the third as you did before with the 2. Fractions that had diverse denominations, and so you shall found that ⅔ and ¾ being first brought to one denomination, and then added together do make 17/12. whereto if you will add ⅘. than you must bring them again to one self denomination, and so you shall find that 17/12. and ⅘ do make 85/60. and 48/60. which last 2. Numerators being added together do make in all 133/60. which is 2. Integrums and 13/60.  

Subtraction of Fractions.  
HOw are Fractions to be Subtracted, one out of another, or out of Integrums?  
First make the denominators like as you did before in Addition, then take the lesser Numerator out of the greater, and under the remainder thereof writ the common denominator, so shall you found that 3/7. being taken out of 6/7. there remaineth 3/7. but if you would Subtract Fractions out of Integrums, than you must take one of the Integrums and break it into parts. As for example, to Subtract 3/7. out of 9 Integrums take one from 9 and break it into parts making it 7/7. for that is one Integrum, for whensoever the Numerator is made like and equal to the denominator, it signifieth one Integrum, then take 3/7. out of 7/7. and there remaineth 4/7. and 8. integrums.  
What is to be observed in breaking the integrum?  
In breaking the Integrum, you are to be directed always by the denominator of the Fraction which you have to Subtract, for the Integrum which should be the Numerator, must be equal in value to the said denominator, as before is said.   

multiplication of Fractions.  
HOw are Fractions to be Multiplied by Fractions, or Integrums by Fractions?  
As touching Fractions, Multiply the Numerators one into another, and the product thereof shall be the Numerator, then Multiply the denominators in like manner, and the product thereof shall be the denominator, so shall you found that 5/7. being Multiplied by ¾. do make 15/28. But if you would Multiply integrums by Fractions, then Multiply the integrums by the Numerator of the Fraction, and under the product thereof, set down the denominator of the same Fraction, drawing a line betwixt them, so shall you found that 20. integrums being Multiplied by 5/12. do make 100/12. that is to say 8. integrums and 4/12. which is ⅓, etc.   

Division of Fractions.  
HOw are Fractions to be divided by Fractions, or integrums by Fractions, or Fractions by integrums?  
The divison of Fractions is done by multiplication thus, first set down your dividend on the left hand, and the Divisor always on the right hand, and then draw 2. cross lines like a Saint Andrews cross betwixt them, which shall direct you in your working: As for example, if you would divide ⅔. by ⅘. set them down dividend thus ⅔ ⅘ divisor work as followeth, first Multiply the Numerator of the dividend, by the denominator of the Divisor, and the product there of shallbe the Numerator, then Multiply the denominator of the dividend, by the Numerator of the Divisor, and that shall be the denominator, so shall you found that ⅔. being divided by ⅘. there remaineth 10/12. that is ⅚. but if you would divide Integrums by Fractions, or contrariwise Fractions by Integrums, then make of the Integrums a Fraction, by setting down 1. in the place of a denominator under the Integrums, and work as before, so shall you found that 7/1. Integrums being divided by ¾. do make 28/3. contrarily if you divide ¾. by 7/1. Integrums there will arise 3/28.  
What if I have to divide Fractions annexed to integrums?  
Than you must first reduce each Integrum with his Fraction annexed, into one self Fraction by the fourth rule of Fractions before taught. As for example, if you would divide 9 Integrums having ¾. thereunto annexed, by 3. Integrums and ⅓. here the fourth rule of Fractions teacheth you first to Multiply the Integrum 9 by 4. which is the denominator of the Fraction annexed, the product whereof is 36. whereunto by adding the Numerator of the same Fraction which is 3. you shall make the Numerator of the Dividend to be 39 under which you must set the denominator 4. thus 39/4. and that is your whole dividend, then having in like manner brought the Divisor which is 3. Integrums and ⅓. into one self Fraction, work as the former rule of Division teacheth you, and you shall produce 117/40. which is 2. Integrums and 37/40.    

The rule of three belonging to Fractions. Cap. 9  
WHat order is to be observed in this rule having to deal with Fractions only?  
The self same order that hath been taught before touching Integrums: for in working with Fractions you must have also 3. several numbers, and you must see that the first and third numbers be of one self denomination, and that number to be placed always in the third place whereunto the question is annexed, and then to Multiply the second by the third, and to divide by the first, and so the fourth number which you seek to know, shall appear: As for example if ¾. of an ell of fine Holland cost me ⅔. of an English crown, in value 15. groats, what shall ⅚. of an ell cost me, here first you must set down your 3. several numbers in order thus, ¾ ell ⅔ ⅚ ell. so as the first and third may be of one self denomination, then Multiply the second and third Fraction the one into the other, which will make 10/18. and that being divided by ¾. which is the first Fraction will produce 40/54. of a crown, the value of which Fraction if you seek to know by help of the sixth rule of Fractions, teaching you to Multiply the Numerator of the Fraction, by the known parts of a crown which are 5 s or 15. groats, you shall find the value of that Fraction to be 11. groats and 6/54. of a groat, which is one farthing and somewhat more, supposing the least known parts of a groat to be 16. farthings.   

The golden Rule reverse called in Latin regula eversa that is to say turned backward. Cap. 10.  
WHat is the order of this rule?  
Multiply the first by the second, and divide the product thereof by the third, as if a penny Loaf must weigh 2.l. Wheat being at 3. s. the bushel, what shall a penny Loaf weigh when Wheat is at 2. s. the hushell, the question must be framed thus, if 3. s. require 2. l weight, what shall 2. s. have, then by working according to this rule you shall find that the penny Loaf must weigh. 3.l.  

Another example of the same rule.  
I would know how many yards of Bays bearing in breadth 7/4. will suffice to line 7. yards of Silk bearing in breadth 3. quarters & a half. Here you must frame your question thus, if ¾. and ½ require 7. yards, what shall 7/4. require, but because the first and third number of this question are not of one self denomination by reason of the Fraction annexed to the first number, you shall do well to reduce the first and third number both into half quarters, and then to work as though they were all Integrums, which is more easy then to make all the numbers Fractions, wherefore say thus: If 7. half quarters do require 7. yards, what shall 14. half quarters require, and in working by the rule Reverse, you shall find in the quotient 3. yards of Bays and a half.    

The double rule called in Latin regula duplex. Cap. 11.  
WHereto serves this rule and what order is to be observed therein?  
This rule serveth to unfold two questions wrapped in one, as thus. If I pay 4.d. for the carriage of 20.l. weight 30. miles, what shall I pay for the carriage of 50.l. weight 60. miles, here of this and such like demands, you must make 2. sundry questions, and the fourth Somme of the first question being found, shall be the second or middle number of the second question: wherefore frame your first question thus, if 20.l. cost 4. d what shall 50. l cost, and shall found that it will cost you 10.d. then say if 30. mile's cost 10.d. what shall 60. mile's cost and you shall find that it will cost 20.d. And note that each of these 2. questions is to be wrought by the common rule of 3. that is to say by Multiplying the second into the third, and by dividing the product thereof by the first, and the fourth found number of the first question must be the second, or middle number of the second question, as in the former example, you see that 10.d. which was the fourth found number, is here the middle number of the second question.  

Another example.  
If 25.l. do gain me 8.l. in 4. years, how much shall a 100l. win one in 10. years, both these questions are also to be wrought by the common rule of 3. Wherhfore set down the first question thus, if 25.l. yieldeth 8.l. what shall 100l. yield, and you shall find 32.l. then say 4. years yieldeth 32.l. what shall 10. years yield, and you shall found 80.l. But note that these double questions, may be put in such sort as you must work the first or second question, sometimes by the rule reverse. As in this question here following, if 6.l. win 8. Crowns in 10. years, in how many years shall 3.l. win 12. Crowns, here frame your first question thus, if 6.l. require 10. years how many years shall 3.l. require: And in working this question by the rule Reverse, you shall find 20. years, then for the second question say thus, if 8. Crowns require 20. years, how many years shall 12. Crowns require: Here if you work by the common rule of 3. you shall find 30 years.   

Another example.  
If 7. horses do eat 12. bushels of Dates in 20. days, how many bushels shall 14. horses eat in 15. days, here frame your first question thus, if 7. horses do eat 12. bushels, what will 14 horses eat, and in working by the common rule of 3. you shall find in the quotient 24. bushels, then frame your question thus, if 20. days require 24. bushels, what will 15. days require, here in working by the common rule of 3. you shall find in the quotient 18. bushels.   

Another example.  
If ten reapers reap 15. Acres in 7. days, in how many days shall 16. reapers reap 20. Acres: Here frame your first question thus, if 10. reapers require 7. days, how many days shall 16. reapers require, which question must be wrought by the rule Reverse, and so you shall find 4. days and ⅜. of a day which is 9 hours, then say, if 15. Acres require 4. days and ⅜. of a day, how many days shall 20. Acres require, and in working this second question by the common rule of 3. belonging of Fractions as is before taught, you shall find that 20. Acres will require 5. days and 10/12. or ⅚. of a day which is 20 hours. But it were more easy in this second question to reduce the days into hours by Multiplying the 4. days by 24. hours, the product whereof will be 96. hours, whereunto if you add the odd 9 hours it will make in all 105. hours, which being Multiplied by the third number of this second question, which is 20. the product shall be 2100. hours, which divided by the first number of the said question which is 15. you shall find in the quotient 140. hours, which if you divide again by 24. you shall find in the quotient 5. days, and the remainder to be 20. hours, which agreeth in all points with the first manner of working by Fractions, and is the easier way of the two.  
How shall I know having to work by this Double rule, when to use the rule reverse?  
By considering whether the third number requireth more or less of time, or of any other measure or quantity, as in the former example, of Bays for lining, the more breadth it had, the less did serve for lining. Again in the example of the gain by years, of 6.l. and 3.l. you did see that 3.l. require more years than 6.l. and therefore that first question was wrought by the rule reverse: Also in this last example of the reapers, the more reapers, the less time they require, & therefore that question was wrought by the rule reverse.    

The rule of Fellowship Cap. 12.  
WHat doth this rule teach?  
To know the gain or loss of such as do make a stock and do occupy together in the trade of Merchandise: As for example, four Merchants did put their money in lot in this manner. The first brought 30. Crowns, the second 50. the third 60 and the fourth 100 and with these portions they gained 3000. Crowns, the question is how much every one shall have to his share of that gain according to the portion which he brought. To know this, you must first gather all the several portions together by Addition into one Somme, and you shall find the some of the portions to be 240. Crowns, then say if 240. Crowns do gain 3000. Crowns, what shall 30. gain, and after this manner work by the common rule of 3. with all the rest of the portions, and so you shall find that he which brought 30. aught to have 375. Crowns, and he that brought 50. aught to have 625. Crowns, and he that brought 60. aught to have 750. Crowns, and he that brought 100 aught to have 1250. which maketh in all 3000. Crowns, for that is the very Somme of the gain before set down, the order of working whereof, this figure plainly showeth.  
240.— 3000.— 30.— 375. 50.— 625. 60.— 750. 100— 1250. The common Divisor which is the Somme of the particular portions added together. the general gain. the particular portions. every man's single share.  

A like example of loss received by Shipwreck.  
Three Merchants do venture their goods in one Ship, the goods of the first were worth 300. Crowns, the second 400. the third 500 there were as much goods cast out as was worth 100 Crowns, the question is how much every one should lose according to his portion, here work as before, and you shall find that every one shall lose so much as this figure following showeth.  
1200.— 100— 300.— 25. 400.— 33. ⅓ which is 5. groats 500— 41. ⅔ which is 10. groats The common divisor which is the Some of the particular portions added together. the general loss. the particular portions. every man's several loss  
Here to know whether the 3. several losses do make up the general loss, do thus, first add the Integrums of the several losses together, the Somme whereof will be 99 whereunto if you add the 2. Fractions which do make one whole Crown, the Somme will be 100 a number like unto the general loss.   

Another example.  
Three Merchants have bought 1000l. of Pepper for 300. crowns, the first taketh 200.l. the second 350.l. and the third 450.l. what shall every man pay according to the portion which he hath received, then say if 1000l. be worth 300. Crowns, what is 200.l. worth, here working by the common rule of 3. with every man's several portion received, you shall find that the first must pay 60. Crowns, the second 105. and the third 135. Crowns, as this figure here following showeth.  

l.  1000— 300.— 200.— 60. 350.— 105. 450.— 135. The common Divisor. the general price. the several portions received. every man's particular payment.   

Another like example of diverse distance of time.  
Three Merchants occupying together did gain 2345. Crowns, the first put in 40. Crowns for the space of 14. Months, the second put in 50. Crowns for the space of 8. Months, and the third put in 85. Crowns for the space of 6. Months, the question is, how much every one shall have after the rate of his money, and according of the quantity of time: This is to be wrought according to the rule of Fellowship thus, Multiply every man's money by his time, either of which, that is to say, the money and the time must be of one self denomination, then add the Sums of those several portions together, and the total somme of such Addition shall be the first number, and the common gain shall be the second number, and the third number shall be every man's money Multiplied by his time, then in working by the common rule of 3. you shall find that every man shall have such share as this figure here following showeth.  
1470.— 2345.— 560.— 893. ⅓ 400.— 638. 2 / 21. 510.— 813. 4 / 7 The common Divisor which is the total Somme of the several portions of money added together. the general Somme of the gain. the particular portions of the money multiplied by his time. every man's single share.  
Now if you would know whether the single shares in this example do make up the general Somme of the gain, then add together the Integrums or whole Sums of every man's single share, and you shall find the Somme to be but 2344. which wanteth one whole Integrum of the second or middle number of the question, which you shall easily supply by adding the 3. several Fractions together according to the rule of Addition of Fractions before taught, for so shall you find the said 3. Fractions to make in all one whole Integrum, which being added to 2344. will be answerable to the second number of the question, which as you see in the example is 2345. And remember in adding together the said 3. Fractions, to set them in this order ⅓ / 2/21 21/4/7. and then to bring them to one self denomination, as the seventh rule of Fractions teacheth you. But first because you shall found the 3. remainders after the first 3. Divisor of the 3. several portions, made by the first common Divisor of the question to be written in many figures, you must set them down in lesser figures according as the fift rule of Fractions teacheth you, so shall you find the first remainder containing 490/1470. to be no more but ⅓. and the second remainder containing 140/1470. to be 2/21. and the third remainder containing 840/1470. to be 4/7. which 3. Fractions being added together according to the seventh rule of Fractions, will make 441/441. which is one whole Integrum and must be added by the name of 1 to the middle or second number of the question, as I have said before: you may also make up the foresaid number by seeking to know the value of every Fraction annexed to the Integrums, according as the sixth rule of Fractions teacheth, for so shall you find the value of the first Fraction to be five groats, & the value of the second Fraction to be one groat 6. farthings, and 18/21. of a farthing thing, & you shall found the value of the third Fraction to be 8. groats 9 farthings and ½. of a farthing, & if you add 18/2● & 1/7 of a farthing together, you shall found that it will make 147/147. which is one whole Integrum or one whole farthing. Now if you add 6. 9 and 1. farthings together, it will make in all 16. farthings, and that is one groat which being added to the 14 groats before found out, will make in all 15. groats which is one Crown, and that being added to the Some 2344. will make it 2345. Crowns, which is a number agreeable to the middle Somme of the question. Truly if you exercise yourself in this and such like questions, it will make you perfect not only in Addition, Subtraction, multiplication, and Division of whole numbers, but also of Fractions, and almost in all the other rules belonging as well to Fractions, as to Integrums: Wherhfore I would wish you often to use your pen therein.  
Having hitherto treated of the 4. special kinds of Arithmetic, that is of Addition, Subtraction, multiplication and Division, as well belonging to whole numbers, as to Fractions, and also showed the use of the Golden rule, otherwise called the rule of 3. and of all the three kinds thereof, that is the common rule, the rule Reverse, and the Double rule, and given examples how and when every one is to be used, together with the rule of Fellowship, necessary for them that use any trade of Merchandise, I think good now to speak somewhat of Arithmetical and Geometrical progression, and also of Proportion, and of the three kinds thereof, that is Arithmetical, Geometrical, and Musical proportion, and then of the extraction of roots both square and cubical, and last of all, of Astronomical Fractions, showing how they are to be added, Subtracted, Multiplied & divided.    

Of Progression and how manifold it is. Cap. 13.  
WHat is Progression?  
It is a certain order of proceeding with diverse numbers in such sort as every one may exceed each other, either by like difference of quantity, or else by likeness of Proportion, whereof springeth 2. kinds of Progression, the one called Arithemeticall, and the other Geometrical.  
What is Arithmetical Progression?  
It is that which proceedeth by like difference of quantity, as thus, 3. 5. 7. 9 11. 13. whereof every one exceedeth his fellow by the difference of 2.  
What is Progression Geometrical?  
It is that wherein every number exceedeth his fellow by like proportion, for as 6. containeth 3. twice, so doth 12. contain 6. twice, etc.   

Of Addition belonging to Progression Arithmetical. Cap. 14.  
How are numbers being set in order according to Progression Arithmetical, to be added together and to be reduced into one total some?  
Thus, first see how many several numbers there be in all, and note the Somme by itself, then add the first number of the Progression to the last thereof, and note that Somme by itself, that done, Multiply the one of these 2. reserved Sums by the one half of the other, and you shall have the total Somme. As for example let this be your Progression 6. 10. 14 / 18/22 22/26 26/30 30/34 / wherein every number exceedeth his fellow by the difference of 4. here having told the numbers, I find them to be 8. which I set down by itself, than I add 6. which is the first number to 34. which is the last, and the Somme thereof is 40. which I also set down by itself, then in Multiplying 40. by 4. which is the one half of 8. before set down a part by itself, I find the total Somme of the Progression to be 160.   

Of Addition belonging to progression Geometrical. Cap. 15.  
HOw are numbers being set in order according to progression Geometrical, to be added and to be brought into one Somme?  
Thus, Multiply the last Somme of the progression by the number of the proportion whereby such progression proceedeth, and from the product of that multiplication, Subtract the first number of the progression, that done, divide that which remaineth by a number which is less than the proportion by one, and the quotient of such Division shall show you the total Somme of the said progression. As for example, let this be the progression 2. 6. 18. 54. 162. 486. 1458. the proportion of which progression is triple, wherefore according to the rule, I Multiply the last number of the progression by 3. and the product of such multiplication amounteth to 4/374. out of which Somme I Subtract the first number which is 2. and there remaineth 4372. which I divide by 2. (for it is less than be proportion 3. by 1.) and so I find in the quotient 2186. which is the total Somme of the progression Geometrical.  
Is there no brieffer way of adding such kinds of progression?  
Yes indeed, but not so plain as this way is, and therefore I think not good to trouble your memory therewith.   

Of Proportion. Cap. 16.  
WHat is Proportion?  
Proportion is taken generally for the comparing of 2. diverse quantities or numbers together, showing what likeness is betwixt them. But before we deal with Proportion, and with the 3. kinds thereof, that is Proportion Arithmetical, Geometrical, and Musical: You have first to understand, that of numbers some are said to be abstract, and some concreate.  
The abstract are such as are not tied to any denomination, and such are twofold, that is absolute and relative.  
The absolute, are simply pronounced without having any relation to any other number, measure, or quantity, as 2. 3. 4. etc. and all numbers whatsoever, that are without denomination and are not attributed to any other thing.  
The relative are those which have relation one to another, which may be three manner of ways. First in respect of difference which is found by Subtraction, secondly in respect of the quotient found by Division, thirdly in respect of both. Of the first way springeth Arithmetical proportion: Of the second way Geometrical proportion: And of the third Musical proportion.   

Cap. 17.  
WHat is Arithmetical proportion?  
Arithmetical proportion unproperly so called, (because it is no proportion indeed) is when many several numbers have one self and like difference, as 8. 12. 16. which do only differ one from another by 4. and this Proportion is twofold, that is continual and disjunct.  
Continual is when many numbers proceed with like difference, as hath been said before when we spoke of Arithmetical Progression, as 8. 12. 16. 20. etc. whose difference betwixt every 2. numbers is 4.  
The Disiunct, is when many numbers being severally propounded, the difference of the first 2. numbers is not like to the difference that is betwixt the second and the third, and so forth as 5. 8. 4. 7. for 8. differeth from 5. by 3. and 4. from 8. by 4. and 7. from 4. by 3. Now of the second way of comparing which is done by Division, springeth as hath been said before, Geometrical proportion.   

Cap. 18.  
WHat is Geometrical proportion?  
Geometrical proportion is that which showeth what part or parts one number is of another number, as 3. is the half of 6. which proportion is found by Division, wherein you have to note that if the Divisor be greater than the dividend, than it is to be made a Fraction, as in the former example, if you would divide 3. by 6. then you must make it a Fraction thus, 3/6. or ½, and this kind of proportion which may be truly and properly called a proportion indeed or rather a proportionality, is said to be twofold, that is, direct, and reverse, and the direct is either conjunct, or Disiunct  
conjunct differeth not from Geometrical Progression before taught.  
disjunct proportion Geometrical, consisting most commonly of 4. numbers or of 3. at the lest, is when there is not like proportion betwixt the second and the third, that is betwixt the first and the second, or betwixt the third and the fourth, as 3. 6. 4. 8. for here 6. containeth 4. once, and one half thereof, which is called Proportio Sesquialtera, and 6. containeth 3. twice, which is called Proportio dupla: and so is 8. to 4.   

Cap. 19  
Again, proportion Reverse differeth not from the rule of 3. called Regula eversa. But you have to understand that the two chief and special kinds of Geometrical proportion, are these, that is, proportion of equality, and proportion of inequality.  
Proportion of equality, is when 2. numbers compared togethtr, are equal the one to the other, as 3. to 3. 4. to 4.  
The proportion of inequality, is when 2. unequal numbers are compared together, as 6. to 5, 4. to 9 and of this there are two kinds, that is proportion of the greater inequality, and proportion of lesser inequality.   

Cap. 20.  
PRoportion of the greater inequality, is when the greater number is compared to the lesser numbers, as 6. to 5.  
Proportion of the lesser inequality, is when the lesser is compared to the greater, as 5. to 6.  

Of proportion of the greater inequality there be two kinds, Simplex and Multiplex, that is to say, simple and manifold.  
SImplex, is when the Antecedent, that is to say the former number containeth the consequent, that is to say the latter number once and somewhat more, which overplus must always be less than the consequent itself, as 5. containeth 4. once and one part thereof more, for if you divide 5. by 4. the quotient will be 1. and ¼ over. Again, this proportion is twofold, that is Superparticuler, and Superpartient: Superparticular, is when the Antecedent containeth the consequent once and some one part thereof, as 3. to 2. for 3. containeth 2. once and one half thereof, which is called Sesquialtera, or as 4. to 3. for 4. containeth 3. once and one third part thereof, and is called Sesquitertia, the like is to be said of Sesquiquarta, Sesquiquinta, Sesquisexta, and so forth infinitely as this Table showeth.  
Superparticular proportions are these & such like. Sesquialtera as 3. to 2. 6. to 4. 9 to 6. which is as much as. 1 ½ Sesquitertia as 4. to 3. 8. to 6. 12. to 9 1 ⅓ Sesquiquarta as 5. to 4. 10. to 8. 15. to 12. 1 ¼ Sesquiquinta as 6. to 5. 12. to 10. 18. to 15. 1 ⅕ Sesquisexta as 7. to 8. 14. to 12. 21. to 18. 1 ⅙ Sesquiseptima as 8. to 7. 16. to 14. 24. to 21. 1 1/7 Sesquioctava as 9 to 8. 18. to 16. 27. to 24. 1 ⅛ Sesquinona as 10. to 9 20. to 18. 30. to 27. 1 1/9 Sesquidecima as 11. to 10. 22. to 20. 33. to 30 1 1/10 Sesquiundecima as 12. to 11. 24. to 22. 36. to 33 1 1/11 Sesqiuduodec. as 13. to 12. 26. to 24. 39 to 36 1 1/12  
But superpartient is when the Antecedent containeth the consequent once and some parts thereof, that is to say more parts than one, as 5. to 3. for 5. containeth 3. once and 2. third parts thereof, which is called Superbipartiens tertias, of which kinds are these set down in the Table following.  
Proportions superpartient are these and such like. Superbipartiens. Tertias as 5. to 3. 10. to 6. 15. to 9 which is as much as. 1 ⅔ Quintas as 7. to 5. 14. to 10. 21. to 15. 2 ⅖ Septimas as 9 to 7. 18. to 14. 27. to 21. 1 2/7 Nonas as 11. to 9 22. to 18. 33. to 27. 1 2/9 Vndecimas as 13. to 11. 26. to 22. 39 to 33. 1 2/11 Decimas tertias as 15. to 13. 30. to 26. 45. to 39 1 2/13 Supertripartiens. Quartas as 7. to 4. 14. to 8. 21. to 12. which is as much as. 1 ¾ Quintas as 8. to 5. 16. to 10. 24. to 15. 1 ⅗ Septimas as 10. to 7. 20. to 14. 30. to 21. 1 3/7 Octavas as 11. to 8. 22. to 16. 33. to 24. 1 ⅜ Decimas as 13. to 10. 26, to 20. 39 to 30. 1 3/10 Vndecimas as 14. to 11. 28. to 22. 42. to 33. 1 3/11 Superquadripartiens. Quintas as 9 to 5. 18. to 10. 27. to 15. which is as much as. 1 ⅘ Septimas as 11. to 7. 22. to 14. 33. to 21. 1 4/7 Nonas as 13. to 9 26. to 18. 49. to 27. 1 4/9 Vndecimas as 15. to 11. 30. to 22. 45. to 33. 1 4/11 Decimas tertias as 17. to 13. 34. to 26. 51. to 39 1 4/13 Decim quintas as 19 to 15. 38. to 30. 57 to 45. 1 4/15 Superquintupartiens. Sextas as 11. to 6. 22. to 12. 33. to 18. which is as much as. 1 ⅚ Septimas as 12. to 7. 24. to 14. 36. to 21. 1 5/7 Octavas as 13. to 8. 26. to 16. 39 to 24. 1 ⅝ Nonas as 14. to 9 28. to 18. 42. to 27. 1 5/9 Vndecimas as 16. to 11. 32. to 22. 48. to 33. 1 5/11 Duodecimas as 17. to 12. 34. to 24. 51. to 36. 1 5/12 Supersextupartiens. Seqtimas as 13. to 7. 26. to 14. 39 to 21. which is as much as. 1 6/7 Vndecimas as 17. to 11. 34. to 22. 51. to 33. 1 6/11 Decimas tertias as 19 to 13. 38. to 26. 57 to 39 1 6/13 Decimas septimas as 23. to 17. 46. to 34. 60. to 51. 1 6/17 Decimas nonas as 25. to 19 50. to 38. 75. to 57 1 6/19 Vicessimas tertias as 29. to 23. 58. to 46. 1 6/23  
Hitherto of Simplex proportio. Now of Multiplex proportio.  
Multiplex proportio is when the Antecedent containeth the consequent more than once, as 6. to 2. for 6. containeth 2. three tunes, which is called Tripla proportio, Also 12. to 5. for 12. comprehendeth 5. twice and ⅖. And this Multiplex proportio is twofold, that is either exact or not exact.  
Multiplex exact, is when the Antecedent containeth the consequet more than once, and nothing remaineth, as 4. to 2. 6. to 3. etc. whereof are infinite kinds, as Dupla, Tripla, and so forth as this Table showeth.  
The kinds of Multiplex exact are these and such like. Dupla as 4. to 2. 6. to 3. 8. to 4. which is as much as. 2/1 Tripla as 6. to 2. 9 to 3. 12. to 4. 3/1 Quadrupla as 8. to 2. 12. to 3. 16. to 4. 4/1 Quintupla as 10. to 2. 15. to 3. 20. to 4. 5/1 Sextupla as 12. to 2. 18. to 3. 24. to 4. 6/1 Septupla as 14. to 2. 21. to 3. 56. to 8. 7/1 Octupla as 16. to 2. 24. to 3. 32. to 4. 8/1 Nondupla as 18. to 2. 27. to 3. 36. to 4. 9/1 Decupla as 20. to 2. 30. to 3. 40. to 4. 10/1 Vndecupla as 22. to 2. 33. to 3. 495. to 45. 11/1  
But Multiplex not exact, is when the Antecedent containeth the consequent more than once and some thing remaineth over, as 5. to 2. for 5. containeth. 2. twice and one remaineth, and this is also twofold, that is, Multiplex superparticularis, and Multiplex superpartiens  
Multiplex superparticularis, is when the Antecedent containeth the consequent more than once & one only remaineth, as 7. to 3. for 7. containeth 3. twice and one only remaineth, whereof are divers kinds, as Duplex sesquialtera, Duplex sesquitertia, Triplex sesquisexta, & so forth as the table hereafter following showeth.  
But Multiplex superpartiens, is when the Antecedent containeth the consequent more than once & the remainder is more than 1. as 8. to 3. for 8. containeth 3. twice and 2 thirds over, whereof there be many, kinds, as Dupla superbipartiens tertias, Dupla supertripartiens quartas, and so forth as this Table following showeth, which comprehendeth both kinds, that is Multiplex superparticularis, and Multiplex superpartiens  
Multiplex not exact, is either 

Multiplex Superparticularis or else. 

Duplex. 

Sesquialtera as 5. to 2. 15. to 6.  
Sesquitertia as 7. to 3. 21. to 9  
Sesquiquinta as 11. to 5, 22. to 10.    
Triplex. 

Sesquisexta as 19 to 6. 38. to 12.  
Sesquiseptima as 22. to 7. 44. to 14.  
Sesquioctava as 25. to 8. 50. to 16.    
Quadruplex. 

Sesquinona as 37. to 9 148. to 36.  
Sesquidecima as 41. to 10. 82. to 20.  
Sesquiundecima as 45. to 11. 90. to 22.     which is. 

2 ½  
2 ⅓  
2 ⅕  
3 ⅙  
3 1/7  
3 ⅛  
4 1/9  
4 1/10  
4 1/11    
Multiplex Superpartiens. 

Dupla. 

Superbipartiens. 

Tertias as 8. to 3.  
Quintas as 12. to 5.    
Supertripartiens. 

Quartas as 11. to 4.  
Quintas as 13. to 5.    
Superquadripartiens. 

Quintas as 14. to 5.  
Septimas as 18. to 7.      
Tripla. 

Superquintupartiens. 

Octavas as 29. to 8.  
Nonas as 34. to 9    
Supersextupartiens. 

Vndecimas as 39 to 11.  
Decim, tertias as 45. to 13    
Superseptupartiens. 

Octavas as 31. to 8.  
Nonas as 34. to 9      
Quadrupla. 

superbipartiens. 

Tertias as 14. to 3.  
Quintas as 22. to 5.    
Superquadripartiens. 

Quintas as 24. to 5.  
Nonas as 40. to 9    
Superquintupartiens. 

Sextas as 29. to 6.  
Septimas as 33. to 7.       which is. 

2 ⅔  
2 ⅖  
2 ¾  
2 ⅗  
2 ⅘  
2 4/7  
3 ⅝  
3 5/9  
3 6/11  
3 6/13  
3 ⅞  
3 7/9  
4 ⅔  
4 ⅖  
4 ⅘  
4 4/9  
4 ⅚  
4 5/7      
Thus much of proportion of the greater inequality: Now we will speak somewhat of proportion of the lesser inequality.    

Cap. 21.  
PRoportion of the lesser inequality, is when the Antecedent is less than the consequent, as 2. to 3. 5. to 7. for if you divide 2. by 3. the quotient is ⅔. if 5. by 7. then the quotient is 5/7 and this Proportion hath the same names which the Proportion of the great inequality hath, saving that you must add to the beginning of every name, this word Sub. as Subdupla, Subtripla, Subsesquialtera, etc. as 1 to 2 is Subdupla proportio, 3. to 9 is Subtripla proportio, etc.   

Of Musical proportion called in Latin Harmoniaca proportio. Cap. 22.  
Musical proportion which requireth 3. numbers at the lest, is when the first number hath the same proportion unto the third, which the difference betwixt the first and the second, hath to the difference which is betwixt the second and the third, as 3. 4. and 6. for look what proportion 3. hath to 6. which is Subdupla, the same hath the difference betwixt 3. and 4. which is 1. to the difference betwixt 4. and 6. which is 2. for 1. to 2. is Subdupla, and this is called Musical proportion, because the numbers therein have the same proportions one to another, which are found to be in Musical consorts, as 6. 4. and 3. for the proportion which 6. hath to 4. is Sesquialtera, called of the musicans Diapente, or a fift, and the proportion which 4. hath to 3. is Sesquitertia, called of the musicans Diatesseron, or a fourth, and the proportion which 6. hath to 3. is Dupla, called a Diapason, or an Eight, and of this Musical proportion there be two kinds, that is simple and compound.  
It is said to be simple when it consisteth only of 3. numbers.  
And it is called compound when it consisteth of more than 3. numbers: And of compound there be 2. kinds, that is unproper and proper.  
The unproper is, when to 3. numbers given, 2. other several numbers are joined, which do contain the same proportions with the third, which the first 3. numbers have one to another: As for example, let be given these 3. numbers, 3. 4. 6. unto which if you join 8. and 12. here 6. 8. 12. have the same proportion one to another which 3. 4. and 6. have amongst themselves, for like as here betwixt 6. and 8. the proportion is Subsesquitertia, and betwixt 6. and 12. it is Subdupla, and betwixt 8. and 12. it is Subsesquialtera, so betwixt 3. and 4. is Subsesquitertia, and betwixt 3. and 6. is Subdupla, and betwixt 4. and 6. is Subsesquialtera,  
Compound proper, is when diverse numbers in Musical proportion standing together, and the first being omitted, the 3. next do continued still in Musical proportion. Also when omitting the 2. first numbers the 3. next following are in Musical proportion. And when omitting the first 3. the next 3. are in Musical proportion, and so forth how many numbers so ever there be: As for example 10. 12. 15. 20. 30. these are numbers belonging to the proper kind of compound Musical proportion, for 10. 12. and 15. are in Musical proportion, then omitting 10. which is the first number, 12. 15. and 20 are still in Musical proportion, and if you omit the 2. first numbers, that is 10 and 12. then the next 3. that is 15. 20. and 30. are still in Musical proportion, and so forth how many so ever there be of them, so as they be in Musical proportion: But our musicans do make no more but 8. Musical proportions in all, that is.  


Dupla.  
Tripla.  
Quadrupla.  
Sesquialtera.  
Sesquitertia.  
Sesquiquarta.  
Dupla superbipartiens.  
Sesquioctava.   which are thus named 

Diapason.  
Diapason diapente.  
Bis diapason.  
Diapente.  
Diatesseron.  
Diatonus semitonus.  
Diapason diatesseron.  
Tonus.    
The use whereof is to be learned at the hands of the musicans.   

Cap. 23.  
But now because it is not enough to know the foresaid proportions and their names, unless you can both add, Subtract, Multiply, and divide them when need is, I will therefore briefly here set down the order thereof, and first show how to set them down in writing, and then how they are to be added, Subtracted, Multiplied, and divided.  
They are to be set down in writing in like manner that Fractions are wont to be set down, saving that as in Fractions the upper number is called Numerator, and the inferior number Denominator, so in proportions the upper number is called Antecedent, and the inferior number Consequent, but whereas in Fractions, there is wont to be drawn a little line betwixt the Numerator and the Denominator, in proportions no line is drawn betwixt the Antecedent and the Consequent, but the Antecedent is set over the Consequent, without any line drawn betwixt them thus. 3. 4. 2. 5. 7. 1.  
And look as Fractions are to be written in as few figures as may be, so are proportions to be set down in so small numbers as may be, which is to be done by the self same rule that Fractions are. Again as you can neither add nor Subtract Fractions, having diverse denominations, before that you have brought them to one self denomination: So can you neither add nor subtract proportions having diverse consequents, until you have brought them to one self consequent by the same rule that Fractions are reduced to one self denomination: And look what order is to be observed in adding, subtracting, multiplying and dividing of Fractions, the same is to be kept in adding, subtracting, multiplying and dividing of proportions. Wherhfore I wholly refer you to the rules of Fractions before most plainly taught. And thus I end with proportions, the knowledge whereof is very necessary to such as have to deal in matters either of Geometry or of Astronomy or with Music.   

How to find out the square root of any number. Cap. 24.  
WHat is a square root?  
It is any Digit or any other number, which being multiplied into itself bringeth forth a square number, as 4. being multiplied in itself, in saying 4. times 4. maketh 16. which is a square number, the root whereof is 4. And to find out the square root of any number be it square or not square, you must do thus. First having set down the number propounded which must consist of 3. figures at the lest, set a prick under the first Digit or cipher of the said number on the right hand, that done, prick every other figure thereof towards the left hand, leaving always one void space betwixt every 2. pricks as you see here done in this number ⁴ .⁷ ⁵ .⁶ ⁴ .⁵ and look how many pricks there be, so many figures shall you have in the quotient, and this differeth nothing in a manner from Division, the order in working is thus, first seek out a Digit, which being multiplied in itself, may take away that which standeth over the last prick on the left hand, or as much thereof as may be, and set the Digit in the quotient, that done, double the said quotient, then consider whether the double do consist of one figure or of many, if of one only, than place that in the next void space on the right hand, but if the double do consist of many figures, then take the first figure thereof, that is to say, that figure or cipher which standeth in the first place of the double, and place that in the foresaid next void space towards the right hand, and next to that, place all the rest of the figures of the said double orderly towards the left hand, that done, seek another Digit, which being multiplied in itself together with the doubled number, may take away all that which standeth right over it, or as much thereof as may be, which Digit you must not only put in the quotient to his fellow, but also set down the same right under the next prick on the right hand, and then multiply that Digit together with the double whereto it is joined, by the said last found Digit set down in the quotient, the product whereof you must subtract out of the upper number, standing right over the foresaid prick, and if there be any remainder, writ it above, canceling the other, and look how many pricks there be, so many times must you double the quotient, observing always like order of working as before, which you shall more plainly understand by these examples following, and for your better understanding thereof, I will first give you an example of 3. figures only as thus, 464. here to find out the square root of this number, you must first prick the said number in such order as is before taught, and then seek out some one of the 9 Digits, which being multiplied in itself, may take away the first 4. on the left hand, or as much thereof as may be, which you shall find to be 2. which 2. being set in the quotient, multiply the same in itself, in saying 2. times 2. maketh 4. which 4. doth clean take away the first 4. standing over the last prick on the left hand, and therefore cancel the 4. that done, double the quotient which maketh 4. and set the same in the next void space on the right hand under the figure 6. thus.  
And then seek out some Digit, which being multiplied in itself together with the doubled, may take away the Somme of 64. that remaineth, which by ask how many times 4. is comprehended in 6. you shall find to be but one, which you must not only put in the quotient, but also set it down under the first prick on the right hand thus.  
So as you shall make the lower Somme to be 41. which is to be multiplied by the last found quotient which is 1. wherefore you must say that 1. times 41. is 41 which being taken out of 64. there remaineth 23. so as the root is 21. and the remainder 23. as this example showeth.  
And to know whether you have done well or not, multiply the quotient in itself, and if there be any remainder left, add that unto the product of the multiplied quotient, and if you find the sum thereof to be like to the first, than you have done well, if it be not like, than you have erred, the product of this quotient being multiplied in itself amounteth unto 441. whereto by adding the remainder which is 23. you shall find the sum to be like unto the first number that is 464. whereof you sought the square root. But if such number do consist of many figures, then in working you must double the quotient once, twice or thrice, according as the number doth require, as you shall more plainly perceive by this example following: as let 50/467. be the number whereof you would know the square root, here having pricked this number in such order as before, first seek out some Digit which being multiplied in itself may take away the Digit or number standing right over the last prick on the left hand, which you shall find to be 2. and having set down the said 2. in the quotient, say 2. times 2. maketh 4. which being subtracted from 5. there remaineth 1. which you must set over 5. and cancel the said 5. as this example showeth.  
That done, double the quotient which maketh 4. and set down the 4. in the next space towards the right hand, under the cipher as you see in the former example, then seek out a Digit, which being multiplied in itself together with the double quotient, may take away all that which standeth over the second prick towards the left hand, or as much thereof as may be, which Digit by ask how many times 4. is comprehended in 10. you shall find to be 2. then set down the said 2. not only in the quotient, but also under the second prick towards the right hand, which together with the former 4. maketh 42. as you see in this example.  
Than multiply that 42. in a place by itself by the figure 2. last set down in the quotient, and it will make 84. which 84. being taken out of 104. there remaineth 20 which you must set over the second prick and cancel the 104. and also the 42. beneath, as you see in the former example: that done, double the whole quotient, which maketh 44. whereof set the first 4. in the next void space towards the right hand under the figure 6. and the other 4. under the second prick towards the left hand, them ask how many times 4. is comprehended in 20. and you shall found 4. which you must not only set down in the quotient, but also under the next and last prick on the right hand, and so the neither Sum shall be 444. which being multiplied by the Digit 4. last set down in the quotient, will make in all 1776. which being taken out of the upper number which is 2067. there will remain 291. so as the square root of the first number is 224. and the remainder 291. as you may see in this example.  
The square root whereof being multiplied in itself amounteth unto 50176. whereto if you add the remainder which is 291. the Sum thereof will be like unto the first number which is 50467. Moreover you have to note that if you find any number out of the which your quotient being doubled cannot be subtracted, than you must set down a cipher in the quotient, and proceed to the next prick on your right hand, as in this example following.  
In which example the first found Digit set in the quotient is 6. which being multiplied in itself maketh 36. and clean taketh away the first number standing over the last prick on the left hand, that done, I double the quotient, which maketh 12 and I place the 2. in the next void space towards the right hand, and the 1. next to that towards the left hand orderly as in this example.  
Over which 12. there is but 6. remaining, so as I can not take 12. out of 6. and therefore I cancel the 12. and set down a cipher in the quotient, so as the quotient is now 60. which being doubled maketh 120. the cipher whereof I set down in the next void space towards the right hand, and the other two Digits orderly towards the left hand, as you see in this example.  
Than I ask how many times one is contained in 6, and I find 5. which I set down in the quotient and also under the last prick on the right hand, so as the neither number is 1205. which I multiply apart by itself by the 5. last set down in the quotient, the product whereof is 6025. which being taken out of the upper number, which is also 6025. there remaineth nothing, as this example showeth.  
Wherhfore I find the foresaid number to be a just square number, the root whereof is 605. which being multiplied in itself maketh 366025. a number like unto the first number.   

The use of the square root in setting of battles. Cap. 25.  
THe knowledge of finding out the square root of any number, is very necessary for a Sergeant maior in the field, that he may the more readily set and range his Squadrans of Battle: And therefore I think it not amiss, to give them here certain examples of such manner of squares as the Italians were wont to use in my time, which is four manner of ways.  
Whereof the first teacheth how to set the Battle square of ground, the second a square battle of men, the third a long square battle which we call the Hearse battle, and the fourth teacheth how to set a battle of so much and a third.  
Show by what rules these battles are to be set?  
Before I set down any rules you have to understand that all numbers given or supposed, are not meet for such purpose, but only such as may be divided into 3. equal parts without leaving any remainder: for otherwise all the rules following, take no place, and therefore when any number is given you to be brought into any one of the foresaid 4. forms of battle, you shall do well first to divide that number by 3. and if there be any remainder, to reject it, and to take the rest for your number, As for example, A Sergeant Maior is commanded by his General to set a battle square of ground, appointing him thereunto 1345. men, here the Sergeant by dividing that number by 3. findeth in the remainder one man too many, which being only rejected, all the rest of the number which is 1344. is fit for his purpose.  

How to set a battle square of ground, called in Italian Battaglia quadra di terreno.  
NOw show the order of setting such kinds of battle?  
The order is thus, first double the number given, then take the square root thereof, and that root shall be the Front, that done, divide the number given by the said root, and the number in the quotient shall be the flank, now if you multiply the front and the flank the one into the other, you shall have your whole number first given, unless there be some remainder left in the former Division, which remainder must be added to the product to make up the some. But if you would know how many men are to be put in a rank, and also how many such ranks you must have, then divide the front into 3. parts, and the quotient will show you how many men you are to have in a rank, that done, multiply the flank by 3. and the product thereof shall be the number of the ranks: As for example, take the number before given, which was 1345. which being divided by 3. you find the remainder to be 1. which you must reject, and take all the rest which is 1344. and that being doubled maketh 2688. the square root whereof is 51. by which root you must divide the number given, Videlicet 1344. and the quotient will be 26. which shall be the flank, now if you divide the foresaid front 51. by 3. the quotient will show how many shall be in a rank, that is 17. Finally, if you multiply the flank, which is 26. by 3. the product thereof will be 78. which is the just number of the ranks which you must have.   

How to set a battle square of men called in Italian Battaglia quadra d'huomini.  
TAke the square root of the number given, and that shall be both the front and the flank, which if you multiply together, the product thereof will make up the number first given, and if you would know how many men are to be put in a rank, and how many such ranks you must have, then do as you did before in dividing the front by 3. and multiplying the flank by 3.   

How to set a long square battle which we call the Hearse battle, and is called in Italian, Battaglia don tanto emezzo.  
TO the number given, add the just half thereof, and the square root of that Somme shall be the front, by which root, or front you divide the first number given, the quotient shall be the flank. As for example, let the number given be 6144. the half thereof is 3072. which being added to the given number 6144. maketh in all 9/216. the square root whereof is 96. which shall be the front, by which root or front if you divide the given number 6144. you shall find in the quotient 64. and that shall be the flank, that done, divide the foresaid front 96. by 3. and you shall find in the quotient 32. which number showeth how many men shall be in a rank, then multiply the foresaid flank 64. by 3. and the product thereof shall be 192. which is the number of the ranks.   

How to set a battle of so much and a third, called in Italian, Battaglia don tanto & don terzo.  
TO the number given, add the third part thereof and the square root of that Somme shall be the front, then divide the first number given by the foresaid root or front, and the number in the quotient shall be the flank: As for example, let the number given be 1575. the third part whereof is 525. which being added to 1575. maketh in all 2100. the root whereof is 45 which must be the front, by which root or front, if you divide the number given which was 1575. the quotient will be 35. which shall be your flank, then by dividing 45. which is the front by 3. the quotient will show you how many men you must have in a rank that is 15. Again by multiplying the flank which is 35. by 3. the product thereof will be 105. which is the number of your ranks.    

How to find out the Cubique root of any number. Cap. 26.  
AS the square root is said to be a number which being multiplied in itself, doth make a square superficial number, having only length and breadth, so as the Cubique root is a number which being first multiplied in itself, and the product thereof being again multiplied by the first number, doth make a Cubique or Corporal number having both length breadth, and depth: As for example, 2. times 2. maketh 4. and 2. times 4. maketh 8. Again 3. times 3. maketh 9 and 3. times 9 maketh 27. and so you may deal with all the rest of the 9 Digits, and make thereby a Table containing both the square numbers and Cubique numbers of every root, consisting of any one of the 9 Digits, like unto this table here following. Roots Squares cubics 1. 1. 1. 2. 4. 8. 3. 9 27. 4. 16. 64. 5. 25. 125. 6. 36. 216. 7. 49. 343. 8. 64. 512. 9 81. 729.  
Than having to find out the Cubique root of any number that is greater than 1000 (for lesser it cannot be to work upon) then set a prick under the first figure on your right hand, and so proceed towards the left hand, omitting always 2. figures betwixt every two pricks, as in this example 41063625. and look how many pricks there be, so many figures shall you have in the quotient: That done, you have to find out three several numbers, first such a Cubique number as will clean take away the number which standeth right over the head of the last prick on the left hand, or so much thereof as may be, which you may easily find in the foresaid table: The root of which Cubique number, you must set down in the quotient, the second number which you have to found is a number called the Triple, which is easily had by tripling the quotient, and the third number is called the Divisor which you shall most readily find by multiplying the quotient into the Triple, both which numbers are to be placed in such order as followeth. First then having found the Cubique number as is before said, and taken the same out of the number standing over the last prick on the left hand, writ the remainder (if there be any) over his head, canceling that which is under it, and then place the root of the said Cubique number in the quotient: that done, Triple the root set down in the quotient, and that shall be the Triple which you must place in the second void space on the left side of the next prick which is on your right hand, then multiply that Triple by the quotient, and the product thereof shall be the Divisor which must be placed right under the Triple one figure shorter towards the left hand: that done, draw a line as you see in the example hereafter following and work thus: First ask how many times the first figure of the said Divisor is contained in the number standing right over his head: and having found an apt Digit for the purpose, put that Digit in the quotient to his fellow, that done, multiply the said Digit into the Divisor and set the product thereof right under the Divisor: Secondly multiply the said Digit in himself, and again the product thereof into the triple, & set the somme that comes thereof right under the triple: Thirdly multiply the said Digit in itself cubically, and set the product thereof lowest of all, right under the next prick on the right hand: that done, draw a line, and by Addition bring all the foresaid three products into one some placing every figure so as you may easily take the said some out of the upper number, whereof you seek the Cubique root, and writ the remainder (if there be any) over the head, and if such some will not be subtracted out of the upper number, than you must seek out a lesser Digit, reserving still the former triple and Divisor, and if there be more pricks in the number then two, than you must for every prick find out a new Triple, and a new Divisor by tripling the whole quotient, and by working continually in like order as the rule before teacheth, which rule you shall more plainly perceive by example thus. First then having to find out the Cubique root of the number before mentioned, which is 4 1 0 6 3 6 2 5 marked with pricks as before is taught: resort to the Table wherein you shall find such a Cubique number as will take away as much of the 41. as may be, which is 27. the root whereof is 3. for all the other Cubique numbers in the table are either too small or too great, and therefore you must always have due consideration thereof. Than take 27. which is the Cubique number, out of 41. and there remaineth 14. which remainder you must set over the last prick on your left hand canceling the 41. as in this example.  
Than triple the quotient which maketh 9 which you must set down under the figure 6. next unto the second prick which is on your right hand: Than to found out the Divisor multiply the Triple 9 by 3. which is the quotient, and the product thereof shall be the Divisor: which divisor you must place right under the triple one figure shorter towards the left hand: that done, draw a line as you see in this example.  
Now having orderly set down the Triple and the Divisor, ask how many times 2. which is the first figure of the Divisor is contained in 14. standing right over his head, and remember to make choice of such a Digit as may not clean take away the whole number at the first, but rather leave so much as the said quotient having afterward to be multiplied diverse ways as shall be said hereafter, may take away the rest or as much thereof as may be, as in this example following, you shall find the aptest Digit for this purpose to be 4. which you must put in the quotient, whereby the quotient shall be now 34. that done, you must first multiply this last quotient which is 4. into the Divisor 27. the product whereof is 108. which you must set down right under the Divisor, beneath the line already drawn, as you see in this example.  
Secondly you must multiply the said 4. in itself quadratly which maketh 16. then multiply that 16. by the triple 9 the product whereof is 144. which is to be set right under the triple as in this example.  
Thirdly you must multiply the said 4. in itself cubically, which maketh 64. which is to be set right under the next prick on the right hand beneath the former sums as you see in this other example.  
So as every one of the foresaid three products do extend one further than another by one figure towards the right hand, as you see in the former example: Now these products being thus placed, draw another line and bring all the three several products contained betwixt the 2. lines, into one sum by Addition, and you shall find the total sum to be 12304. which being subtracted out of the number standing right over it which is 14063. there will remain 1759. which you must set down above as you see in this example.  
Now to proceed with the next, you must find out a new Triple and a new Divisor.  
How is that done?  
Thus, multiply the whole quotient which is 34. by 3. the product whereof is 102. and that is the Triple which is to be placed in the next void space hard by the next prick that is on your right hand: Than multiply the whole quotient into the last found Triple 102. and the product thereof which is 3468. shall be your Divisor, which is to be placed under the Triple one figure shorter towards the left hand as you see in the example following: which two numbers being thus found and rightly placed, draw a line, then ask how many times the first figure of your Divisor which is 3. is contained in the number right over his head which is 17. and you shall find it to be 5. times contained therein, wherefore set down 5. in the quotient, that done, multiply the 5. into the Divisor, and place the product thereof which is 17/340. right under the Divisor beneath the line, as you see in this example.  
Secondly square the same 5 that is to say, multiply it in itself, and that maketh 25. which 25. you must multiply again into the Triple 102. the product whereof is 2550. which number you must set right under the said triple beneath the line. Thirdly multiply the said 5. in itself cubically which maketh 125. and place that right under the first prick on the right hand, and draw a line, that done, add all the foresaid 3. products together, and you shall find the total sum thereof to be 1759625. which being subtracted out of the upper number, there remaineth nothing, as this example plainly showeth.  
Whereby you may conclude that the foresaid number whereof you sought the root, is a perfect Cubique number, for if you multiply the whole quotient in itself cubically, it will produce the self same number whereof you sought the root. But note that if in making the first Subtraction, the first Divisor is not to be found in the upper number, than you must set a cipher in the quotient, that done, triple the whole quotient, and place the said triple under the figure which is next to the next prick on the right hand, working in such order as before, and to avoid confusion, having to deal with a multitude of numbers it shall not be amiss at the finding out of every new triple, & divisor to set the remainders in several places by themselves, & to work then in the self same order that is before taught, also note that if you have to deal with a few numbers, and that the Divisor cannot be subtracted out of the number standing right over his head, than you must set a cipher in the quotient and so you have done, as for example, having to take the cubical root of 8597. here I found 2. to be the quotient, which being cubically multiplied, doth clean take away the 8. and now according to my former rule 12. must be my Divisor, which because I cannot take it out of 5. I set down therefore a cipher in the quotient, so as the quotient is now 20. which is the cubical root of the foresaid number, for if you multiply 20. cubically and add thereunto 5 6 7. which is the remainder, the sum thereof will be like unto the first number, as you may see in this example.   

Of Astronomical Fractions. Cap. 27  
BEcause the use of these Fractions is very necessary for those that have to calculate the motions of the Stars and the difference of time, I thought good to show here how the same are to be added, subtracted, multiplied and divided, for the measure of time falleth not out always to be a whole year, month, day, or hour, nor the moving of the celestial bodies are to be measured always by whole circles, signs, or whole degrees, & therefore to have an exact measure of such things it was thought best by the ancient writers to divide all whole things called Integra into the lessest parts that might be, for which purpose no number was thought so meet as 60. for there is no number under 100 that receiveth so many Divisions as 60. which may be divided many sundry ways, that is by 2. 3. 4. 5. 6. 10. 12. 15. 20. and by 30. and therefore they divided every whole thing that had no usual parts into 60. minutes, and every minute into 60. seconds, and every second into 60. thirds, and so forth unto 60. fourthes, fifts, sixts, sevenths, eights, ninthes and tenths, and further if need were but that seldom chanceth. And you have to note that minutes are marked with one stréeke over the head, seconds with two stréekes, thirds with three stréekes, and so forth thus, 23′· 6″· 7‴· 8''''· etc. which do signify 23. minutes 6. seconds 7. thirds, and 8. fourths.  

Of Addition.  
WHat is to be observed in adding of these kinds of Fractions?  
First that you bring Integrums to Integrums, and Fractions to Fractions, that be of like denomination, beginning always with the lest on the right hand, and if the sum of such Addition do amount any where to the number of 60. or above 60 then you must look how many 60. are comprehended therein, and for every 60. add one to the next greater Fraction that is on the left hand, observing still that order until you come to the Integrums, of which Integrums, it is also necessary to know their values, that is to say, what parts they contain, and what denomination those parts have: As for example, if you add common signs such as the twelve signs of the zodiac be, than every sign containeth 30. degrees, so as every sum exceeding 30 is to be divided by 30. but if they be Physical signs whereof 6. do make a whole circle, such as be set down in the table of Alphonsus, than the sum of those degrees is to be divided by 6. Moreover so often as the sum of the common signs do exceed 12. or the sum of the Physical signs do exceed 6. the overplus is always to be rejected, and the remainder to be set in the place of the signs, as you may see in this example following wherein seconds are reduced to minutes, minutes to degrees, and finally degrees, to signs.   

An example of Addition consisting of signs, degrees, minutes and seconds.  
 Signs Degrees Minutes Seconds 9 19 1. 19 0. 0. 35. 16. 11. 29. 33. 5. 9 29. 38. 11. 0. 11. 49. 40.   4. 56. The total sum. 8. 0. 42. 27.  
In which example beginning first with the seconds because they are here the lest Fractions, you shall find by Addition that the sum of them amounteth to 147. which being divided by 60. you shall find in the quotient 2. and the remainder to be 27″· which remainder you must set down under the collum of seconds, keeping the quotient which is 2. in mind to be added to the collum of minutes, the sum whereof is 162. which being divided by 60. you shall found in the quotient 2. degrees, and the remainder to be 42′· which is to be set down under the collum of minutes, and the quotient 2. kept in mind to be added to the collum of degrees, the sum whereof is 90. degrees, which being divided by 30. (because that 30. degrees do make one common whole sign) you shall found in the quotient 3. signs and no remainder: wherefore you must set down a cipher under the collum of degrees, and add the 3. signs to the collum of signs, the sum whereof is 32. which it you divide by twice 12. which maketh 24. you shall find the remainder to be 8. signs, which is to be set down under the collum of signs, for the quotient here is to be rejected according to the rule before given, so as the total sum of this Addition is 8. signs, 0. degrees 42′· 27″·   

Another example of days, hours, minutes and seconds to be added together.  
 Days Hours Minutes Seconds 21. 14. 32. 11. 16. 16. 19 41. 8. 16. 30. 30. Summa totalis. 46. 23. 22. 22.  
In this example one self order as in the other before is to be observed as touching the seconds and mynutes, for the exceeding number in both, are divided by 60. but when you come to the hours you must divide that number by 24. because that so many hours do make a whole day, and having set down the remainder under the collum of hours, add that one day which was in the quotient, unto the collum of days, and so you shall found the total sum to be as the example above showeth.   

Of Subtraction.  
WHat order is to be observed in Subtraction?  
The self same that was before observed in Addition, so as you always remember that when you have to take a greater number of Fractions, as of minutes, seconds, thirds and such like, out of a lesser number of Fractions, to borrow 60. and having set down the remainder to add the one borrowed, unto the next collum on the left hand, for there the 60. borrowed, is but one, but if you have to deal with degrees, which are counted Integrums, than you must borrow but 30. for so many degrees do make one sign, and if you have to subtract hours, than you must borrow 24. for so many hours do make one day, as by the example here following you shall more plainly perceive. Signs Degrees Minutes seconds 8. 0. 42. 27. 0. 1. 9 53. 7. 29. 32. 34.  
In this example because you can not take 53″· out of 27″· you must borrow one minute from the next collum on the left hand, which one minute is 60″· which being added to 27. do make in all 87″· out of which if you take 53″· there will remain 34″· that done, you must pay home the minute which you borrowed by adding the same unto the next 9 on the left hand, which maketh 10. then say, take 10′· out of 42′· and there remaineth 32′· which being set down, proceed to the next. But here to take one out of none that cannot be, and therefore borrow one whole sign from the collum of signs which is 30. degrees, from whence take one, and there remaineth 29. which being set down, take the one which you borrowed out of 8. and there remaineth 7. so as the whole remainder of this Subtraction is 7 / signs, 29 / degrees, 32′ / and 34″· as the former example showeth.   

Of multiplication.  
THough there be more difficulty in multiplying and dividing Astronomical Fractions then in adding or subtracting them, yet the greatest difficulty thereof chief consisteth in the finding out of the true denomination of the products, for first as touching multiplication you must multiply every number of the multiplyer into all the particular numbers of the sum that is to be multiplied, and then severally to add together the products that be of one self denomination, and whatsoever in that Addition ariseth to the number of 60. or exceedeth 60. it is to be reduced by the sexaginarie Division into the greater sum, so shall you collect the whole sum of the multiplication. But you have to note by the way, that if there be any Integrums of divers denominations in your multiplyer: That then such Integrums must be reduced to one self kind of Integrums: As for example, suppose that you would multiply the daily motion of the Moon which according to Alphonsus' tables, is 13. degrees, 10′· 35″· and 1‴· by 29. days, 12. hours, 44′, and 3″· here because there be in this multiplyer Integrums of divers denominations, that is to say days and hours, you must therefore reduce the same into one self denomination, before that you can make your multiplication.  
How is that to be done?  
By multiplying every number of the said multiplyer by 5. and then by halfing the product thereof, by which halfing you shall reduce mynutes to seconds, and seconds to thirds, and so forth to the smallest Fractions of all, and if any product do amount to 60. or exceed 60. than you must divide that product by 60. the remainder whereof is to be set under his proper denomination, and you must keep the quotient in mind to add the same to the next greater number as this table showeth: In the front whereof I have set down in several spaces not only the denominations of the two Integrums, as days and hours, but also the denominations of so many Fractions as I think meet to serve my turn, under which from I place the foresaid multiplyer, and then draw a line as you see in this example following.  
Denomination. Days Hours ′ ″ ‴ ' ' ' ' The multiplyer to be reduced into one self denomination. 29. 12. 44 3   The products of the reduction.  29. 31 50 7 30  
Now beginning first on the right hand with the lest Fraction of the said multiplyer which is 3″· I multiply 3. by 5. which maketh 15. the one half whereof is 7‴· and half a third which is 3''''0· wherefore I set down the said 7‴· and 3''''0· under their proper denominations as you see in the example above, than I multiply 44′ by 5. the product whereof is 220. and the half thereof is 110. which being divided by 60. the quotient is 1′· and the remainder 50″· which remainder I set under his proper denomination, keeping the quotient still in mind, that done, I multiply 12. hours by 5. the product whereof is 60. and the half of that is 30′· whereunto by adding the one which I had in mind, I make it 31′· and so I set it down under his proper denomination, and because there be no more Fractions to be multiplied, I set down on the left hand the Integrum 29. and by this means I have brought the foresaid multiplyer to one self denomination and to one kind of Integrum, that is to say to 29. days 31′· 50″· 7‴· 3''''0· which now being the greater number is to be set above, and to be Multiplied by the foresaid daily motion of the Moon, that is 13. degrees, 10′ 35″· and 1‴· but to the intent that in multiplying these numbers together you may set every product in his true place, that is to say, under his proper denomination, it shall not be amiss in the front of your work to set down two rows of numbers, whereof the first must contain many denominations or Fractions as you think good, as minutes, seconds, thirds, fourth's, and so forth, marked with stréekes and vulgar numbers, and the second row shall be the natural order of numbers written in Arithmetical figures as this table showeth. Integra i ii iii iiii u· vi· seven viii The denominations. 0. 1. 2. 3. 4. 5. 6. 7. 8. The natural numbers.  
Under which table you must first set the number that is to be multiplied, and right under that the multiplyer in such sort as every particular product may be placed under his proper denomination, and then draw a line as you see in this example following, and when you have multiplied 2. numbers the one into the other, and know not where to place the product, then mark under which of the natural numbers in the front, the said two numbers, that is to say the multiplyer and the multiplycand do stand: That done, add those two natural numbers of the front together, and the sum thereof will them you under what denomination the product is to be placed, as in this example.  
Int●gra ′ ″ ‴ ' ' ' ' ᵛ vi seven Denomination 0 1 2 3 4 5 6 7 Natural numbers. 29 31 50 7 30    The multiplycand. 13 10 35 1     The multiplyer    29 31 50 7 30 The several products.   1015 1085 1750 245 1050   290 310 500 70 300   377 403 650 91 390    389 6 24 2 31 12 37 30 The general product or total sum. 
The remainder.   
In which example I first multiply 1‴· into 3''''0· the product whereof is 3 vii0·s which must be set under the denomination seven. because the two natural numbers that is 3. standing over 1. the multiplyer, and 4. standing over 30. the multiplycand being added together do make 7. appointing to the product his proper denomination, then multiply again the same 1‴· into 7‴· the product whereof is 7 vi·s which must be set under the denomination vi. because 3. standeth over both their heads, and therefore must be taken twice, that is to say for each number 3. which being added together do make 6. appointing thereby to the product his proper place of denomination, that done, multiply the said 1‴· into 50″· the product whereof is 50″· which must be set under the denomination v. for the 3 which standeth over 1. and the 2. over 50″·S being added together maketh 5. appointing to his proper place of denomination, then multiply the said 1‴· into 31′· the product whereof is 1''''1· here the two natural numbers that is to say 3. standing over 1‴· and one over 31′·S being added together, do make 4. appointing to the product his proper place of denomination, then multiply the foresaid 1‴·S into 29. Integrums, the product whereof is 2‴9· and must be set under the denomination ‴ because 3. standeth over 1‴· but 29. being an Integrum, hath no natural number standing over him but a cipher, thus having gone through out all the numbers of the multiplycand, with the first number of the multiplyer, proceed in like order with the second number of the multiplyer, which is 35″· which being multiplied into 3''''0· maketh 10 vi50·s to be set under the denomination vi. because 2. standeth over 35″· the multiplyer, and 4. over 3''''0· the multiplycand, which 4. and 2. being added together maketh 6. then multiply 35″· by 7‴· which maketh 245 u·s which you must set down under the denomination v. because 2. and 3. maketh 5. that done, multiply 35″· by 50″· which maketh 1750''''· which you must set down under the denomination' ' ' ' because the multiplycand and the multiplyer are both under the denomination ″ which being twice repeated, maketh 4. then multiply 35″· by 31′· and that maketh 1085‴· which you must set down under the denomination ‴ because 1. and 2. maketh 3. finally multiply 29. Integrums, by 35″· and that maketh 1015″· thus as you have gone through with two numbers of your multiplyer, so proceed in like order with the other two numbers of the multiplyer which is 10. and 13. and when you have ended your multiplication, and set every product in his proper place, and so as every figure may stand one right under another, to avoid confusion when you come to Addition, (to which end the spaces of collums had need to be the larger) then draw a line under all the products and beginning on the right hand, add all the products contained in every sever all collum together, and if the sum of any such particular Addition do arise to the sum of 60. or exceed the number of 60. then divide that sum by 60. and set down the remainder, keeping the quotient in mind to be added to the product of the next collum on the lefthand, so shall you found the total sum of your multiplication to be 389. degrees, 6′· 24″· 2‴· 31''''· 12 u· 37 vi· and 30 vii·s as the former example plainly showeth. Now if you dsuide 389. degrees, by 30. because every common sign containeth 30. degrees, you shall found your total sum to be 12. signs, 29. degrees, 6′· 24″· 2‴· 31''''· 12 u· 37 vi· and 30 vii·s and so much the Moon runneth in the space of 2. 9 days 12. hours, 44′ and 3″· of an hour, which is her full revolution betwixt every two changes, but for as much as it chanceth as well in this example as in many others like, that Integrums of two sundry denominations are propounded in the question, it may be very well doubted with what denomination the product of such multiplication is to be named, as in this example having multiplied time by motion, a man may ask whether the product shall be named days or degrees, the resolving of which doubt dependeth upon the nature of the question propounded, for in the foresaid example, because time or days do comprehend any certain appointed motion, therefore the product of the multiplication is to be referred to the degrees of motion which are comprehended under time, and not to time which comprehendeth motion, wherefore this product of Integrums videliz. 389. signifieth here degrees and not days, so likewise when degrees and minutes are multiplied by miles and minutes, the product of such multiplication taketh his name from miles & not from degrees, because degrees do comprehend miles, for we say in matters of Geography that every degree of the great circle comprehendeth 60 miles, thus having spoken sufficiently of the multiplication of Astronomical Fractions, we will now proceed to the Division of such Fractions.   

Of the Division of Fractions Astronomical.  
WHat is to be observed therein?  
First you must consider whether your Divisor be compound, or simple, I call that compound which containeth Fractions of divers denominations, and that simple which consisteth of Integrums, or is one whole number of one self denomination, wherein there is no difficulty, for than you have no more to do but to divide every particular number contained in the dividend by the same Divisor and to place the product of every one under such denomination, as the little table of denominations showeth, & therefore it shall not be amiss to set the foresaid little table over your dividend even as you did in multiplication: Also the Sexagenary progression is always to be used, as well in Division as in multiplication. Moreover if your Divisor be not exactly contained in the dividend, then having multiplied the dividend by 60. you must add to the product thereof the next Fraction following: As for example, knowing by Alphonsus' tables that the daily motion of the Moon is 13. degrees, 10′· 35″· 1‴· 15''''· you would know how much the goeth in the space of an hour, here because that one day containeth 24. hours, the number must be 24. your Divisor which is simple and not compound, first then set down in the front of your work the row of denominations only, and not the natural numbers, because they are not to be used in this way of Division, that done, right under the row of denominations place your dividend, and right under th● your Divisor, as you see in this example.  
Degree● ′ ″ ‴ ' ' ' ' ᵛ The denominations. 13 10 35 1 15  The dividend. 24      The divisor.  32 56 27 33 7 The several sum of every quotient.  
In which example because the Divisor 24. is not contained in 13. therefore I multiply 13. by 60. which maketh 780. whereunto by adding the next Fraction on the right hand which is 10′· the whole sum is 790′· which being divided by 24. the quotient is 32′· which because they are minutes, I place them under the denomination of minutes, and the remainder is 22′· which being multiplied by 60. maketh 1320″· whereunto I add the next figure which is 35. and so the whole sum is 1355″· which being divided by 24. the quotient is 56″· which I place under the denomination of seconds, and the remainder of this Division is 11″· which being multiplied by 60. maketh 660‴· whereto I add the next Fraction which is 1‴· so that now the whole sum is 661‴· which being divided by 24. the quotient is 27‴· which I set down under the denomination of thirds, and the remainder is 13‴· which being multiplied by 60. maketh 780''''· whereunto I add the next Fraction which is 15''''· which maketh in all 795''''· which being divided by 24. the quotient is 33''''· which I place under the denomination of fourths. and the remainder is 3''''· which being multiplied by 60. maketh 180 u·s whereunto having no Fraction to add, I divide the same by 24. and so I found in the quotient 7 u·s which I set under the denomination of fifts, so as I found the hourly motion of the Moon to be 32′· 56″· 27‴· 33''''· 7 u·s and somewhat more, for I leave to deal any further with the smaller Fractions that would still grow by multiplying the remainders by 60. thinking this sufficient to show you in what order you have to work, to divide your dividend by a simple Divisor, into as many small parts as you will: but if your Divisor be compound, than the Division is to be done either by reduction into the smallest Fractions, or without reduction: which last way is very hard and tedious, and therefore I will only show you how to make your division whereof the Divisor is compound by reduction, and that by this one example here following. Suppose then that the Moon according to her own course which is from West to East, is distant from some fixed Star 36. degrees. 30′· 24″· 50‴· and 15''''· and that you would know in what time she will run that distance, according to her daily moving which as hath been said before is 13. degrees, 10′· 35″· 1‴· and 15''''· here to make this division by reduction, you must do thus. First reduce all the numbers of your dividend into the smallest Fractions thereof by the Sexagenarie multiplication and Addition of the next Fraction unto the product of that multiplication: that done, reduce all the numbers of your Divisor by like multiplication and Addition, into the smallest Fractions, so as the dividend & the Divisor may be both of one self denomination, and divide the one by the other, even as they were Integrums, as in this example you must first multiply 36. degrees, by 60. and it will make 2160′· whereto by adding 30′· you make the whole sum of minutes to be 2190′· which being multiplied again by 60. do make 131400″· whereto if you add the 24″· the sum of seconds will be 131424″· and so proceeding still with the Sexagenarie multiplication and Addition of the next Fraction as you did before, you shall found the dividend to be 473129415''''· Than in like order reduce your Divisor into the smallest Fraction, and you shall found the total sum thereof to be 170766075''''· this reduction being made, divide the dividend by the divisor, so shall you found in the quotient 2. Integrums, that is to say 2. days, and the remainder to be 132597365''''· which remainder if you multiply by 60. and divide the product by the self same Divisor, you shall have in the quotient minutes, then multiply again that remainder by 60. and divide the product thereof by the same Divisor, and you shall have in the quotient seconds, and so by observing still that order you shall bring it into as small Fractions as you will, thus shall you find that the Moon according to her daily motion, will run the foresaid space of distance that was betwixt her and the fixed Star in 2. days, 46′·S and 14″·    

How to divide Astronomical Fractions when the Divisor is greater than the dividennd. Cap. 28.  
THough by the last chapter you may learn how to divide any number in Astronomical Fractions, whereof the Divisor is greater than the dividend, yet I mind once again, to set down a general rule to serve for such purpose, because it cometh often in use in having to deal with Astronomical tables, and to give you example thereof: First then having to divide any number, whose Divisor is greater than the dividend, do thus; multiply the greatest denomination of the dividend by 60. and if there be any Fractions annexed thereunto which are of the next inferior kind, as minutes are to degrees, or to hours, and seconds to minutes, and thirds to seconds etc. Than add them to the former product, but if such Fractions be not of the next inferior kind, then let them stand as they are until you come to deal with them, and having divided according to the common rule of Division the first sum of the dividend by the Divisor, multiply the quotient into the whole Divisor, and subtract that product out of the upper number if it may be, if not, then make the quotient lesser and lesser, until you can find such a number as will be subtracted out of the said upper number, and if there be any remainder left, then multiply that remainder by 60. not leaving to follow the former order of working, until you have found the nearest exact quotient that may be. And you have to note that the denomination of the first quotient must be of the next inferior kind, to that denomination which the Divisor hath, and to make this rule the plainer, I will set down and example used by Standius ut the 115 page of his Ephemerideses who to know the very instant of the full Moon the second of March 1569. biddeth to divide the distance of the opposition which was 8. degrees, 46′· by the diurnal excess of the moons motion from the sùnne which was then 13. degrees, 48′·S which Divisor because it is greater than the dividend, you must according to the rule before given, work thus: First multiply the greatest denomination of the dividend which is 8. degrees by 60′· the product whereof will be 480′· whereunto by adding the Fraction annexed videliz. 46′· it maketh in all 526′· which is to be divided by 13. degrees, 48′· here in dividing the last product first by 13. degrees, I find in the quotient 39 which is one too many, considering that I must take out of the foresaid dividend 48′· as often as I did take out thereof 13. degrees, wherefore I set down but 38. in the quotient, and then the remainder will be 96. which because I may easily divide by the common Divisor 13. degrees, and 48′·S I divide therefore that 96. first by 13. whereof the quotient is 7. and the remainder is 5. which I reduce into seconds by multiplying that 5. by 60. the product whereof is 300″· that done, I multiply 48′· by 7. the product whereof is 336. which though it be somewhat too great a number to be taken out of 300. yet I let it stand because it approacheth to a very nigh exactness, and by this means I find the whole quotient to be 38′· and 7″· and you have to note that if after the first quotient be set down, there happen any remainder which is lesser than the Divisor, than you must set down a cipher in the quotient, and remove your Divisor one place further, even as you do in common Division, and then to work as before.   

How to take the square root of Astronomical Fractions. Cap. 29.  
THe greatest difficulty hereof consisteth in finding out the true denomination of the root, for if the Fraction be seconds, than the root thereof are mynutes, and if the Fraction be fourth's, than the root are seconds, for the Fraction must always have such denomination as may be halfed, as seconds, fourth's, and such like, the one half whereof giveth always name to the root, for if the question be of thirds, you must first reduce them to fourth's before you can take the root, and you must do the like with any other Fraction, whose denomination is odd and not even. As for example, if you would take the root of 43‴· here by multiplying these 43‴· by 60. you shall reduce them into 1580''''· the root whereof is 50″· Moreover the Fractions where. with you have to deal, are either simple or compound, if they be simple and less than minutes, and therewith have even denominations and not odd, than you need to make no further reduction, but to work as if you had to deal with whole numbers. As in seeking the square root of 1600″· you found it to be just 40′·S but if the number be compound, that is to say, consisting ot Integrums and Fractions, or of many Fractions having divers denominations, than you must first reduce them all to the smallest Fraction that hath an even denomination before that you can take the root: As for example, you would know the root of 4. degrees, 25′· here you must by the Sexagenarie multiplication and Addition of the next Fraction, reduce the degrees to minutes, and the minutes to seconds, as you were taught before in Division, and then to work as you were wont to do in taking the square root of whole numbers, and in so doing, you shall find the sum of seconds to be 15900″· the square root whereof is 126′· which if you divide by 60. it will make 2. degrees, 6′· Another example, as to take the root of 13. degrees, 42′·S and 45″· here by reduction as before, you shall bring the degrees and minutes to 49365″·. the square root whereof is 222′· which being divided by 60. maketh 3. degrees, 42′·S And thus I end with the Astronomical Fractions, which kind of Fractions, though they be very learnedly and orderly taught by Reinoldus in the beginning of his Prutenicall tables, yet in mine opinion not in so plain order, and so fit for every man's understanding, as I have here set them down according to the doctrine of Gemma Frisius, which being once learned, you shall the sooner attain to the other. And without the knowledge of these Fractions, you can never truly calculate any thing out of the Astronomical tables, and therefore such Fractions are most necessary to be learned.   

The description and use of the Sexagenarie table.  
THis table consisteth of two figures, whereof the neither figure having four Angles, is called in Latin Trapezium, marked with the letters A. D. E. B. and the upper figure is a Triangle, marked with the letters A. B. C. and each figure containeth particular collums of numbers, serving to find out the products of Astronomical Fractions being multiplied one by another, and also the quotients of the like Fractions being divided one by another, and also the square roots of the said Fractions, for which purpose the first collum on the left hand, containeth 59 Fractions, counting from 1. standing above, and so proceeding down ward to 59 and are contained betwixt A. and D. and the foot of the said Trapezium, containeth 30. counting from the least hand towards the right, which are contained betwixt D. and E. and the rest of the numbers to make up 59 are to be found in the upper most front of the Triangle, proceeding from C. towards A. you have to note also that in the outermost collum of the Trapezium on the right hand, the numbers do proceed downward from 30. to 59 that is from B. to E. and the numbers in the outermost collum of the Triangle on the right hand, do proceed upward from 31. to 59 contained betwixt B. and C. both which do serve to fill up the first multiplyers, and multiplycands, for when you cannot find them in the Trapezium, than the outermost collum of the Triangle on the right hand, serveth to supply that want, and when you cannot find the said numbers in the Triangle, than the outermost collum of the Trapezium on the right hand, serveth to supply that want, and in seeking any multiplyer or multiplycand, either in the Trapezium or in the Triangle, consider always which way they are most readily found out, so as they may directly answer one against another.  
All the rest of the numbers contained betwixt the two outermost collums and are set down in square Angles, called common Angles, do signify either products or dividends or square numbers, according as occasion shall require. And the outermost collums do signify sometime multiplyers, sometime multiplycands. sometime quotients, and sometime roots. All which things you shall better understand by the examples hereafter following.  

The use of the Table.  
By help of this table you may more readily multiply and divide Astronomical Fractions, and also found out the square root of such Fractions, then by those rules which I have heretofore set down according to the doctrine of Gemma Frisius. And first I will set down an example of multiplication, than another of Division, and thirdly one example of finding out the square root of the said Fractions, and let the example of multiplication be thus: Suppose that you would multiply 29. degrees, 31′· 53″· 7‴· 30''''· by 13. degrees, 10′· 35″· 1‴· which you must set down in such order as followeth, (that is to say) first the denominations, than next under them the multiplycand, and next under that the multiplyer, and under them the several products, and lowest of all the total sum of the said products.  
Integra De. ′ ″ ‴ ' ' ' ' ᵛ vi seven Denominations  29 31 50 7 30    The multiplycand.  13 10 35 1     The multiplyer.     29 31 50 7 30 The first product.   17 13 34 14 22 30  The second product.  4 55 18 21 15 0   The third product. 6 23 53 51 37 30    The fourth product, 6 29 6 24 2 31 12 37 30 The total sum  
Here beginning with the first number of the multiplyer on the right hand which is 1‴· say thus, one times 30. is 30 which is to be placed under the denomination of 7. because the denominatino over 1. is 3. and the denomination over 30. is 4. which being added together according to the rule before set down in the Chapter of multiplication of Astronomical Fractions do make 7. then say one times 7. is 7. which you must place under the sixth denomination. Again one times 50. is 50. which is to be set under the fift denomination, because 3. and 2. maketh 5, then say one times 31. is 31. which is to be set under the fourth denomination, for 3. and 1. maketh 4, than one times 29. is 29. which is to be set under the third denomination, and thus you have the first product: then proceed with the next number of the multiplyer towards your left hand which is 35. and is to be multiplied into 30. which to do readily, you must enter the table with these two numbers, and seeking in the first Collum of the Trapezium on the left hand, for 35. in the foot of the said Trapezium, for the number of 30. you shall find in the common angle the product to be 17. and 30. whereof you must place the 30. under the sixth denomination, and keeping the number 17. still in mind to be added to the next product, multiply 35. into 7. and you shall find in the Trapezium the product to be 4. and 5. whereunto if you add the 17. which you had in mind, the product will be 4. and 22. whereof you must set the 22. under the fift denomination, and keeping the 4. in mind, multiply again the same 35. into 50. the product whereof you shall find in the Triangle to be 29. and 10. whereto if you add the 4. in mind, it will make 29. and 14. whereof the 14. is to be placed under the fourth denomination, and keeping the 29. in mind, multiply again the said 35. into 31. and you shall find the product thereof in the Triangle to be 18. and 5. whereunto if you add 29. in mind, it will make in all 18. and 34. which 34. is to be set under the third denomination, then keeping 18. in mind, multiply the foresaid 35. into 29. and you shall find the product thereof in the Trapezium to be 16. and 55. whereunto if you add the 18. in mind, the product will be 17. and 13. which 13. is to be placed under the second denomination. Now because you have gone through all the numbers of the multiplycand with the number 35. you must place the 17. which you had in mind under the denomination of mynutes, and so having ended the second product, proceed to the finding out of the third product by multiplying 10. first into 30. the product whereof you shall found in the Trapezium to be 5. and 0. whereof you must set the cipher under the fift denomination and keeping the 5 in mind, multiply again 10. into 7. and you shall find the product to be 1. and 10. whereunto if you add the 5. in mind, the product will be 1. and 15. whereof you must set the 15. under the fourth denomination, and keeping the one in mind, multiply 10. into 50. and you shall found the product thereof in the Trapezium to be 8. and 20. whereunto if you add the 1. in mind, it will make 8. and 21. whereof you must set the 21. under the third denomination, and keeping 8. in mind, multiply 10. into 31. and you shall found the product thereof in the Trapezium to be 5. and 10. unto which if you add the 8. in mind, the product will be 5. and 18. whereof set down the 18. under the second denomination, and keeping 5. in mind multiply 10. into 29. the product where of you shall find in the Trapezium to be 4. and 50. whereunto if you add the 5. in mind, the product will be 4 and 55. Now because you have gone through all the numbers of the multiplycand, with 10. you must set down the 4. which you had in mind under the denomination of degrees, and so having the third product, proceed to the fourth by multiplying 13. into 30. and you shall found the product in the Trapezium to be 6. and 30. whereof set down 30. under the fourth denomination of Fractions, and keeping 6. in mind multiply 13. into 7. the product whereof you shall find in the Trapezium to be 1. and 31. whereunto if you add the 6. in mind, the product will be 1. and 37. whereof set down the 37. under the third denomination, and keeping the 1. in mind multiply 13. into 50. and you shall found the product in the Trapezium to be 10. and 50. whereunto if you add the one in mind, the product will be 10. and 51. whereof set the 51. under the second denomination of Fractions, and keeping 10. in mind multiply 13 into 31. the product whereof you shall found in the Trapezium to be 6. and 43. whereunto if you add the 10. in mind, the product will be 6. and 53. whereof set down 53. under the first denomination of Fractions, and keeping 6. in mind multiply 13. into 29. the product whereof you shall found in the Trapezium to be 6. and 17. whereunto if you add the 6. in mind, the product will be 6. and 23. whereof set down 23. under the denomination of degrees. Now because you have gone through all the numbers of the multiplycand with the last number of the multiplyer, you must set down the 6. which you had in mind under the denomination of Integrums, that done, add all the four products together, beginning on the right hand, saying thus, 30. and 0 is but 30. which set down under the nethermost line, as you see in the former figure, then say 30. and 7. maketh 37. which you must set down under the nethermost live next unto 30. then say 22. and 50. maketh 72. out of which by subtracting the Sexagenarie number, the remainder is 12 v. which is to be set under the line next unto the 37. keeping still the 60. in mind, which in this account maketh but one, and is to be added to the next rank on the left hand, then say 1. in mind and 5. maketh 6. and 4. maketh 10. and 1. maketh 11. here set down 1. and keep the Article in mind, then say 1. in mind and 3. maketh 4. then 1. and 1. and 3. being added to 4. do make in all 9 out of which 9 you must subtract 6. which is but one 60. and is to be kept in mind and there remaineth 3. which is to be set down by the 1. then say 1 in mind and 7. is 8. and 1. is 9 and 4. is 13. and 9 is 22. here set down 2. and keep the 2. Articles in mind, then say 2. and 3. maketh 5. and 2. is 7. and 3. is 10. and 2. maketh 12. tens, which do make 2. sixties, and are to be kept in mind, then say 2. and 1. maketh 3. and 8. maketh 11. and 3. maketh 14. wherefore set down 4. keeping the 1. Article in mind, then say 1. in mind and 5. is 6. and 1. is 7. and 1. is 8. then take 6. from 8. and there remaineth 2. which set down by the 4. and keep one sixty in mind, then say 1. in mind and 3. is 4. and 5. is 9 and 7. is 16. wherefore set down 6. and keeping one in mind, say that 1. and 5. is 6. and 5. is 11. and 1. is 12. which maketh two sixties to be kept in mind, then say 2. in mind and 3. is 5. and 4. maketh 9 which set down, then say 2. is 2. which set down by the 9 under the denomination of degrees, then say 6. and nothing maketh 6. which set down under the denomination of Integrums, for that 6. in this place signifieth 6. sixties, which is in value 360. degrees, and being divided by 30. because 30. degrees maketh one whole sign, you shall found in the quotient 12. signs, so as the total sum of the four products is 12. signs, 29. degrees, 6′· 24″· 2‴· 31''''· 12 u· 37 vi· 30 vii·s as the former figure showeth, and this is the very same example which I wrought before when I taught you how to multiply Astronomical Fractions according to Gemma Frisius his rule, and they both do wholly agree in every condition, saving that to work by the Sexagenarie table is the readier way of the two.   

An example how to divide Astronomical Fractions by help of the Sexagenarie table.  
SVppose then that you would divide the former total sum or product found by multiplication, which is 6. Integrums, 29. degrees, 6′· 24″· 2‴· 31''''· 12 u· 37 vi· 30 vii·s by this Divisor 29. degrees, 31′· 50″· 7‴· 30''''· which in the former example of multiplication was the multiplycand. Now to divide these two numbers the one by the other, you must do thus, first you must set down the row of denominations as you did before in multiplication, and next under that the dividend, then right under that the divisor, and on the right hand behind a crooked line made like a half Moon, all the several quotients are to be set one by another in a right line, as you may see by the figures hereafter following.  
Denominatiō●▪ Integrum de ′ ″ ‴ ' ' ' ' ᵛ vi seven The dividend. 6 29 6 24 2 31 12 37 30 The divisor.  29 31 50 7 30     
Here having made your row of denominations, and set down your dividend, you have to consider whether the first number of your Divisor be greater than the first number of your dividend, for if it be, than you must place your Divisor one space further towards your right hand, as in this example, because the first number of your Divisor 29. cannot be taken out of 6. you set it under the second number of the dividend, and so all the rest of the numbers successively towards the right hand, as the former example showeth, now the order of working is thus: You must first seek out the first number of your Divisor, in the first collum of the Trapezium on the left hand which is 29. then in the row right against that 29. on the right hand, you have to seek out those numbers of the dividend, which do stand right over the first number of your Divisor which is 6. and 29. and if you cannot find those numbers justly, then seek in the self same row a number which is somewhat less and nearest in value unto it, as in this example, because you cannot find 6. and 29. you take 6. and 17. the quotient, whereof you shall found in the foot of the Trapezium right under the said number 6. and 17. to be 13. degrees, which you must set in the quotient line, and that is your first quotient, having his proper denomination over his head which are degrees, and are to be found by the rule before taught. Thus you see here that the number of the Divisor is to be found in the outermost collum on the left hand, and the number of the dividend in that row which is right against the said Divisor, and the quotient in the foot of the Trapezium right under the number of the dividend last found. Than you have to multiply the whole Divisor (that is to say) every particular number thereof by the first quotient 13. which you may do by help of the Table, as you did before in the example of multiplication, and is to be set down in this manner. And you shall find the whole product to be 6. 23. 53. 51. 37. 30.  
Which being set right under the first dividend, is to be subtracted out of the same, and the remainder to be written over the head of the dividend as you do in common Division, first then to multiply all the particular numbers of the Divisor by the quotient 13. and to found every product thereof, resort to the Table and seek for 13. in the foot of the Trapezium, and for 30. which is the multiplycand in the outermost collum of the Trapezium on the left hand, and the common angle will show the product which is 6. and 30. whereof you must set down 30. under 30. and keeping 6. in mind, multiply again 13. into 7. the product whereof you shall find to be 1. and 31. unto  29 31 50 7 30 The Divisor.      13 The first quotient. 6 23 53 51 37 30 The first product. which add the 6. which you kept in mind, and the product shall be 1. and 37. whereof you must set down 37. and keeping one in mind, multiply again 13. into 50. and you shall found the product to be 10. and 50. whereunto if you add 1. in mind, the product shall be 10. and 51. whereof you must set down 51. and keeping 10. in mind, multiply 13. into 31. the product whereof you shall found in the Trapezium to be 6. and 43. whereunto if you add the 10. which you had in mind, the product shall be 6. and 53. whereof set down 53. and keeping 6. in mind, multiply 13. into 29. and you shall found the product to be 6. and 17. whereunto if you add the 6. in mind, the product shall be 6. & 23. whereof set down 23 under the last number of your multiplycand, and because you have no more numbers of the Divisor to be multiplied, set down 6. in mind on the left hand, so shall the whole product be 6. 23. 53. 51. 37. 30. as the former example showeth, which product is to be subtracted out of the first dividend, & the remainder is to be set down over the head of that dividend, as you see in this example next following, wherein the first dividend is first set down, and right under that the foresaid product which is the first product, and the remainder above the dividend, and the quotient 13. is set in the quotient line which is your first quotient.  
The remainder.  5 12 32 25 1    The first dividend. 6 29 6 24 2 31 12 37 30 (13. The first product, 6 23 53 51 37 30     
And remember (in making your Subtraction) to begin with the first number of the foresaid product which is on the right hand, and when you cannot take it out of the number standing right over his head, to borrow always 60. of the next number on the left hand, and to pay it home again with 1. for there 60. is but one. This done, remove your Divisor one space further towards the right hand (that is to say) set the first number of your Divisor under 12 which is the second number of the second dividend, which together with the first remainder is 5. 12. 32. 25. 1. 12. 37. 30. and all other numbers of the Divisor orderly towards your right hand, as you see in this example.  
The second dividend. 5 12 32 25 1 12 37 3● The Divisor.  29 31 50 7 30    
Than ask how many times 29. is in 5. & 12. which number you must seek in the Trapezium in the row that answereth towards the right hand, to the first number of your Divisor which is 29. standing in the outermost collum of the Trapezium on the left hand, and because you cannot found 5. and 12. in that row, you must take in the same row the number which is nighest unto it, but less, which you shall found to be 4. and 50. and right under that in the foot of the Trapezium you shall found 10. which must be your second quotient, by which quotient you have to multiply all the particular numbers of the Divisor in such order as is before set down and you shall found the product of that multiplication to be 4. 55. 18. 21. 15. 0. which product you must place under the second dividend setting 4. under 5. and 55. under 12. and so forth orderly towards the right hand, that done, subtract the same product out of the numbers of the second dividend, standing right over the said product, and the remainder will be 17. 14. 3. 46. 12. as you see in this example.  
The remainder.  17 14 3 46 12 37 30  The second dividend. 5 12 32 25 1 12 37 30  (13.10. The second product. 4 55 18 25 15 0     
Now remove your Divisor one space further towards the right hand by setting the first number of your Divisor which is 29 under 14. which is the second number of the third dividend, and so all the rest orderly towards the right hand, as you see in this example.  
The third dividend. 17 14 3 46 12 37 30 The Divisor.  29 31 50 7 30   
Than ask how many times 29. is contained in 17. and 14. which dividend because you can not found it in the row that answereth to that 29. which standeth in the outermost collum of the Trapezium on the left hand (for all those numbers are too little) you must seek for it in that collum which standeth right upon 29. in the foot of the Trapezium, neither shall you found it there, but you shall find 16. and 55. which is somewhat lesser yet in value nighest unto 17 and 14. and right against that you shall found in the outermost collum on the right hand 35. which must be your third quotient to be set in the quotient line, whereby you have again to multiply the whole Divisor in such order as before, the product whereof you shall found to be 17. 13. 34. 14. 22. 30. which product being placed under the dividend by setting 17. under 17. and 13. under 14 and so forth orderly towards the right hand, subtract the said product (beginning at the right hand) out of the numbers which stand right over the said product, and the remainder will be 29. 31. 50. 7. 30. which is to be set down over the third dividend, and the rest to be canceled as you see in this example.  
The remainder.   29 31 50 7 30 The third dividend 17 14 3 46 12 37 30 (13.10.35 The third product. 17 13 34 14 22 30   
And now the remainder of the third dividend is come to be the fourth dividend, wherefore remove your Divisor one space further towards the right hand by setting 29. under 29. and the rest orderly towards the right hand as you see in the example following. Than ask how many times 29. is contained in 29. which being but once, your fourth quotient is 1. and is to be set in the quotient line, whereby the whole Divisor being multiplied, the product will be 29. 31. 50. 7. 30. which you must place under the fourth dividend, and being subtracted out of the same, nothing will remain, and so the whole quotient will be 13. degrees, 10′· 35″· 1‴· as you see in this example.  
 0 0 0 0 0   The fourth dividend. 29 31 50 7 30  (13. degrees 10′·35″·1‴· The fourth product. 29 31 50 7 30     

Another example of Division.  
IF the daily motion of the Moon be 13. degrees 3′· 53″· 56‴· 23''''· 58 u·s in what time shall she make her whole revolution allowing 360 degrees to that revolution, which is otherwise called the month of Paragration. Hear for so much as this example is to be wrought by Division, & that your dividend is a simple and whole number (that is to say) 360. degrees, without any Fractions of divers denominations annexed thereunto: you must first set down the 360. degrees, and next to that towards your right hand set down a long row of Ciphers with so many denominations over their heads as you shall think needful to serve your turn, & right under your dividend set your Divisor as you see in this example.  
The denominations. ●nte●rum De. ′ ″ ‴ ' ' ' ' ᵛ vi seven viii ix The dividend.  360 0 0 0 0 0 0 0 0 0 The Divisor.  13 3 53 56 23 58     (27. The first product.  352 45 16 22 47 6      
Hear you must first ask how many times 13. is in 360. and by the common rule of Division you shall find the quotient to be 27. which you must set down in the quotient line, and by that quotient you have to multiply every particular number of the Divisor beginning on the right hand, as you do in multiplication, saying that 27. times 58. is 26. and 6. as the Trapezium showeth, for by seeking 27. in the foot of the Trapezium and for 58. in the outermost collum on the right hand, you shall found in the common Angle 26. and 6. wherefore set down 6. under 58. and keeping 26. in mind, multiply 23. which is the second number of the Divisor by 27. and by the Trapezium you shall found the product thereof to be 10. and 21. whereunto if you add the 26. in mind, it will make 10. and 47. whereof set down 47. under 23. and keeping 10. in mind, multiply 56. which is the third number of the Divisor by 27. and in the Trapezium you shall find the product to be 25. and 12. whereto if you add the 10. in mind, it will make 25. and 22. whereof set down 22. under 56. and keeping 25. in mind, multiply 53. which is the fourth number of the Divisor by 27. the product whereof you shall found by the Trapezium to be 23. and 51. whereto if you add 25. in mind, it will make 76 which is one 60. and 16. whereof set down 16. under 53. and add the one in mind to 23. and that will make 24. which you must keep in mind, then multiply 3. which is the fift number of the Divisor by 27. the product whereof you shall found in the Trapezium to be 1. and 21. whereto if you add the 24. in mind, it will make 1. and 45. whereof set down 45. under 3. and keeping 1. in mind, multiply 13. which is the last number of the Divisor by 27. the product whereof you shall found in the Trapezium to be 5. and 51. whereunto if you add one in mind, it will make 5. and 52. here because the first 5. is 5. sixties, it maketh in all 352. and is to be set under 360. so as the first product of this multiplication containeth these numbers 352. 45. 16. 22. 47. 6. as you see them set down in the former example, and this product is to be subtracted out of 360 which is the first dividend. And to avoid confusion, it shall not be amiss to set down the first Dividend and the first product apart by themselves thus.  
The first remainder. 7 14 43 37 12 54 0 0 0 The first dividend. 360 0 0 0 0 0 0 0 0 (27. The first product. 352 45 10 22 47 6     
Here beginning on the right hand say thus, take 6. out of nothing which will not be, wherefore you must borrow 60 then by taking 6. out of 60. there will remain 54. which you must set above that cipher which standeth right over 6. and caucell the 6. keeping still the one 60. which you borrowed in mind, then say 47. and one in mind maketh 48. which will not be taken out of nothing, and therefore you must borrow again one 60. as you did before, so shall the remainder be 12. which is to be set above the cipher which standeth right over 47. and cancel 47. and so proceed with like order in subtracting all the rest of the numbers of the first product out of the first dividend, so shall the remainder be 7. 14. 43. 37. 12. 54. as you see them set down in the former figure. Now having to remove your Divisor one space further towards the right hand, you shall do well to make your first remainder which is 7. 14. 43. 37. 12. 54. to be your second dividend, and under that to set your Divisor as you see in this example.  
The second dividend. 7 14 43 37 12 54 0 0 0 (27.33. The Divisor.  13 3 53 56 23 58    
Than ask how many times 13. is contained in 7. and 14. & having found 13. in the foot of the Trapezium, seek in that collum for 7. and 14. and not finding it there, take in the self same collum that number which is nighest in value unto it & less, which you shall found to be 7. and 9 right against which in the outermost collum on the right hand is 33. which must be your second quotient, and is to be set in the quotient line, next unto 27. and by this quotient you have to multiply every number of the Divisor as you did before, the product whereof you shall found to be 7. 11. 8. 40. 1. 10. 54. wherefore you must first set down your second dividend, and then the second product right under the same as you see in this example.  
The second remainder.  3 34 57 11 43 6    The second dividend. 7 14 43 37 12 54 0 0 0 0 (27.33. The second product. 7 11 8 40 1 10 54     
Which product being subtracted out of the second dividend, the remainder will be 3. 34. 57 11. 43. 6. which is to be set above the second dividend, and the product to be canceled as you see in the former example. Here having to remoonue agine your common Divisor one space further towards the right hand, set down first the last remainder which now must be your third dividend, and under that set the common Divisor as you see in this example.  
The third dividend. 3 34 57 11 43 6 0 0 0 (27. 33. 16. The Divisor.  13 3 53 56 23 58    
Than ask how many times 13. is contained in 3. and 34. here by seeking in the foot of the Trapezium for 13. though you cannot found in that collum 3. and 34. yet you shall found 3. and 28. which is the nighest, right against which in the outermost collum on the left hand you shall found 16. which must be your third quotient, and is to be set in the quotient line, by which quotient you must multiply the whole Divisor as before, the product whereof you shall found to be 3. 29. 2. 23. 2. 23. 28. which being the third product you must set down under the third dividend which was your last remainder and to be subtracted out of the same as you see in this example.  
  5 54 48 40 42 32 0 0 The third dividend. 3 34 57 11 43 6 0 0 0 The third product. 3 20 2 23 2 23 28    
And so the remainder will be 5. 54. 48. 40. 42. 32. which you must set above the third dividend, and all the inferior numbers are to be canceled as you see in the former example. Here having again to remove your Divisor one space further towards your right hand, the last remainder must be your fourth dividend, under which the common Divisor is to be set thus.  
The fourth dividend. 5 54 48 40 42 32 0 0  The Divisor.  13 3 53 56 23 58  27 days / 33′/16″ 16″/26‴·  
Here ask how many times 13. is contained in 5. and 54. seek for 13. in the foot of the Trapezium, in whose collum you shall not find 5. and 54. but 5. and 51. which is nighest unto it, and right against that in the outermost collum on the left hand you shall found the quotient to be 27. by which if you should multiply the whole Divisor, the whole product thereof would be 5. 56 21. 24. 1. 47. 6. which is more than the dividend, and therefore you must make your quotient one less, setting down no more but 26. in the quotient line, by which if you multiply the Divisor, the product will be 5. 26. 41. 22. 25. 14. 8. which being subtracted out of the fourth dividend, the remainder will be 28. 7. 18. 17. 17. 52. which remainder if you will, you may make to be a fift dividend, and then to work as before, if you would have your quotient to extend to smaller denominations, which I leave to do because I think that thirds be small enough. And as often as the product of any particular quotient shall be greater than the dividend, remember to take a less quotient even as you do in common Division. But now you have to note that though this whole quotient here signifieth time, for the first quotient signifieth days, the second quotient minutes of days, the third quotient seconds, and the fourth quotient thirds, yet for so much as the day is to be counted by 24. hours and not by mynutes, you must therefore reduce all the particular quotients saving the first, into hours and parts of hours, even to so small denominations as you shall think good yourself, by help of this rule which in division of Astronomical Fractions biddeth to multiply the quotient by 2. and to divide the product thereof by 5. as here if you multiply the 33. which is the second quotient by 2. the product will be 66. which being divided by 5. the quotient will be 13. hours and one sixty remaining to be kept in mind, wherefore set down in the place of 33′· 13. hours, then multiply the third quotient which is 16. by 2. and that will make 32. whereunto if you add the one 60. in mind it will make in all 92. which being divided by 5. the quotient will be 18. and two sixties, which is 120. remaining to be kept in mind, wherefore in steed of 16″·S which was the third quotient, set 18′ of an hour, and then proceed to the fourth quotient which is 26‴· which being multiplied by 2. maketh 52. whereunto if you add, 120. it will make 172. which if you divide by 5. the quotient will be 34″· of an hour, which is to be set down in the place of 26. so shall your whole former quotient contain 27. days, 13. hours, 18′· of an hour and 34″· of an hour. And note that as in the Division of Astronomical Fractions to bring the quotients to like denomination, you do multiply by 2. and divide by 5. so in multiplication to reduce the numbers of the multiplyer being of divers denominations to one self denomination, you must by order reverse, multiply by 5. and divide by 2. whereof I have given you an example before, whereas I show you how to multiply Astronomical Fractions according to Gemma Frisius, without the help of the Sexagenarie table.   

An example showing how to extract the square root out of Astronomical Fractions.  
SVppose the number given to be 17′· 12″· 33‴· 4''''· whereof you have to take the square root, here having set down the said numbers with their proper denominations over their heads, as you see in the example following, first set a prick under the last number on the right hand, and then prick every other number leaving one void space betwixt every two pricks, as you do when you seek the square root of whole numbers or jutegrums, as you see here in this example.  
The number given 17′· 12″· 33‴· 4 ' ' ' '  
Than resort to the table & seek amongst the products which are placed next to the line A. B. as well in the Trapezium as in the Triangle, and see whether you can found the number standing over the first prick on the left hand which is 17. and 12. but not finding it there, you must take that which is nighest unto it and less, which you shall found in the Triangle towards B. to be 17. 4. the root whereof you shall find both in the head, and also in the outward collum of the Triangle on the right hand to be 32. answering to the foresaid square root, which root you must place behind the quotient line, then subtract 17. and 4. out of 17. and 12. and there will remain 8. which is to be set over 12. and the 17. and 12. to be canceled, that done, double the root 32. which will be 64. that is to say 1. and 4. and setting 1. under 8. and 4. under 33. ask how many times 1. is in 8. and there is 8. which is the second quotient by which you must multiply 1. and 4. the product whereof will be 8. and 32. which being subtracted from the upper number which is 8. and 33. the remainder is 1. which is to be set over 33. and the 8. and 33. to be canceled. Finally multiply the second quotient 8. in itself, and the product will be 1. and 4. which being subtracted out of the former remainder of the given number nothing remaineth, so as you shall find 32′· 8″·S. to be the root of the given number, as this example showeth.  
Which root if you multiply into itself squarely, the product will be like unto the number given, and by this means you shall found the given number to be a just square number. But you have to note that if the last denomination standing on the right hand be odd & not even, as thirds or fifts, than you must set down a cipher beyond the last denomination towards your right hand, and under that cipher set your first prick, as in this example 3′· 2″·. 9‴· 17''''·. 8 u· here because the last denomination on the right hand is odd, that is to say fifts, you must therefore set down next unto it towards the right hand a cipher having over his head the next denomination which is vi. and is even, wherefore set your first prick under that cipher and so proceed towards your left hand pricking every other number, and then work as followeth.  
First then seek in the table amongst the square numbers nigh unto the line A. B. for 3. and 2. which should stand in one self square Angle next unto the line A. B. for those only are square numbers, and not finding it amongst the square numbers, take that square which is nighest unto it and less, which you shall found to be 2. and 49. the root whereof is 13. and is to be found as well in the foot of the Trapezium right under the foresaid square number, as also in the outermost collum of the said Trapezium on the left hand standing right against the same square number, which root is to be set in the quotient line, that done, subtract the foresaid square number out of the number standing right over the last prick on the left hand which is 3. and 2. and the remainder will be 13. which is to be set over 2. and the 3. and 2. to be canceled as you see in the former example, that done, double the foresaid found root 13. which will make 26. by which you have to divide 13. and 9 wherefore resort to the Trapezium, in the foot whereof you shall found 26. in whose collum not finding 13. and 9 take that which is nighest unto it and less, which is 13. and 0. right against which you shall found in the outermost collum on the right hand 30. which should be your quotient, but because it is too great and that the square thereof cannot be taken out of the remainder, you must make the quotient one less, and set in the quotient line no more but 29. by which 29. if you multiply 26. the product will be 12. and 34. as the Trapezium showeth, which being subtracted out of 13. and 9, standing over 26. the remainder will be 35. which 35. is to be set over 9 then by multiplying the second quotient which is 29. into itself, you shall found by the Trapezium the product to be 14. and 1. which being subtracted out of 35. and 17. the remainder will be 21. and 16. which is to be set over 35. and 17. as you see in this example.  
That done, double both the quotients or whole root, and the product will be 26. 58. and having set 58. under 8. and 26. under 16. divide 21. 16. and 8 by 26. 58. wherefore look for 26. in the foot of the Trapezium, and for 21. 16. in that collum, and not finding it there, take that which is nighest unto it and less, which you shall found to be 21 and 14, and right against that in the outermost collum on the right hand standeth 49. which being too great to be your quotient by 2. therefore make your quotient no more but 47. which being set down in the quotient line, multiply thereby 26. 58. and the product will be 21. 7. 26. which being subtracted out of 21. 16. and 8. the remainder will be 8. and 42. then setting 42. over 8. and 8. over 16. cancel 21. 16. and 8. then multiply 47. into itself squarely, and you shall find the product thereof in the Triangle to be 36. 49. which being subtracted out of 8. 42. and 0. the remainder will be 8. 6. 11. as you see in this example.  
And thus you have found the root of your given number to be 13. 29 and 47. and whether you have done rightly or not, you may know by multiplying the said root into itself squarely, and by adding to the product thereof the remainder, and if you would find a more exact root, then add more Ciphers to your given number towards your right hand, together with their denominations over their heads as they increase successively, and be sure always that the number of the Ciphers be even and not odd as 2. 4. 6. and so forth, & the more Ciphers that you add to the given number (so as the number of them be even) the more exact root you shall have.  
Now to give to every number of the root his proper denomination, do thus, take half of those denominations that are over every prick, and those shall be the denominations of the root, as in the former example, whereas seconds do stand over the last prick on the left hand take half of that which is 1′· wherefore the first number of your root must be minutes, and over the rest of the numbers of the root set seconds, thirds and so forth successively until you have brought your root to as small denominations as you desire.  
FINIS.    

A BRIEF DEscription of the tables of the three special right lines belonging to a Circle, called Sines, lines Tangent, and lines Secant. Not only showing why they were first invented, and defining the proper terms thereto belonging: But also showing divers necessary uses thereof, written by Master Blundevile 1593.  
LONDON. Printed by john Windet, 1594.   

THE DESCRIPTION and use of the Table of Synes.  
BEcause there is no proportion, comparison, or likeness betwixt a right line and a crooked, the ancient Philosophers, as Ptolomey and divers other, were much troubled in seeking to know the measures of a Circle or of any portion thereof by his Diameter, and by knowing the Diameter to found out the length of any chord in a circle, which is always lesser than the Diameter itself, and finding that the more parts whereinto the Diameter was divided, the nearer they approached to the truth: Some of them therefore, as Ptolomey, divided the Diameter of a circle into 120. parts, and the Semidiameter into 60. parts, and every such part into 60′· and every minute into 60. seconds etc. And in like manner did Arzahel, an ancient Arabian, who divided the Diameter into 300. parts and the Semidiameter into 150. and every of those parts into 60′· and so forth as before, according to which computation they made their tables: but because the working by those tables was very tedious and troublesome, by reason that it was needful continually to use the art of numbering by Astronomical Fractions: therefore Georgius Purbachius, and Regio montanus his Scholar to avoid that trouble of calculating by Astronomical Fractions, divided the Diameter of a Circle into a far greater number of parts, and made such tables as are used at this present, the description and use whereof doth hereafter follow, first of those that are set down by Monte Regio in Folio, and then of those that were lately Corrected and made perfect by Clavius the jesuite which are Printed in quarto.  
And because that the way to found out the proportion which any chord hath to the whole Diameter, was very hard, therefore the said Purbachius and Monte Regio having direction from certain propositions of Euclyd, as from the 47. proposition of his first book, and from the third proposition of his third book, and also from the 15. proposition of his fift book, they made choice of the half chord and Semidiameter of the Circle, calling the half chord, Sinum rectum, and the Semidiameter Sinum totum. And because that the proportion of any circumference to his Diameter never changeth, how great or how little so ever the Circle be: after that they had calculated for one Circle, they made such tables as might serve for all Circles, and though these tables of sins do suffice to work thereby all manner of conclusions, as well of Astronomy, as of Geometry, yet for more ease, our modern Geometricians have of late invented two other right lines belonging to a Circle called lines Tangent, and lines Secant, and have made like tables for them that were made for sins, and both tables, that is to say as well of the sins, as of the lines Tangent and Secant, have one self manner of working thereby, as shall plainly appear hereafter when we come to describe the same. But first we will begin with the tables of sins, and plainly define every term or vocable of Art, belonging thereunto: The terms are these here following: An arch, a Chord, Sinus rectus Sinus versus, Quadrants, Complementum, and sinus Complementi.  

The definitions of the foresaid terms.  
AN Arch is any part or portion of the circumference of a Circle, which in this practice doth not commonly extend beyond 180. degrees which is one half of the circumference of any Circle how great or small so ever it be, for every Circle containeth 360 degrees.  
A Chord is a right line drawn from one end of the Arch to the other end thereof, and note that all chords are always lesser than the Diameter itself, for that is the greatest chord in any Circle.  
Sinus rectus is the one half of a Chord or string of any Ark which is double to the Ark that is given or supposed, and falleth with right Angles upon that Semidiameter which divideth the double Ark into two equal parts.  
Sinus versus that is to say turned the contrary way, is a right line, and that part of the Semidiamiter, which is intercepted betwixt the beginning of the given Ark and the right Sine of the same Ark, and this is also called in Latin Sagitta, in English a Shaft or Arrow, for the Demonstrative figure thereof hereafter following, is not unlike to the string of a bow ready bend having a Shaft in the midst thereof.  
Quadrants is the fourth part of a Circle containing 90. degrees.  
Complementum arcus, is that portion of the Circle, which showeth how much the given Ark is lesser than the quadrant, if the given Ark do contain fewer degrees than the Quadrant, but if it contain more degrees than the Quadrant, than the difference betwixt such given arch, and the half Circle is the compliment of the said given Ark.  
Sinus complementi, is the right Sine of that Arch which is the compliment of the given Ark.  
Sinus totus, is the Semidiamiter of the Circle, & is the greatest Sine that may be in the Quadrant of a Circle, which according to the first tables of Monte Regio containeth 6/000/000. and according to the last tables 10/000 000/000 parts, for the more parts that the total Sine hath, the more true and exact shall your work be, notwithstanding sometime it shall suffice to attribute unto the total Sine but 60/000. parts, which number Appian observeth in teaching the way to found out the distance of two places differing both in Longitude and Latitude by the tables of Sines, and some do make the total Sine to contain 100/000 parts, as Wittikindus in his treatise of Dial's, and divers other do the like. Also Clavius himself saith that in the tables set down by him in quarto, you may sometime make the total Sine to be but 100/000. so as you cut off the two last figures on the right hand in every Sine, but you shall better understand every thing here above mentioned, by the figure Demonstrative here following.  
The figure Demonstrative.  
In this figure you see first a whole Circle drawn upon the Centre E. and marked with the letters A. B. C D. which Circle by two cross Diameters marked with the letters A C. and B. D. & passing both through the Centre E. is divided into four Quadrantes or quarters, the upper Quadrante whereof on the left hand is marked with the letters A. B. E. in which Quadrant, the right perpendicular line marked with the letters F. H. betokeneth the right Sine of the given Ark A. F. which right Sine is the one half of the chord or string F. G. and the given Ark A. F. is the one half of the double Ark or bow G. A. F. and A. H. is the Shaft called in Latin Sinus versus: Again the letters F. B. do show the compliment which together with the given Ark A. F. do make the whole Quadrant A. F. B which is divided into 9 spaces, every space containing 10. degrees, whereby you may plainly perceive that in this demonstration, the given Ark A. F. is 50. degrees, and the compliment F. B. is 40. degrees, both which being added together do make up the whole Quadrant of 90. degrees, marked with the letters A. F. B. Now Sinus complementi is the cross line marked with the letters F. K. the total Sine which is the whole Semidiameter and greatest right Sine, is marked with the letters B. E. But because it is not enough to know the signification of the things above specified to use the foresaid tables when need is, unless you know also how to found out those things in the said tables, I think it good therefore to show you the order of the said tables by describing the same as followeth.  
You have then to understand that the tables of Monte Regio printed in Folio, are contained in 18. Pages, and every Page containeth eleven partitions, called collums, whereof the first on the left hand containeth 60. minutes, which are to be counted from head to foot, as they stand in order one right under another in several places, proceeding from 1. to 60. The second collum containeth Sines. The third containeth only a portion or part of one second, and from thence forth proceeding towards the right hand all the other collums do contain in like manner Sines and the portion of one second. And right over the head of every Sine (the first collum of Sines only excepted, having nothing but a cipher over his head) are set down the degrees of the whole Quadrant called arches, in such order as from the first Page to the last, there are in all 89. degrees, or arches, as by perusing the said tables you may plainly see. Now to found out in these tables the things above mentioned, you must do as followeth.  
First to found out the right Sine of any given Ark, you must seek out the number of the said Ark in the front of the tables, and if the given Ark hath no minutes joined thereunto, than the first number of Sines right under the said Ark, is the right Sine thereof. But if it hath any minutes joined thereunto, than you must seek out in that Page, where you found the given Ark, the number of the minutes in the first collum of the said Page, on the left hand, and right against those minutes on the right hand, in the square Angle right under the said arch, you shall found the right Sine. As for example, you would found out the right Sine of a given Ark containing 8. degrees, and 20′· here having found out in the front of the second Page the figure of 8. standing right over the eight collum, seek in the first collum on the left hand of the said Page, for 20. minutes, and right against the 20. minutes you shall found on the right hand in the common Angle or square 8695 93. which is the right Sine of the foresaid given Ark, so as you make 6/000/000. to be the total Sine: but if you make 60/000. the total Sine, than you must always reject the two last figures standing on the right hand of the said right Sine, and the rest of the figures shall be the right Sine.  
Now to found out the compliment, there is nothing to be done, but only to subtract the given Ark out of the whole Quadrant which is 90. degrees, and the remainder shall be the compliment: as in the former example by subtracting 8. degrees, 20′· out of 90 degrees, you found that there remaineth 81. degrees, 40′·S which is the compliment of that arch. Again to found out the Sine of the compliment you must do thus, seek the compliment in the front of the tables of Sines, even as you do to found out any given ark: as in the former example, the compliment being 81. degrees 40′· you must seek 81. in the front of the 17. Page of the first tables, which being found, seek out also the 40′· in the first collum of the said Page on the left hand, and right against those 40′· in the common Angle right under the Ark 81. you shall found 5/936/649. which number is the right Sine of the foresaid compliment, so as you make 6/000/000. to be the total Sine, for if 60/000. be the total Sine, than you must reject (as I said before) the two last figures on the right hand, and the number remaining shall be the right Sine of the foresaid compliment, and therefore in working by these tables, you must always remember what number you make the total Sine to be.  
Sinus versus cometh seldom in use, notwithstanding if you would know how to found it out, you need to do no more but to subtract Sinum complementi of the given Ark, out of the total Sine, and the remainder shall be Sinus versus, as in the former example your Sinus complementi was 5/936/649. which being subtracted out of the total Sine 6/000/000. there remaineth this 63/351. and that number is Sinus versus: for if you add this remainder to the number which you subtracted, it will make up the total Sine 6/000/000. But there is one thing more necessary to be known then this, because it cometh oftener in use, and that is upon some division made how to found out the Ark of any quotient, which is to be done thus: Enter with the quotient into the body of the tables, and leave not seeking amongst the squares of the Sins, until you have found out the just number of the quotient (if it be there) if not, you must take the number of that Sine which is in value most nigh unto it, whether it be a little more or less, it maketh no matter, and having found that number, look in the front of that collum, and you shall found the Ark of your quotient, standing right over the head of that collum, and also the mynutes thereof in the first collum of the said Page on the left hand. As for example, having divided one number by another, I found the quotient to be 469/012. whereof I would know the arch, now in seeking this quotient amongst the Sins, I can not found that just number, but I found in the first Page, and in the tenth collum 469/015. which is the nighest number unto it that I can see. In the front of which collum I found the Ark to be 4. degrees, and directly against that Sine on the left hand, I found 29′· belonging to that arch, whereof that quotient is the Sinus, so as I gather hereof that the arch of the foresaid quotient is 4, degrees, 29′· But you have to note by the way that the number of your quotient must never be much less than 1745. for otherwise it is not to be found in these tables, unless you make the total Sine to be but 60/000. for then by rejecting the last two figures on the right hand, as I have said before, the first right Sine of these tables shall be no more but 17. and by that account a very small quotient may be found in these tables. And whatsoever hath been said here touching the order that is to be observed in the first tables of Monte Regio, whose total Sine is 6/000/000. the like in all points is to be observed in the last tables, whose total Sine is 10/000/000. Thus much touching the order of the foresaid tables of Monte Regio Printed in Folio: but for so much as those tables be not altogether truly Printed, and for that they have been lately corrected, and made more perfect by Clavius, who doth set down the said tables in quarto and not in folio, whereby they are the more portable, and the more commodious, as well for that they are more truly Printed, as also for that the compliment of every Ark is set down in every Page at the foot of every collum, so as you need to spend no time in subtracting the Ark from 90. I think it good therefore to make a brief description of those tables, and the rather for that I have requested the Printer to print the like here in quarto, and I do work all such conclusions as hereafter follow, by the said tables, the total Sine whereof is 10/000/000. according to the last tables of Monte Regio. But for so much as some may have already the tables of Monte Regio Printed in Folio, not knowing perhaps the use thereof, I will set down two conclusions to be wrought by those tables, and all the rest of the conclusions are to be wrought by those tables which I have here caused to be Printed in quarto like to those of Clavius: and though the two conclusions next following, which are to show the use of the foresaid tables, may be wrought by the tables of Sins in what form so ever they be truly Printed if Folio, or in quarto, yet because I had appointed them to be done by the tables of Monte Regio, Printed in folio before that ever I saw Clavius his book, I mind not now to altar them but to let them stand still as they are.   

How to found out by the said tables, the distance betwixt two places differing both in Longitude and Latitude, making the total Sine to be no more but 60/000.  
THis is done by finding out two numbers, whereof the one is called in Latin Primum inventum: that is to say, the first found number, and the other is called Secundum inventum, that is the second found number in such order as followeth.  
First then knowing the Longitude of either place, take the difference of their Longitudes by subtracting the lesser Longitude out of the greater, that done, multiply the right Sine of that difference into the Sine of the compliment of the lesser Latitude, and divide the product of that multiplication by the total Sine, and then seek out the arch of that quotient according to the rule before taught, so shall you have the first found number: That done, multiply the right Sine of the lesser Latitude by the total Sine, and having divided the product thereof by the right Sine of the compliment of the first found number, subtract the arch of that quotient out of the greater Latitude, and you shall have the second found number. Than multiply the right Sine of the compliment of the first found number into the right Sine of the compliment of the second found number, and having divided the product of that multiplication by the total Sine, seek the Ark of that quotient in the tables, and take that Ark out of the whole Quadrant, and the degrees that do remain, are degrees of the great Circle, which if you multiply by 60. the product of that multiplication will show you how many Italian miles the one place is distant from the other, or if you would have German miles, them multiply the foresaid degrees of the great Circle by 15. or else divide the product of the Italian miles by 4. and you shall have your desire. As for example, you would know what distance is betwixt Jerusalem and Noremberg a famous town in Germany, Jerusalem according to Appian his tables, hath in Longitude 66. degrees 0′· and in Latitude 31. degrees, 40′· Again Noremberg hath in Longitude 28. degrees, 20′·S and in Latitude 49. degrees, 24′· the difference of their Longitudes is 37. degrees, 40′· the right Sine whereof is 36/664. for in this example Appian maketh 60/000. to be the total Sine, and therefore he rejecteth the two last figures en the right hand found in the first tables of Monte Regio. Now you must multiply 36/664. into the right Sine of the compliment of the lesser Latitude which Sine is 51/067. the product of which two Sins being multiplied the one by the other, amounteth to 1/872 872/320/488. which if you divide by the total Sine. 60/000. you shall found in the quotient 31/205. whose arch is 31. degrees, 20′·S and this shall be your first found number. This done, multiply the right Sine of the lesser Latitude which is 31/498. by the total Sine 60/000. and the product thereof will be 1/889 889/880/000. which sum if you divide by the Sine of the compliment of the first found number which Sine is 51/249. you shall found in the quotient 36876. the Ark whereof is 37. degrees 55′· which arch being subtracted out of the greater Latitude, there will remain 11. degrees, 29′·S and that shall be your second found number, then multiply the foresaid Sine of the compliment of the first found number which is 51/249. by the Sine of the compliment of the second found number which is 58/798. and the product thereof will amount to 3/013 013/338/702. which if you divide by the total Sine, you shall found in the quotient 50/222. the arch whereof is 56. degrees, 50′·S which being subtracted out of the whole Quadrant which is 90. degrees, there will remain 33. degrees, 10′· of the greater Circle, which 33. degrees, if you multiply by 60. it will make 1980. miles, whereunto you must add for the 10′· 10. miles, so shall you found the distance betwixt the two foresaid places to be 1990. Italian miles, which if you would reduce into German miles, then divide that number by 4. for 4. Italian miles do make but one German mile, so shall you have 497. German miles, and two Italian miles remaining, which is half a German mile, which sum agreeth with that which Appian setteth down in his Geography, whereas he useth the self same example, and worketh it in like manner Per tabulas sinuum.   

The Altitude of the sun being known how to found out the Longitude of the shadow both right and verse of any body yielding shadow by help of the foresaid tables.  
FIrst you have to understand that every bodily thing yielding shadow, is divided into 12. equal parts, and every such part into 60. minutes, and every minute into 60. seconds, and so forth: Again, of shadows there be 2. kinds, that is, Vmbra recta, and Vmbra versa, Vmbra recta is that which proceedeth from some body rightly erected upon the upper face of the Horizon, as from some tower or post standing right up upon a level ground: And that shadow is called umbra versa which proceedeth from some right style or perch being thrust into a wall or post standing right up and not leaning, in such sort as the said style or perch may be a just parallel to the upper face of the Horizon. Now to found out the length of either the foresaid shadows, you must do thus.  
Multiply the right Sine of the compliment of the given Solar altitude, by 12. and divide the product by the right Sine of the said Solar altitude, and you shall have the Longitude of the right shadow of the said body. Again if you multiply the right Sine of the foresaid altitude by 12. and divide the product by the Sine of the compliment of the said altitude, you shall have the Longitude of Vmbra versa, of the said body. As for example, suppose the given Solar altitude to be 25. degrees, the compliment whereof is 65. and the right Sine of that compliment is 54/378. if you make the total Sine to be 60/000. Than in multiplying the foresaid right Sine by 12. the product will be 652/536. which if you divide by the Sine of the altitude which is 25/357. you shall found the Longitude of Vmbra recta to be 25. parts 44′ 42″ and 6‴· Now if you multiply the Sine of the altitude which is 25/357. by 12. and divide the product by the Sine of the compliment which is 54/378. you shall found the Longitude of Vmbra versa to be 5. parts, 35′·S and in saying here parts, I mean always such parts as are the 12. parts, whereinto the body yielding shadow is divided: but if you work this example by the first tables of Sines making the total Sine 6/000/000. though you find it true in the parts and minutes, yet shall you not found it so in the seconds and thirds, and if you work the same by the second tables making the total Sine 10/000/000. you shall found it to agree only in the parts, but neither in minutes nor seconds, which maketh me to suspect that the Printer hath committed some error therein, for both the tables were made by one self rule.   

A brief description of the tables of Sines Printed in quarto like unto those which Clavius setteth down in his commentaries upon Theodosius his Spheriques.  
Having here before plainly described unto you the tables of Sines made by Monte Regio, which are Printed in folio, and how to use the same I will now briefly describe the said tables lately corrected by Clavius, and were Printed in quarto at Rome Anno 1586. the total Sine of which tables according to the last table of Monte Regio, is 10/000/000. by which tables are to be wrought all the conclusions hereafter following.  
First than you have to understand that these tables are contained in 36. Pages, in the front whereof are set down the degrees of the Quadrant proceeding from 1. to 89. but because the whole number of minutes belonging to the said degrees, which is 60′· can not be all placed in one self Page, but only 30′· in the left outermost collum of the left hand Page: and other 30′· in the left outermost collum of the right hand Page: therefore the degrees or arches of the Quadrant are feign to be twice repeated in the front of every two Pages, as you may plainly see by viewing the said tables, and every Page containeth seven collums, whereof the first on the left hand containeth the minutes belonging to the degrees or arches of the Quadrant, which minutes do proceed downward from 1. to 30′·S and the seventh collum on the right hand in every Page containeth the minutes belonging to the compliment of every arch, which minutes do proceed backward, that is to say from 1. set down at the lowest end of the last collum of the second Page, and so proceeding upward to 30′· which 30′· is also set down at the lowest end of the last collum of the left hand Page, and so proceedeth upward to 60′· which 60. minutes do help to make up the compliment that is answerable to every arch whereunto no minutes be annexed, for if the arch hath no minutes, than you must add to the compliment thereof 60. minutes, which is one whole degree, to make up the compliment. As for example, suppose the Ark to be 46. degrees, without any minutes joined thereunto, the compliment whereof set down at the foot of the said arch, is but 43. degrees, wherefore you must add thereunto 60′· which is one degree, so shall the compliment be 44. degrees, which is the true compliment indeed, but if you suppose the foresaid Ark to have 13′· joined thereunto you shall found the compliment to be 43. degrees, 47′·S which is answerable to the foresaid Ark 46. degrees, and 13′· for if you take 46. degrees, 13′· out of 90. degrees, the remainder will be 43. degrees, 47′·S which is the compliment, so as you need not to make any Subtraction out of 90. to found the compliment of any arch, that hath any minutes annexed thereunto: but when so ever you have to found out the right Sine of any compliment in these tables, you must then make the compliment an arch, seeking for the same in the front and not at the foot of the tables, & if the said compliment have any minutes annexed thereunto, you must seek those minutes in the left outermost collum of every Page, and not in the outermost right collum belonging to compliments, for in this case the compliment is an arch and not a compliment. The order of working by these tables in all other things differeth not one jot from that which we have observed in working the two former conclusions by the tables of Monte Regio Printed in folio, as you shall easily perceive by the examples here following.   

1 How to found out the declination of the Sun at any time his place in the zodiac being given, Per tabulas Sinuum.  
KNowing the place of the Sun for the day, consider how much the said place is distant from the first point of Aries if the place of the Sun be nigher to Aries then to Libra. But if it be nigher to Libra, then take his distance from the first point of Libra, which distance must not exceed 90. degrees, and seek that distance amongst the arks in the front of the tables if they be degrees, if minutes, you shall found them in the first collum on the left hand, then multiply the Sine of that distance by the Sine of the greatest declination which is 23. degrees, 28′·S and divide the product thereof by the total Sine, and the quotient will show you the Sine of the declination for that day, the arch whereof, is the very number of the declination itself. As for example, you would know the declination of the Sun the eleventh of April 1591. when as the Sun was entered 33′· into Taurus unto which you must add according to the rule before given 30. degrees, for that is his distance from the first point of Aries, then seek out the said 30. degrees, in the front of the last tables among the arks and the 33′ in the first collum on the left hand right against which you shall found the Sine of the said distance to be 5082/901. which being multiplied by 3/982/155. (which is the Sine of the greatest declination) the product thereof will be 20/240 240/890 890/631/655. which if you divide by the total Sine which is 10/000/000. you shall found the quotient to be 2/024/089. which quotient must be sought out in the said tables, and if you found no such number, then take the nearest number thereunto, which is 2025025. and the Ark thereof together with the minutes that stand right against the said quotient in the first collum on the left hand, is the declination of the Sun for that day which is 11. degrees, 41′·S   

2 How to know the right ascension of the Sun, Per tabulas Sinuum.  
FIrst knowing the suns place, you shall learn the right ascension thereof thus. First consider how far his place is from the Equinoctial point, as is said before in the last proposition, and multiply the Sine of the compliment of that distance by the total Sine, then knowing the declination of the Sun for that day by the last rule, divide the former product by the Sine of the compliment of the said declination, and the quotient will show the Sine of that Ark whereof the compliment is the right ascension. As for example, the Sun being in the 33′· of Taurus and his declination 11. degrees, 41′· as you found by the last proposition and his distance from the first point of Aries to be in all 30. degrees, 33′ the compliment whereof is 59 degrees, 27′· the Sine of which compliment is 8/611/860. which being multiplied by the total Sine which is 10/000/000. the product will be 86/118 118/600 600/000/000. which being divided by the Sine of the compliment of the suns declination which is 78. degrees, 19′·S whose Sine is 9792818. by which Sine if you divide the former product, the quotient will be 8/773/600. the Ark of which quotient is 61. degrees, and 19′·S and the compliment of that, is 28. degrees, and 41′· which is the right ascension of the place of the Sun for the foresaid day.  
And here note that if the Ark from Aries to the given point, do contain just 90. degrees, the right ascension thereof is also 90. degrees, but if the said Ark be more than 90. degrees, and lesser than 180. degrees, then subtract the same out of 180. and seek out the right ascension of the remainder as before, whose right ascension if you take out of 180. the remainder shall be the ascension of the propounded Ark. But if the said arch which is comprehended betwixt Aries and the given point be greater than 180. degrees, or 6. signs, and lesser than 270. degrees, or 9 signs, then having subtracted 180. degrees from the same arch, calculate the right ascension of the arch from the beginning of Libra, as before, and the right ascension thereof being added to 180. degrees, shall be the ascension desired.  
Lastly if the arch comprehended betwixt Aries and the point given, be greater than 270. degrees, or 9 signs, then subtract the same out of 360. degrees, and seek out the right ascension of that remainder as before, which if you subtract out of 360. degrees, the remainder shall be the right ascension desired.   

3 How to find out the ascentionall difference, Per tabulas Sinuum.  
MVltiply the Sine of the Latitude given by the total Sine, and divide the product by the compliment of the said Latitude, that done, multiply the quotient by the Sine of the suns declination, and divide the product by the Sine of the compliment of the declination, and the quotient thereof will show the sign of the ascentionall difference: and by working according to this rule, you shall found that when the Sun is entered 33′·S into Taurus, at which time his declination is 11. degrees, 41′·S (as hath been said before) the ascension all difference will be 15. degrees, 21′·S And here note that the ascentional differences for one quarter of the Circle serveth also for all the rest, so that the Latitude be not altered, and that the declination of the points in the later quarters be equal to the declination of the points in the first quarter.   

4 How to found out the obliqne ascension of any point of the Ecliptic in any Latitude assigned.  
FIrst found the right ascension of the given point by the second proposition, & also the ascentional difference thereof by the third proposition: then consider whether the declination of the said point be Northward or Southward, for if it be Northward, then subtract the ascentionall difference out of the right ascension, and the remainder shall be the obliqne ascension desired: but if the declination be Southward, then add the ascentionall difference unto the right ascension, and the sum shall be the obliqne ascension. In working thus for the Latitude 52. and supposing the Sun to be in the 33′· of Taurus, & having found by the former proposition his right ascension to be 28. degrees, 41′·S and the ascentionall difference to be 15. degrees, 21′·S you shall found his obliqne ascension in the foresaid Latitude to be 13. degrees, 20′·S   

5 How to find out the time of the suns rising and setand thereby the length of the artificial day.  
FIrst you must know the ascentionall difference, and convert the same into hours & minutes, then if the Sun be in any of the Northern signs, add those hours to 6. hours which is the one half of an Equinoctial day, and the sum of such Addition shall be the one half of the artificial day, which being subtracted out of 12. hours, the remainder shall be the hour of the suns rising: As for example, the Sun being in the 33′· of Taurus, you found the ascension all difference to be 15. degrees, and 21′· which being turned into hours maketh one hour 1′· 2″· and somewhat more, which being added to 6. maketh 7. hours 1′· 2″· and a little more which is the one half of the artificial day which being doubled, maketh in all 14. hours 2′· and 4″· but if the Sun be in any of the Southern signs, you must remember to subtract the hours of the ascentionall difference out of 6. hours, and the remainder shall be the one half of the artificial day, and by subtracting the half length of the artificial day out of 12. you shall know the hour of the suns rising, and having the time of his rising, you must needs know the time of his setting.   

6 How to found out the Meridian altitude of the Sun in any day though he doth not shine at all, the elevation of the Pole being given.  
SVbtract the elevation of the pole from 90. and the remainder shall be the elevation of the Equinoctial, then if the Sun be in any of the Northern signs, add the declination of the Sun, unto the altitude of the Equinoctial, or else if he be in any of the Southern signs, subtract the declination, and the sum of the Addition, or remainder of the Subtraction, shall show the Meridian altitude. As for example, the second of May 1591. the Sun being in the 20. degrees, 42′· of Taurus, and his declination Northward 17. degrees, 56′· 21″· here by subtracting 52. degrees, which is our Latitude from 90. you shall found the remainder to be 38. which is the altitude of the Equinoctial, whereunto if you add the suns declination for that day which is 17. degrees, 56′·S and 21″· the sum will be 55. degrees, 56′·S and 21″· and that is the Meridian altitude of the Sun for that day.  
Let these few conclusions serve to show you the use of the tables of Sines, for it would make a long book to set down so many conclusions as are to be wrought by these tables, and therefore I leave to trouble you any further therewith, minding now briefly to declare unto you the use of the tables of lines Tangent & Secant by one example or two as followeth. But first I think it necessary to show you what those lines be, and whereto they serve.    

THE DESCRIPTION and use of the Tables of Tangents and Secants.  
EVclid in the second proposition of his third book defineth the line Tangent in this sort. A right line (saith he) is said to touch a Circle when it toucheth it so, as being drawn out in length, it would never cut the said Circle.  
The line Secant is not by him any where defined, but what these two lines are, you shall better understand by this figure Demonstrative here following, then by any definition that can be made thereof: for a definition aught to be plain and brief, and not long, intricate or doubtful, which will be hardly performed in showing the nature of these two lines by way of definition, and therefore mark well this figure following.  
In this figure you see first a Circle drawn upon the Centre C. from which Centre is extended to the circumference of the Circle a right line, called the Semidiameter, marked with the letters A. C. then there is another right line which toucheth the said Circle, and also the outermost end of the said Semidiameter making therewith a right Angle in the point A. and is called the line Tangent, then there is a third line which proceeding from the Centre C. doth cut the circumference of the Circle in the point B. and also meeteth with the line Tangent in the point D. and therefore is called the line Secant, betwixt which two lines, I mean the Tangent and Secant, is intercepted or included a certain portion or arch of the foresaid Circle less than a Quadrant marked with the letters A. B. of which Ark the line A. D. is the Tangent, and the line C. D. is the Secant thereof, which must needs meet with the Tangent in the point D. because that the two Angles C. A. D. and D. C. A. are lesser than two right Angles, for the one is right, and the other sharp, by reason that the Ark is less than a Quadrant. And some do call the line Tangent the line Ascript, because it is ascribed to the Circle, and they call the line Secant the Hipothenuse, because it subtendeth the right Angle A. & they call the Semidiameter or total Sine, the base of the rectangle Triangle C. A. D which is called a rectangle Triangle because it containeth one right Angle marked with the letter A. and note that whensoever any manner of Angle is propounded by three letters, that the middle letter doth always signify the Angle propounded, be it right, sharp, or blunt. Now if you would know to what end the foresaid two lines were invented, & whereto they serve, you have to understand that they chief serve in calculating the quantity of Angles and their sides, as well in right lined Triangles, as in spherical Triangles, for the sides of Triangles are either right or crooked, and if they have right sides, such Triangles are either right angled Triangles, or obliqne angled Triangles, and you have to note that the quantity of every Angle is to be measured by the arch of a Circle subtending that Angle, for the point of every Angle is imagined to be the Centre of a whole Circle, which you may suppose to be so great or little as you will, for every Circle (be it great or little) is divided into 360. degrees, and look how many degrees and minutes the arch subtending that Angle containeth, so much is the quantity of that Angle, the practice whereof is very well set down by Clavius in his commentaries upon Theodosius, which I mind (God willing) hereafter to translate into our mother tongue: In the mean time my intention here, is only to show you by one example or two, the use of the tables made for the foresaid lines Tangent and Secant.  

The use of the said tables according to Clavius, is thus.  
IN seeking out the Tangent or Secant of any Ark given, or of the compliment of any Ark by either of these tables, you have to observe the self same order which you did before in finding out the right Sine of any Ark given, or of the compliment of any Ark, Per tabulas sinuum. As for example, if you would found out the Tangent of an Ark containing 50. degrees, 24′· then resort to the table of Tangents in the front, whereof look first for the Ark 50. degrees, and then in the first collum on the left hand of the said table, for 24′· right against which on the right hand under the Ark 50. you shall found in the common Angle the Tangent to be 12/087/923. the total Sine whereof is 10/000/000. but if you would found out the Secant of the foresaid Ark 50. and 24′·S than you must resort to the table of Secants, and having found out the Ark 50. in the front of the said tables, and the 24′· on the left hand as before, you shall found in the common Angle the Secant to be 15688144. And if you would have the Tangent of the compliment of the said Ark which is 39 degrees, and 36′· you shall found the 39 degrees of the compliment in the foot of the table of Tangents right under the Ark 50. & the 36′·. in the outermost collum on the right hand of the said table, with which compliment you must enter the table of Tangents, seeking for 39 degrees in the front of the table, and 36′· in the first collum on the left hand of the said table, right against which in the common Angle you shall found 8272720. to be the Tangent of 39 degrees, 36′·S which is the compliment of 50 degrees, 24′·S  
And you must work in like manner with the table of Secants: As for example, if you would found the Secant of 72. degrees, 36′· first then enter the table of Secants, looking for 72. degrees above in the front of the table, and 36′· in the first collum on the left hand of the said table, and in the common Angle you shall found 33/440/240. which is the Secant of 72. degrees, 36′· But if you would have the Secant of the compliment of the said arch 72. degrees, 36′· then looking in the foot of the table right under 72. degrees, you shall found 17. degrees, and in the outermost collum on the right hand, just against 36′· you shall found 24′· so as you see that 17. degrees 24′· is the compliment of 72. degrees, 36′· with which compliment you must enter the table of Secants, looking for 17. degrees, above in the front of the table, and for 24′· in the first collum on the left hand of the said tables, & in the common Angle you shall found 10/479/542. to be the Secant of the arch 17. degrees, 24′·S which is the compliment of the arch 72. degrees, 36′·   

The use of which tables in Astronomical matters, I have here set down as followeth.  

1 To found out the declination of the Sun, the place thereof being known.  
MVltiply the Secant of the compliment of the greatest declination by the total sine, and divide the product by the sine of the suns distance from one of the equinoctial points, the quotient is the Secant of an arch, whose compliment is the declination of the sun: for example, suppose that the sun be entering into ♉ to find the declination thereof, first I multiply 25112030 the Secant of 66. degrees 32′· (which is the compliment of 23. degrees 38′ the greatest declination) by 10000000. the product is 251120300000000. which being divided by 5000000. (the sine of 30. degrees the suns distance from the equinoctial point) the quotient is 50224060. for which number I seek in the table of Secants, the arch answering unto it, is 78. degrees 32′· the compliment whereof is 11. degrees 29′· which is the declination of the sun.   

2 Knowing the declination of the sun how to find his distance from the equinoctial point, and so consequently his place in the Zodiac.  
MVltiply the Secant of the compliment of the greatest declination by the total sine, and divide the product by the Secant of the compliment of the declination given, the quotient is the distance of the sun from the equinoctial point. As for example, the declination of the sun is supposed to be 11. degrees 29′· then to find his distance from the equinoctial, I multiply 25112030. the Secant of 66. degrees 32′· (which is the compliment of the greatest declination) by the total sine the product is 251120300000000. which I divide by 50224350. the Secant of 78. degrees 31′ the compliment of 11. degrees 29′· the supposed declination, the quotient is 5000000. the sine whereof is 30. dedegrées 0′ which is the distance of the sun from the equinoctial: them for his place you must take the same according to the season of the year: for if it be in April, than the sun is entering into Taurus, but if it be in August it is entering into Virgo, and being in October, it is entering into Scorpio, and being in February it is in the beginning of Pisces.   

3 To find out the right ascension of the sun.  
MVltiply the Tangent of the distance of the sun from the equinoctial point which is nearest unto it, by the sine of the compliment of the greatest declination, and divide the product by the total sine, the quotient is the Tangent of the right ascension of the sun, for which if you seek in the table of Tangents, the arch answering unto it is your desire: For example the sun being in the first of Taurus, to know the right ascension thereof, I multiply 5773502. the Tangent of 30. degrees (for that 30. degrees is the distance of the sun from the equinoctial point) by 9172920. the sine of 66. degrees 32′· which is the compliment of 23 degrees 28′· the greatest declination, the product is 52959871965840. which being divided by 10000000. the total sine, the quotient is 5295987. which is the Tangent of 27. degrees, 54′·S which is the right ascension of the sun being entered into Taurus.   

4 How to find out the declination of the sun knowing only the right ascension thereof.  
MVltiply the Tangent of the compliment of the greatest declination by the total sine, and divide the product by the sine of the right ascension given, the quotient is the Tangent of the compliment of the suns declination: for example the right ascension of the sun being 27. degrees 54′· I would know the declination thereof, multiplying 23035062. the Tangent of 66. degrees 32′· the compliment of 23. degrees 28′ by 10000000. the product is 230350620000000. which being divided by 4679298. the sine of the given ascension the quotient is 49224856. the arch of which Tangent is 78. degrees 32′· which being subducted out of 90. the remainder is 11. degrees 28′· and so much is the declination of the sun.   

5 How to find the place of the sun knowing only the right ascension thereof.  
SVbduct the right ascension given out of 90. if it be less than 90. but if the same be more than 90. subtract the ascension given out of 180. & being greater than 180. subduct the same out of 270. or being greater than 270. subduct the same from 360. and multiply the Tangent of the remainder by the sine of the compliment of the greatest declination, and divide the product by the total sine, the quotient is the Tangent of the compliment of the sun's distance from one of the Equinoctial points: which distance being known, the place of the sun can not be unknown. For example, supposing the right ascension of the sun to be 27. degrees 54′· the compliment thereof is 62. degrees 6′· the Tangent whereof is 18886715. which being multiplied by 9172920. the product is 173245985757800. which being divided by 10000000. the quotient is 17324598. for which I look in the table of Tangents, and I find the arch thereof to be 60. the compliment whereof is 30. which is the distance of the sun from the equinoctial point, that is, from Aries, for that the right ascension is less than 90. I say then that the sun is in the beginning of Taurus. But if the right ascension had been more than 90. and less than 180. the place of the sun had been betwixt Cancer and Libra 30. degrees from Libra, so should it have been in the first point of Virgo, but if the right ascension had been more than 180. the place of the sun should be betwixt Libra and Capricorn, that is in the beginning of Scorpio, but being more than 270. the place of the sun should be betwixt Capricorn and Aries, that in the beginning of Pisces.   

6 To found out the ascentionall difference of the sun or any star in the firmament, knowing the declination thereof, and also the latitude of your region.  
MVltiply the Tangent of the declination of the Sun or star by the Tangent of the latitude of the place, and divide the product by the total sine: the sine of the quotient is the sine of the ascentionall difference, the arch whereof is the desired ascentionall difference. For example let the declination of the sun, star, or other point in the firmament be 10. degrees 3′· and suppose the latitude to be 52. degrees, the Tangent of 10. degrees 3′· is 1772268. the tangent of 52. degrees is 12799416. which being multiplied together, the product will be 22674995395488. which being divided by 10000000. the total sine, the quotient is 2267499. for which number I look amongst the sins, the arch answering thereunto is 13. degrees 6′· which is the ascentionall difference desired.   

7 To found out the obliqne ascension of the sun.  
KNowing the place of the sun, find out the right ascension of the same by the third proposition, and find also the ascentionall difference of the same point, then if the declination of the sun be North, subduct the ascentionall difference out of the right ascension, the remainder is the obliqne ascension. For example, suppose the sun to be in the beginning of Taurus, now to find the obliqne ascension thereof in the latitude 52. degrees, first by the third proposition I find out the right ascension of the beginning of Taurus, which I find to be 27. degrees 54′· then by the sixth proposition I find the ascentionall difference to be 15. degrees 4′· which being subducted from 27. degrees 54′· (for that the declination is North) the right ascension the remainder is 12. degrees 50′· and so much is the obliqne ascension for the latitude of 52. degrees. But if the sun be in any of the Southern sins, and that the declination be South, than the ascentionall difference is to be added unto the right ascension before given.   

8 To find out the obliqne descension of the Sun at any time.  
FIrst find out the right ascension of the sun by the third proposition, the same shall be the right descension thereof, then find the ascentionall difference by the sixth proposition, and if the sun be in any of the Northern signs, add the ascentional difference unto the right ascension, the sum shall be the obliqne descension of the sun: But if it be in any of the Southern signs subduct the ascentionall difference out of the right ascension, the remainder is the desired descension: as for example the sun being in the beginning of Taurus, the right ascension thereof by the third proposition is 27. degrees 54′· the ascentional difference thereof by the sixth proposition is 15. degrees 4′· for as much then as Taurus is a Northern sign, I add the ascentionall difference unto the right ascension, the sum is 42. degrees 58′· and so much is the obliqne descension of the sun being in the beginning of Taurus.   

9 To find out the length of the day or night.  
Having found out the ascentionall difference by the sixth proposition, add the same unto 90. if the sun be in any of the Northern signs, but if it be in any of the Southern signs, subduct the ascentionall difference out of 90. then divide the sum of the addition or the remainder of the subtraction by 15. the quotient will show the half length of the day in hours and minutes, which being doubled you shall have the whole length of the artificial day: For example the sun being in the beginning of Taurus, the ascentionall difference thereof is 15. degrees 4′· which for that the sun is in a Northern sign, I add unto 90. degrees, the sum is 105. degrees 4′ which being divided by 15. the quotient is 7. hours o′· the half length of the day which being doubled will be 14. hours the whole length of the artificial day in the latitude of 52. degrees.   

10 To find the hour of the Sun his rising or setting in any latitude assigned.  
FIrst found out the half length of the artificial day by the 9 proposition, and subtract the same from 12. hours, the remainder will show the hour of the suns rising, for example the sun being in the beginning of Taurus to know the hour of his rising in the latitude of 52. degrees, by the ninth proposition, the half length of the artificial day I found to be 7. hours, which I subtract from 12. hours the remainder is 5. which showeth that the sun riseth at 5. of the clock in the morning, but the half length of the day itself is the hour of the suns setting.   

11 To found out the length of the planetary hours and to found what Planet reigneth at any hour of the day.  
FIrst found out the length of the artificial day by the ninth proposition, and divide the same by 12. the quotient is the length of one planetary hour: or thus, having an hour of the artificial day given, look what hour the same is from the sun rising, and multiply the same by 12. divide the product by the length of the artificial day, the quotient is the number of the Planetarye hour. For example the sun being in the beginning of Taurus, and our latitude being 52. degrees, the length of the artificial day by the ninth proposition is 14. hours: then do I divide 14. by 12. the quotient is 1 ⅙. that is 1. hour 10′· the length of one planetary hour. But if an hour of the artificial day be given as that I would know what planetary hour it is at 9 of the clock the sun being in the first of Taurus in the latitude of 52. degrees, having found that the sun riseth at five of the clock by the tenth proposition, I see that 4. hours of the artificial day are gone at nine of the clock, I therefore multiply 12. by 4. the product is 48. which I divide by 14. the length of the artificial day, the quotient is 3 3/7. which is the planetary hour at the time set down: likewise shall you find the planetary hour of the night, finding the length thereof, and then work with it as was showed before for the day.  
Then to know the Planet which reigneth at the appointed time, you must consider the Planet whereof the day taketh his name, for that Planet ruleth the first hour of that day, the next Planet the second hour, and so forth. But for your ease I have set down a table whereby to find the Planet which reigneth at any time.  
Hours 1 2 3 4 5 6 7 of the day. 8 9 10 11 12   Hours 10 11 12   1 2 of the night. 3 4 5 6 7 8 9 Sunday. ☉ ♀ ☿ ☽ ♄ ♃ ♂ Monday. ☽ ♄ ♃ ♂ ☉ ♀ ☿ Tuesday. ♂ ☉ ♀ ♂ ☽ ♄ ♃ Wednesday ☿ ☽ ♄ ♃ ♂ ☉ ♀ Thursday. ♃ ♂ ☉ ♀ ♂ ☽ ♄ Friday. ♀ ♂ ☽ ♄ ♃ ♂ ☉ Saturday. ♄ ♃ ♂ ☉ ♀ ☿ ☽  
The use whereof is this, having found the number of the Planetary hour, look for the same in the head of the Table, whether it be in the day or night, and right under it just against the name of the day, you shall have the Planet which raineth at that time. For example the sun being in the beginning of Taurus the 11. of April, being Thursday at nine of the clock in the morning, I found the number of the Planetary hour at that time to be 3 3/7. them looking for three amongst the hours of the day, I descend in that collum until I come to be just against thursday, where I see the Character of Sol to be, I conclude then that the sun reigneth at that time.   

12 To find the arch of the Equinoctial, comprehended betwixt the Meridian and any Circle of position according unto Campanus and Gazula.  
MVltiply the Sine of the compliment of your latitude by the Tangent of the distance of the given Circle of position from the Zenith, & divide the product by the total Sine, the quotient is the tangent of the arch of the Equinoctial, which is comprehended betwixt the given Circle of position, and the Meridian. For example in the latitude of 52. I would know what part of the Equinoctial is comprehended betwixt the Meridian and that Circle of position, which is 30. degrees from the Zenith: the latitude being 52. degrees, the compliment thereof is 38. degrees, the Sine whereof 6156615. and the tangent of 30. degrees, (for that 30. is the distance betwixt the given Circle of position and the Zenith) is 5773502. which being multiplied by 6156615. the product is 35545229015730. which being divided by 10000000. the total Sine, the quotient is 3554522. for which I seek amongst the tangents, and I find the Ark answering thereunto, to be 19 degrees 34′ the Arch of the Equinoctial, betwixt the Meridian and the given Circle of position.   

13 Knowing the Latitude of your Region, and also the elevation of the Pole above any Circle of position, how to find the inclination of the said Circle of position unto the Meridian, and so consequently the Arch of the Equinoctial, which is betwixt the said Circle of position & the Meridian.  
MVltiply the Secant of the compliment of the elevation of the Pole above the Circle of position by the Sine of your Latitude, and divide the product by the total Sine, the quotient is the Secant of the compliment of the inclination of the Circle of position unto the Meridian, and that is the distance betwixt the Circle of Position and the Zenith, by help whereof you shall found the Arch of the Equinoctial, betwixt the Circle of Position and the Meridian, as in the former proposition. As for example, suppose the elevation of the Pole above a Circle of position, in our Latitude of 52. to be 23. degrees 12′· Now to found out the Inclination of that Circle of Position unto the Meridian, first I multiply 25384445. the Secant of 66. degrees 48′· (for that is the compliment of 23. degrees 12′· the elevation of the Pole above the Circle of position) by 7880108. the Sine of 52. the Latitude of our Region, the product is 200032168120060. which being divided by 10000000. the quotient is 20003216. for which I look in the Table of Secants, and the Arch thereof is 60. degrees 0′· the compliment whereof is 30. degrees 0′· which is the Inclination of the Circle of position unto the Meridian, or the distance of the Zenith from the said Circle, then to found the Arch of the Equinoctial betwixt the said Circle of Position and the Miridian, repeat the work of the former proposition.   

14 To find out the Elevation of the Pole above any assigned Circle of Position in any given Latitude.  
KNowing the Inclination of the assigned Circle of Position unto the Meridian, multiply the Secant of the compliment thereof by the total Sine, and divide the product by the Sine of your Latitude, the quotient is the Secant of the compliment of the Elevation of the Pole above the given Circle of Position: as for example, suppose the inclination of a Circle of position to be 30. degrees 0′· Now to find the elevation of the Pole above the same for the Latitude of 52. degrees 0′· First take the compliment of 30. degrees 0′· which is 60. degrees 0′· the Secant whereof is 20000000. which being multiplied by the total Sine, the product is 200000000000000. which being divided by 7880108. the Sine of 52. the assigned Latitude the quotient is 25380362. for which I seek in the Table of Secants, and the Arch answering thereunto I find to be 66. degrees 48′· the compliment whereof is, 23. degrees 12′· and so much is the elevation of the Pole above the assigned Circle of Position in your Latitude.  
And thus much shall be sufficient to have been spoken for the use of the lines Tangent and Secant at this present, of whose ample and infinite use, you shall have further taste in our Directory tables and Horologies, whereof a beginning is made, and shall be ended so soon as may be possible: in the mean time I shall desire the Reader favourably to accept of these, until better leisure and more fit opportunity will be offered. And now follow the Tables themselves.     

HERE FOLLOWETH THE TABLE OF RIGHT SINS FOR EVERY MINUTE OF THE QVADRANT, FIRST CALCULATED BY I REGIO MONTANUS, BUT NOW EXAMINED AND IN MANY PLACES CORRECTED AND AMENDED BY CLAVIUS.  
The Table of Sines.  
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Quadrant. 0 1 2 3 4 The minutes of degrees of the Quadrant for right Sins of the compliment of the Arches of the same Quadrant. 0 0000 174524 348995 523360 697565 60 1 2909 177433 351902 526265 700467 59 2 5818 180341 354809 529170 703369 58 3 8727 183250 357716 532075 706270 57 4 11636 186158 360623 534980 709172 56 5 14544 189066 363530 537884 712073 55 6 17453 191975 366437 540789 714975 54 7 20362 194883 369344 543694 717876 53 8 23271 197792 372251 546598 720777 52 9 26180 200700 375158 549503 723678 51 10 29088 203608 378064 552407 726579 50 11 31997 206517 380971 555312 729480 49 12 34906 209425 383878 558216 732381 48 13 37815 212333 386785 561120 735282 47 14 40724 215241 389692 564024 738183 46 15 43632 218149 392598 566928 741084 45 16 46541 221057 395505 569832 743985 44 17 49450 223965 398412 572736 746886 43 18 52359 226873 401318 575640 749787 42 19 55268 229781 404225 578544 752688 41 20 58177 232689 407131 581448 755588 40 21 61086 235597 410038 584352 758489 39 22 63995 238505 412944 587256 761389 38 23 66904 241413 415851 590160 764290 37 24 69813 244321 418757 593064 767190 36 25 72721 247229 421663 595967 770090 35 26 75630 250137 424570 598871 772991 34 27 78539 253045 427476 601775 775891 33 28 81448 255953 430382 604678 778791 32 29 84357 258861 433288 607582 781691 31 30 87265 261769 436194 610485 784591 30 31 90174 264677 439100 613389 787491 29 32 93083 267585 442006 616292 790391 28 33 95992 270493 444912 619196 793291 27 34 98901 273401 447818 622099 796191 26 35 101809 276308 450724 625002 799090 25 36 104718 279216 453630 627905 801990 24 37 107627 282124 456536 630808 804889 23 38 110536 285032 459442 633711 807789 22 39 113445 287940 462348 636614 810688 21 40 116353 290847 465253 639517 813587 20 41 119262 293755 468159 642420 816486 19 42 122171 296663 471065 645323 819385 18 43 125079 299570 473970 648226 822284 17 44 127988 302478 476876 651129 825183 16 45 130896 305385 479781 654031 828082 15 46 133805 308293 482687 656934 830981 14 47 136714 311200 485592 659837 833880 13 48 139622 314108 488498 662739 836778 12 49 142531 317015 491403 665642 839677 11 50 145439 319922 494308 668544 842576 10 51 148348 322830 497214 671447 845474 9 52 151257 325737 500119 674349 848372 8 53 154165 328645 503024 677251 851271 7 54 157074 331552 505929 680153 854169 6 55 159982 334459 508834 683055 857067 5 56 162891 337367 511740 685957 859965 4 57 165799 340274 514645 688859 862863 3 58 168708 343181 517550 691761 865761 2 59 171616 346088 520455 694663 868659 1 60 174524 348995 523360 697565 871557 0  89 88 87 86 85   The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Quadrant. 5 6 7 8 9 The minutes of the degrees of the Quadrant for right Sins of the compliments of the same Quadrant. 0 871557 1045285 1218693 1391731 1564345 60 1 874455 1048178 1221580 1394612 1567218 59 2 877353 1051071 1224467 1397492 1570091 58 3 880250 1053964 1227354 1400373 1572964 57 4 883148 1056857 1230241 1403253 1575837 56 5 886045 1059749 1233128 1406133 1578709 55 6 888943 1062642 1236015 1409013 1581581 54 7 891840 1065534 1238901 1411893 1584453 53 8 894737 1068426 1241788 1414772 1587325 52 9 897634 1071318 1244674 1417652 1590197 51 10 900531 1074210 1247560 1420531 1593069 50 11 903428 1077102 1250446 1423410 1595941 49 12 906325 1079994 1253332 1426289 1598812 48 13 909222 1082886 1256218 1429168 1601684 47 14 912119 1085778 1259104 1432047 1604555 46 15 915016 1088669 1261990 1434926 1607426 45 16 917913 1091561 1264876 1437805 1610297 44 17 920809 1094452 1267761 1440684 1613168 43 18 923706 1097344 1270647 1443562 1616038 42 19 926602 1100235 1273532 1446441 1618909 41 20 929498 1103126 1276417 1449319 1621779 40 21 932395 1106017 1279302 1452197 1624649 39 22 935291 1108908 1282187 1455075 1627519 38 23 938187 1111799 1285072 1457953 1630389 37 24 941083 1114690 1287957 1460831 1633259 36 25 943979 1117580 1290841 1463708 1636129 35 26 946875 1120471 1293726 1466586 1638999 34 27 949771 1123361 1296610 1469463 1641868 33 28 952667 1126252 1299495 1472340 1644738 32 29 955563 1129142 1302378 1475217 1647607 31 30 958458 1132032 1305262 1478094 1650476 30 31 961354 1134922 1308146 1480971 1653345 29 32 964249 1137812 1311030 1483848 1656214 28 33 967144 1140702 1313914 1486724 1659082 27 34 970039 1143592 1316798 1489601 1661951 26 35 972934 1146482 1319681 1492477 1664819 25 36 975829 1149372 1322564 1495353 1667687 24 37 978724 1152261 1325447 1498229 1670555 23 38 981619 1155151 1328330 1501105 1673423 22 39 984514 1158040 1331213 1503981 1676291 21 40 987408 1160929 1334096 1506857 1679159 20 41 990303 1163818 1336979 1509733 1682027 19 42 993198 1166707 1339862 1512608 1684894 18 43 996092 1169596 1342744 1515484 1687761 17 44 998987 1172485 1345627 1518359 1690628 16 45 1001881 1175374 1348509 1521234 1693495 15 46 1004775 1178263 1351392 1524109 1696362 14 47 1007669 1181151 1354274 1526984 1699229 13 48 1010563 1184040 1357156 1529859 1702095 12 49 1013457 1186928 1360038 1532734 1704962 11 50 1016351 1189816 1362920 1535608 1707828 10 51 1019245 1192704 1365802 1538482 1710694 9 52 1022139 1195592 1368683 1541356 1713560 8 53 1025032 1198480 1371564 1544230 1716426 7 54 1027926 1201368 1374446 1547104 1719292 6 55 1030819 1204255 1377327 1549978 1722157 5 56 1033713 1207143 1380208 1552852 1725022 4 57 1036606 1210031 1383089 1555725 1727887 3 58 1039499 1212918 1385970 1558599 1730725 2 59 1042392 1215806 1388851 1561472 1733617 1 60 1045285 1218693 1391731 1564345 1736482 0  84 83 82 81 80   The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Quadrant. 10 11 12 13 14 The minutes of the degrees of the Quadrant for right Sins of the compliments of the same Quadrant. 0 1736482 1908090 2079117 2249511 2419219 60 1 1739347 1910945 2081962 2252345 2412041 59 2 1742●11 1913800 2084807 2255179 2424863 58 3 1745075 1916655 2087652 2258013 2427685 57 4 1747939 1919510 2090497 2260847 2430507 56 5 1750803 1922365 2093342 2263680 2433329 55 6 1753667 1925220 2096186 2266513 2436150 54 7 1756531 1928074 2099030 2269346 2438971 53 8 1759394 1950928 2101874 2272179 2441792 52 9 1762258 1933782 2104718 2275012 2444613 51 10 1765121 1936636 2107562 2277844 2447434 50 11 1767984 1939490 2110405 2280676 2450254 49 12 1770847 1942344 2113248 2283508 2453074 48 13 1773710 1945197 2116091 2286340 2455894 47 14 1776573 1948050 2118934 2289172 2458714 46 15 1779435 1950903 2121777 2292004 2461533 45 16 1782298 1953756 2124620 2294835 2464352 44 17 1785160 1956609 2127462 2297666 2467171 43 18 1788022 1959462 2130304 2300497 2469990 42 19 1790884 1962314 2133146 2303328 2472809 41 20 1793746 1965166 2135988 2306159 2475628 40 21 1796608 1968018 2138830 2308989 2478446 39 22 1799469 1970870 2141671 2311819 2481264 38 23 1802331 1973722 2144512 2314649 2484082 37 24 1805192 1976574 2147353 2317479 2486900 36 25 1808053 1979425 2150194 2320309 2489717 35 26 1810914 1982276 2153035 2323138 2492534 34 27 1813774 1985127 2155876 2325967 2495351 33 28 1816634 1987978 2158716 2328796 2498168 32 29 1819495 1990829 2161556 2331625 2500984 31 30 1822355 1993679 2164396 2334454 2503800 30 31 1825215 1996530 2167236 2337282 2506616 29 32 1828075 1999380 2170076 2340110 2509432 28 33 1830935 2002230 2172916 2342938 2512248 7 34 1833795 2005080 2175755 2345766 2515064 26 35 1836654 2007930 2178594 2348594 2517879 25 36 1839513 2010780 2181433 2351421 2520694 24 37 1842372 2013629 2184272 2354248 2523509 23 38 1845231 2016478 2187111 2357075 2526324 22 39 1848090 2019327 2189949 2359902 2529138 21 40 1850949 2022176 2192787 2362729 2531952 20 41 1853808 2025025 2195625 2365555 2534766 19 42 1856666 2027874 2198463 2368381 2537580 18 43 1859524 2030722 2201300 2371207 2540393 17 44 1862382 2033570 2204137 2374033 2543206 16 45 1865240 2036418 2206974 2376859 2546019 15 46 1868098 2039266 2209811 2379684 2548832 14 47 1870956 2042114 2212648 2382509 2551645 13 48 1873813 2044962 2215485 2385334 2554458 12 49 1876670 2047809 2218322 2388159 2557270 11 50 1879527 2050656 2221158 2390983 2560082 10 51 1882384 2053503 2223994 2393808 2562894 9 52 1885241 2056350 2226830 2396632 2565706 8 53 1888098 2059197 2229666 2399456 2568517 7 54 1890954 2062043 2232502 2402280 2571328 6 55 1893810 2064889 2235337 2405104 2574139 5 56 1896666 2067735 2238172 2407927 2576950 4 57 1899522 2070581 2241007 2410750 2579760 3 58 1902378 2073427 2243842 2413573 2582570 2 59 1905234 2076272 2246677 2416396 2585380 1 60 1908090 2079117 2249511 2419219 2588190 0  79 78 77 76 75   The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Quadrant. 15 16 17 18 19 The minutes of the degrees of the Quadrant for right Sins of the compliments of the same Quadrant. 0 2588190 2756373 2923717 3090170 3255682 60 1 2591000 2759169 2926499 3092936 3258532 59 2 2593809 2761965 2929280 3095702 3261182 58 3 2596618 2764761 2932061 3098468 3263931 57 4 2599427 2767556 2934842 3101234 3266681 56 5 2602236 2770351 2937623 3103999 3269430 55 6 2605045 2773146 2940403 3106764 3272179 54 7 2607853 2775941 2943183 3109529 3274927 53 8 2610661 2778735 2945963 3112294 3277675 52 9 2613469 2781529 2948743 3115058 3280423 51 10 2616277 2784323 2951523 3117822 3283171 50 11 2619084 2787117 2954302 3120586 3285918 49 12 2621891 2789911 2957081 3123349 3288665 48 13 2624698 2792704 2959860 3126112 3291412 47 14 2627505 2795497 2962638 3128875 3294159 46 15 2630312 2798290 2965416 3131638 3296906 45 16 2633118 2801082 2968194 3134400 3299652 44 17 2635924 2803874 2970972 3137162 3302398 43 18 2638730 2806666 2973750 3139924 3305144 42 19 2641536 2809458 2976527 3142686 3307889 41 20 2644342 2812250 2979304 3145448 3310634 40 21 2647147 2815041 2982081 3148209 3313379 39 22 2649952 2817832 2984857 3150970 3316123 38 23 2652757 2820623 2987633 3153731 3318867 37 24 2655562 2823414 2990409 3156491 3321611 36 25 2658366 2826204 2993185 3159251 3354355 35 26 2661170 2828994 2995960 3162011 3327098 34 27 2663974 2831784 2998735 3164770 3329841 33 28 2666777 2834574 3001510 3167529 3332585 32 29 2669580 2837364 3004284 3170288 3335327 31 30 2672383 2840153 3007058 3173047 3338069 30 31 2675186 2842942 3009832 3175805 3340811 29 32 2677989 2845731 3012606 3178563 3343553 28 33 2680792 3848520 3015380 3181321 3346294 27 34 2683595 2851308 3018153 3184079 3349035 26 35 2686397 2854096 3020926 3186837 3351776 25 36 2689199 2856884 3023699 3189594 3354516 24 37 2692001 2859672 3026472 3192351 3357256 23 38 2694802 2862459 3029244 3195108 3359996 22 39 2697603 2865246 3032016 3197864 3362736 21 40 2700404 2868033 3034788 3200620 3365475 20 41 2703205 2870819 3037559 3203375 3368214 19 42 2706005 2873605 3040330 3206130 3370953 18 43 2708805 2876391 3043101 3208885 3373691 17 44 2711605 2879177 3045872 3211640 3376429 16 45 2714405 2881963 3048643 3214395 3379167 15 46 2717204 2884748 3051413 3217150 3381905 14 47 2720003 2887533 3054183 3219904 3384642 13 48 2722802 2890318 3056953 3222658 3387379 12 49 2725601 2893103 3059723 3225412 3390116 11 50 2728400 2895888 3062492 3228165 3392852 10 51 2731198 2898672 3065261 3230918 3395588 9 52 2733996 2901456 3068030 3233671 3398324 8 53 2736794 2904240 3070798 3236423 3401060 7 54 2739592 2907023 3073566 3239175 3403795 6 55 2742389 2909806 3076334 3241927 3406530 5 56 2745186 2912589 3079102 3244679 3409265 4 57 2747983 2915371 3081869 3247430 3411999 3 58 2750780 2918153 3084636 3250181 3414733 2 59 2753577 2920935 3087403 3252932 3417467 1 60 2756373 2923717 3090170 3255682 3420201 0  74 73 72 71 70   The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Quadrant. 20 21 22 23 24 The minutes of the degrees of the Quadrant for right Sins of the compliments of the same Quadrant. 0 3420201 3583679 3746066 3907311 4067366 60 1 3422934 3586395 3748763 3909989 4070023 59 2 3425667 3589110 3751460 3912666 4072680 58 3 3428400 3591825 3754156 3915343 4075337 57 4 3431133 3594540 3756852 3918020 4077993 56 5 3433865 3597254 3759548 3920696 4080649 55 6 3436597 3599968 3762243 3923372 4083305 54 7 3439329 3602682 3764938 3926048 4085960 53 8 3442060 3605395 3767633 3928723 4088615 52 9 3444791 3608108 3770327 3931398 4091269 51 10 3447522 3610821 3773021 3934072 4093923 50 11 3450253 3613533 3775715 3936746 4096577 49 12 3452983 3616245 3778408 3939420 4099231 48 13 3455713 3618957 3781101 3942093 4101884 47 14 3458442 3621669 3783794 3944766 4104537 40 15 3461171 3624380 3786486 3947439 4107189 45 16 3463900 3627091 3789178 3950112 4109841 44 17 3466629 3629802 3791870 3952784 4112493 43 18 3469357 3632512 3794562 3955456 4115144 42 19 3472085 3635222 3797253 3958128 4117795 41 20 3474813 3637932 3799944 3960799 4120446 40 21 3477540 3640642 3802635 3963470 4123096 39 22 3480267 3643351 3805325 3966140 4125746 38 23 3482994 3646060 3808015 3968810 4128395 37 24 3485721 3648768 3810704 3971480 4131044 36 25 3488447 3651476 3813393 3974149 4133693 35 26 3491173 3654184 3816082 3976818 4136341 34 27 3493899 3656892 3818771 3979487 4138989 33 28 3496624 3659599 3821459 3982155 4141637 32 29 3499349 3662306 3824147 3984823 4144285 31 30 3502075 3665012 3826834 3987491 4146932 30 31 3504799 3667718 3829521 3990159 4149579 29 32 3507523 3670424 3832208 3992826 4152226 28 33 3510247 3673130 3834895 3995493 4154872 27 34 3512971 3675835 3837581 3998159 4157518 26 35 3515694 3678541 3840267 4000825 4160163 25 36 3518417 3681246 3842953 4003491 4162808 24 37 3521140 3683951 3845638 4006156 4165453 23 38 3523862 3686655 3848323 4008821 4168097 22 39 3526584 3689359 3851008 4011486 4170741 21 40 3529306 3692062 3853692 4014150 4173385 20 41 3532027 3694765 3856376 4016814 4176028 19 42 3534748 3697468 3859060 4019478 4178671 18 43 3537469 3700170 3861743 4022141 4181313 17 44 3540190 3702872 3864426 4024804 4183955 16 45 3542910 3705574 3867109 4027467 4186597 15 46 3545630 3708276 3869791 4030130 4189239 14 47 3548350 3710977 3872473 4032792 4191880 13 48 3551070 3713678 3875155 4035454 4194521 12 49 3553789 3716379 3877837 4038115 4197162 11 50 3556508 3719080 3880518 4040776 4199802 10 51 3559227 3721780 3883199 4043437 4202442 9 52 3561945 3724480 3885880 4046097 4205081 8 53 3564663 3727179 3888560 4048757 4207720 7 54 3567380 3729878 3891240 4051416 4210359 6 55 3570097 3732577 3893919 4054075 4212997 5 56 3572814 3735275 3896598 4056734 4215635 4 57 3575531 3737973 3899277 4059392 4218273 3 58 3578247 3740671 3901955 4062050 4220910 2 59 3580963 3743369 3904633 4064708 4223547 1 60 3583679 3746066 3907311 4067366 4226183 0  69 68 67 66 65   The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Quadrant. 25 26 27 28 29 The minutes of the degrees of the Quadrant for right Sins of the compliments of the same Quadrant. 0 4226183 4383712 4539905 4694710 4848096 60 1 4228819 4386326 4542497 4697284 4850640 59 2 4231455 4388940 4545088 4699852 4853184 58 3 4234090 4391554 4547679 4702419 4855727 57 4 4236725 4394167 4550270 4704986 4858270 56 5 4229360 4396780 4552860 4707553 4860812 55 6 4241994 4399392 4555450 4710119 4863354 54 7 4244628 4402004 4558039 4712685 4865895 53 8 4247262 4404616 4560628 4715250 4868436 52 9 4249895 4407227 4563216 4717815 4870977 51 10 4252528 4409838 4565804 4720380 4873517 50 11 4255161 4412449 4568392 4722944 4876057 49 12 4257793 4415059 4570979 4725508 4878596 48 13 4260425 4417669 4573566 4728071 4881135 47 14 4263056 4420278 4576153 4730634 4883674 46 15 4265687 4422887 4578739 4733197 4886212 45 16 4268318 4425496 4581325 4735759 4888750 44 17 4270949 4428104 4583911 4738321 4891287 43 18 4273579 4430712 4586496 4740882 4893824 42 19 4276209 4433320 4589081 4743443 4896361 41 20 4278838 4435927 4591665 4746004 4898897 40 21 4281467 4438534 4594249 4748564 4901433 39 22 4284096 4441140 4596833 4751124 4903968 38 23 4286724 4443746 4599416 4753683 4906503 37 24 4289352 4446352 4601999 4756242 4909037 36 25 4291979 4448957 4604581 4758801 4911571 35 26 4294606 4451562 4607163 4761359 4914105 34 27 4297233 4454167 4609744 4763917 4916638 33 28 4299859 4456771 4612325 4766474 4919171 32 29 4302485 4459375 4614906 4769031 4921703 31 30 4305111 4461978 4617486 4771588 4924235 30 31 4307736 4464581 4620066 4774144 4926767 29 32 4310361 4467184 4622646 4776700 4929298 28 33 4312986 4469786 4625225 4779255 4931829 27 34 4315610 4472388 4627804 4781810 4934359 26 35 4318234 4474990 4630382 4784365 4936889 25 36 4320858 4477591 4632960 4786919 4939418 24 37 4323481 448019● 4635538 4789473 4941947 23 38 4326104 4482792 4638115 4792026 4944476 22 39 4328726 4485392 4640692 4794579 4947004 21 40 4331348 4487992 4643268 4797132 4949532 20 41 4333970 4490591 4645844 4799684 4952059 19 42 4336591 4493190 4648420 4802236 4954586 18 43 4339212 4495788 4650995 4804787 4957113 17 44 4341833 4498386 4653570 4807338 4959639 16 45 4344453 4500984 4656145 4809888 4962165 15 46 4347073 4503582 4658719 4812438 4964690 14 47 4349693 4506179 4661293 4814988 4967215 13 48 4352312 4508776 4663866 4817537 4969740 12 49 4354931 4511372 4666439 4820086 4972264 11 50 4357549 4513968 4669012 4822635 4974788 10 51 4360167 4516563 4671584 4825183 4977311 9 52 4362785 4519158 4674156 4827731 4979834 8 53 4365402 4521753 4676727 4830278 4982356 7 54 4368019 4524347 4679298 4832825 4984878 6 55 4370635 4526941 4681869 4835371 4987399 5 56 4373251 4529535 4684439 4837917 4989920 4 57 4375867 4532128 4687009 4840462 4992441 3 58 4378482 4534721 4689578 4843007 4994961 2 59 4381097 4537313 4692147 4845552 4997481 1 60 4383712 4539905 4694716 4848096 5000000 0  64 63 62 61 60   The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Qudrant. 30 31 32 33 34 The minutes of the degrees of the Quadrant for right Sins of the compliment of the same Quadrant. 0 5000000 5150381 5299192 5446390 5591929 60 1 5002519 5152874 5301659 5448829 5594340 59 2 5005038 5155367 5304125 5451268 55967●1 58 3 5007556 5157859 5306591 5453707 55991●1 57 4 5010074 5160351 5309056 5456145 5601571 56 5 5012591 5162843 5311521 5458583 5603981 55 6 5015108 5165334 5313985 5461020 5606390 54 7 5017624 5167825 5316449 5463456 5608798 53 8 5020140 5170315 5318913 5465892 5611206 52 9 5022656 5172805 5321376 5468328 5613614 51 10 5025171 5175294 5323839 5470763 5616021 50 11 5027686 5177783 5326301 5473198 5618427 49 12 5030200 5180271 5328763 5475632 5620833 48 13 5032714 5182759 5331224 5478066 5623239 47 14 5035227 5185246 5333685 5480499 5625644 46 15 5037740 5187733 5336145 5482932 5628049 45 16 5040253 5190220 5338605 5485364 5630453 44 17 5042765 5192706 5341065 5487796 5632857 43 18 5045277 5195192 5343524 5490228 5635260 42 19 5047788 5197677 5345983 5492659 5637663 41 20 5050299 5200162 5348441 5495090 5640066 40 21 5052809 5202646 5350898 5497520 5642468 39 22 5055319 5205130 5353355 5499950 5644869 38 23 5057829 5207614 5355812 5502379 5647270 37 24 5060338 5210097 5358268 5504808 5649670 36 25 5062847 5212580 5360724 5507236 5652070 35 26 5065355 5215062 5363179 5509664 5654469 34 27 5067863 5217544 5365634 5512091 5656868 33 28 5070370 5220025 5368088 5514518 5659266 32 29 5072877 5222506 5370542 5516944 5661664 31 30 5075384 5224986 5372996 5519370 5664062 30 31 5077890 5227466 5375449 5521795 5666459 29 32 5080396 5229949 5377902 5524220 5668856 28 33 5082901 5232425 5380354 5526645 5671252 27 34 5085406 5234904 5382806 5529069 5673648 26 35 5087911 5237382 5385258 5531493 5676043 25 36 5090415 5239860 5387709 5533916 5678438 24 37 5092619 5242337 5390159 5536338 5680832 23 38 5095422 5244814 5392609 5538760 5683226 22 39 5097925 5147290 5395058 5541182 5685619 21 40 5100427 5249766 5397507 5543603 5688012 20 41 5102929 525224● 5399955 5546024 5690404 19 42 5105430 5254716 5402403 5548444 5692796 18 43 5107931 5257191 5404851 5550864 5695187 17 44 5110431 5259665 5407298 5553283 5697578 16 45 5112931 5262139 5409745 5555702 5699968 15 46 5115431 5264612 5412191 5558120 5702358 14 47 5117930 5267085 5414637 5560538 5704747 13 48 5120429 5269557 5417082 5562956 5707136 12 49 5122927 5272029 5419527 5565373 5709524 11 50 5125425 5274501 5421972 5567790 5711912 10 51 5127922 5276972 5424416 5570206 5714299 9 52 5130419 5279443 5426859 5572622 5716686 8 53 5132916 5281913 5429302 5575037 5719072 7 54 5135412 5284383 5431745 5577452 5721458 6 55 5137908 5286852 5434187 5579866 5723844 5 56 5140403 5289321 5436620 5582280 5726229 4 57 5142898 5291789 5439070 5584693 5728613 3 58 5145393 5294257 5441510 5587106 5730997 2 59 5147887 5296725 5443950 5589518 5733381 1 60 5150381 5299192 5446390 5591929 5735764 0  59 58 57 56 55   The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Qudrant. 35 36 37 38 39 The minutes of the degrees of the Quadrant for right Sins of the compliment of the same Quadrant. 0 5735764 5877852 6018150 6156615 6293204 60 1 5738147 5880205 6020470 6158907 6295464 59 2 5740529 5882558 6022796 6161198 6297724 58 3 5742911 5884910 6025118 6163489 6299983 57 4 5745292 5887262 6027439 6165780 6302242 56 5 5747672 5889613 6029760 6168070 6304501 55 6 5750052 5891964 6032080 6170359 6306759 54 7 5752432 5894314 6034400 6172648 6309016 53 8 5754811 5896664 6036719 6174936 6311273 52 9 5757190 5899013 6039038 6177224 6313529 51 10 5759568 5901361 6041357 6179512 6315784 50 11 5761946 5903709 6043675 6181799 6318039 49 12 5764323 5906056 6045992 6184085 6320293 48 13 5766700 5908403 6048309 6186371 6322547 47 14 5769076 5910750 6050625 6188656 6324800 46 15 5771452 5913096 6052940 6190940 6327053 45 16 5773827 5915442 6055255 6193224 6329305 44 17 5776202 5917787 6057570 6195508 6331557 43 18 5778576 5920132 6059884 6197791 6333808 42 19 5780950 5922476 6062198 6200074 6336059 41 20 5783324 5924820 6064511 6202356 6338310 40 21 5785697 5927163 6066824 6204638 6340560 39 22 5788069 5929505 6069136 6206919 6342809 38 23 5790441 5931847 6071448 6209199 6345058 37 24 5792812 5934189 6073759 6211479 6347306 36 25 5795183 5936530 6076069 6213758 6349553 35 26 5797553 5938871 6078379 6216037 6351800 34 27 5799923 5941211 6080688 6218315 6354046 33 28 5802292 5943551 6082997 6220593 6356292 32 29 5804●61 5945890 6085306 6222870 6358537 31 30 5807030 5948228 6087614 6225146 6360782 30 31 5809398 595056● 6089922 6227422 6363026 29 32 5811766 5952904 6092229 6229698 6365270 28 33 5814133 5955241 6094536 6231973 6367513 27 34 5816499 5957578 6096842 6234248 6309756 26 35 5818865 5959914 6099147 6236522 6371999 25 36 5821230 5962250 6101452 9238796 6374241 24 37 5823595 5964585 6103756 6241069 6376482 23 38 5825959 5966919 6106060 6243342 6378722 22 39 5828323 5969253 6108364 6245614 6380962 21 40 5830687 5971586 6110667 6247885 6383201 20 41 5833050 5973919 6112970 6250156 6385440 19 42 5835412 5976251 6115272 6252426 6387678 18 43 5837774 5978583 6117573 6254696 6389916 17 44 5840136 5980915 6119873 6256966 6392153 16 45 5842497 5983146 6122173 6259235 6394390 15 46 5844858 5985577 6124473 6261503 6396626 14 47 5847218 5987907 6126772 6263771 6398862 13 48 5849578 5990237 6129071 6266038 6401097 12 49 5851937 5992566 6131369 6268305 6403332 11 50 5854295 5994894 6133667 6270572 6405566 10 51 5856653 5997222 6135964 6272838 6407799 9 52 5859010 5999549 6138261 6275103 6410032 8 53 5861367 6001876 6140557 6277368 6412264 7 54 5863724 6004202 6142853 6279632 6414496 6 55 5866080 6006528 6145148 6281895 6416728 5 56 5868436 6008853 6147442 6284158 6418959 4 57 5870791 6011178 6149736 6286420 6421189 3 58 5873145 6013502 6152030 6288682 6423419 2 59 5875499 6015826 6154323 6290943 6425648 1 60 5877852 6018150 6156615 6293204 6427876 0  54 53 52 51 50   The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Quadrant. 40 41 42 43 44 The minutes of the degrees of the Quadrant for right Sins of the compliments of the same Quadrant. 0 6427876 6560590 6691306 6819984 6946584 60 1 6430104 6562785 6693468 6822111 6948676 59 2 6432331 6564979 6695629 6824237 6950767 58 3 6434558 6567173 9697789 6826363 6952858 57 4 6436785 6569367 6699949 6828489 6954949 56 5 6439011 6571560 6702108 6830614 6957039 55 6 6441236 6573753 6704267 6832738 6959128 54 7 6443461 6575945 6706425 6834861 6961216 53 8 6445685 6578136 6708582 6836984 6963304 52 9 6447909 6580326 6710739 6839107 6965392 51 10 6450132 6582516 6712895 6841229 6967479 50 11 6452355 6584705 6715051 6843350 6969565 49 12 6454577 6586894 6717206 6845471 6971651 48 13 6456799 6589082 6719361 6847591 6973736 47 14 6459020 6591270 6721515 6849711 6975821 46 15 6461240 6593458 6723668 6851830 6977905 45 16 6463460 6595645 6725821 6853949 6979988 44 17 6465679 6597831 6727973 6856067 6982071 43 18 6467898 6600016 6730125 6858184 6984153 42 19 6470116 6602201 6732276 6860301 6986235 41 20 6472333 6604386 6734427 6862417 6988319 40 21 6474550 6606570 6736577 6864533 6990396 39 22 6476766 6608753 6738726 6866648 6992476 38 23 6478982 6610936 6740875 6868762 6994555 37 24 6481198 6613118 6743024 6870876 6996634 36 25 6483413 6615300 6745172 6872989 6998712 35 26 6485628 6617481 6747319 6875102 7000789 34 27 6487842 6619661 6749465 6877214 7002866 33 28 6490055 6621841 6751611 6879325 7004942 32 29 6492268 6624021 6753757 6881436 7007018 31 30 6494480 6626200 6755902 6883546 7009093 30 31 6496692 6628379 6758047 6885656 7011167 29 32 6498903 6630557 6760191 6887765 7013241 28 33 6501114 6632734 6762334 6889874 7015314 27 34 6503324 6634911 6764477 6891982 7017387 26 35 6505533 6637087 6766619 6894089 7019459 25 36 6507742 6639263 6768760 6896196 7021530 24 37 6509950 6641438 6770901 6898302 7023601 23 38 6512158 6643612 6773041 6900408 7025671 22 39 6514365 6645786 6775181 6902513 7027741 21 40 6516572 6647959 6777320 6904617 7029810 20 41 6518778 6650132 6779459 6906721 7031879 19 42 6520984 6652304 6781597 6908824 7033947 18 43 6523189 6654476 6783734 6910927 7036014 17 44 6525394 6656647 6785871 6913029 7038081 16 45 6527598 6658817 6788007 6915131 7040147 15 46 6529801 6660987 6790143 6917232 7042213 14 47 6532004 6663156 6792278 6919332 7044278 13 48 6534206 6665325 6794413 6921432 7046342 12 49 6536408 6667493 6796547 6923531 7048406 11 50 6538609 6669661 6798681 6925630 7050469 10 51 6540809 6671828 6800814 6927728 7052432 9 52 6543009 6673994 6802946 6929725 7054594 8 53 6545208 6676160 6805078 6931922 7056655 7 54 6547407 6678326 6807209 6934018 7058716 6 55 6549606 6680491 6809340 6936114 7060776 5 56 6551804 6682655 6811470 6938209 7062836 4 57 6554001 6684818 6813599 6940303 7064895 3 58 6556198 6686981 6815728 6942397 7066953 2 59 6558394 6689144 6817856 6944491 7069011 1 60 6560590 6691306 6819984 6946584 7071068 0  49 48 47 46 45   The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Quadrant. 45 46 47 48 49 The minutes of the degrees of the Quadrant for right Sins of the compliments of the same Quadrant. 0 7071068 7193398 7313537 7431448 7547096 60 1 7073125 7195418 7315521 7433394 7549004 59 2 7075181 7197438 7317504 7435339 7550911 58 3 7077236 7199457 7319486 7437284 7552818 57 4 7079291 7201476 7321468 7439229 7554724 56 5 7081345 7203494 7323449 7441173 7556630 55 6 7083399 7205511 7325429 7443116 7558535 54 7 7085452 7207527 7327409 7445058 7560439 53 8 7087504 7209543 7329388 7447000 7562343 52 9 7089556 7211559 7331367 7448941 7564246 51 10 7091607 7213574 7333345 7450882 7566148 50 11 7093658 7215588 7335322 7452822 7568050 49 12 7095708 7217601 7337298 7454761 7569951 48 13 7097757 7219614 7339274 7456690 7571851 47 14 7099806 7221627 7341250 7458637 7573751 46 15 7101854 7223639 7343225 7460574 7575650 45 16 710390● 7225661 7345199 7462511 7577548 44 17 7105949 7227662 7347173 7464447 7579446 43 18 7107995 7229672 7349146 7466382 7581343 42 19 7110041 7231681 7351118 7468317 7583240 41 20 7112086 7233689 7353090 7470251 7585136 40 21 7114131 7235697 7355061 7472184 7587031 39 22 7116175 7237704 7357031 7474117 7588925 38 23 7118218 7239711 7359001 7476049 7590819 37 24 7120261 7241718 7360970 7477981 7592714 36 25 7122303 7243724 7362939 7479912 7594606 35 26 7124344 7245729 7364907 7481842 7596498 34 27 7126385 7247733 7366874 7483771 7598389 33 28 7128425 7249737 7368841 7485700 7600280 32 29 7130465 7251741 7370807 7487629 7602170 31 30 7132504 7253744 7372773 7489557 7604960 30 31 7134543 7255746 7374738 7491484 7605949 29 32 7136581 7257747 7376702 7493410 7607837 28 33 7138618 7259748 7378666 7495336 7609725 ●7 34 7140655 7261749 7380629 7497262 7611612 26 35 7142691 7263749 7382592 7499187 7613498 25 36 7144727 7265748 7384554 7501111 7615384 24 37 7146762 7267746 7386515 7503034 7617269 23 38 7148796 7269744 7388475 7504957 7619153 22 39 7150830 7271741 7390435 7506879 7621037 21 40 7152863 7273737 7392394 7508801 7622920 20 41 7154895 7275733 7394353 7510722 7624802 19 42 7156927 7277728 7396311 7512642 7626683 18 43 7158958 7279722 7398268 7514561 7628564 17 44 7160989 7281716 7400225 7516480 7630445 16 45 7163019 7283710 7402181 7518398 7632325 15 46 7165049 7285703 7404137 7520316 7634204 14 47 7167078 7287695 7406092 7522233 7636082 13 48 7169106 7289687 7408046 7524149 7637960 12 49 7171134 7291678 7410000 7526065 7639838 11 50 7173161 7293668 7411953 7527980 7641715 10 51 7175187 7295658 7413905 7529894 7643591 9 52 7177213 7297647 7415856 7531808 7645466 8 53 7179238 7299635 7417807 7533721 7647341 7 54 7181263 7301623 7419758 7535634 7649215 6 55 7183287 7303610 7421708 7537546 7651088 5 56 7185310 7305597 7423657 7539457 7652961 4 57 7187333 7307583 7425605 7541367 7654833 3 58 7189355 7309568 7427552 7543277 7656704 2 59 7191377 7311553 7429501 7545187 7658575 2 06 7193398 7313537 7431448 7547096 7660445 0  44 43 42 41 40   The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Quadrant. 50 51 52 53 54 The minutes of the degrees of the Quadrant for right Sins of the compliments of the same Quadrant. 0 7660445 7771460 7880108 7986355 8090170 60 1 7662314 7773290 7881898 7988105 8091879 59 2 7664183 7775120 7883688 7989855 8093588 58 3 7666051 7776949 7885477 7991604 8095296 57 4 7667919 7778777 7887266 7993352 8097004 56 5 7669786 7780605 7889054 7995100 8098711 55 6 7671652 7782432 7890841 7996847 8100417 54 7 7673517 7784258 7892927 7998593 8102122 53 8 7675382 7786084 7894413 8000339 8103827 52 9 7677246 7787909 7896198 8002084 8105531 51 10 7679110 7789833 7897983 8003828 8107234 50 11 7680973 7791557 7899767 8005571 8108936 49 12 7682835 7793380 7901550 8007314 8110638 48 13 7684697 7795202 7903332 8009056 8112339 47 14 7686558 7797024 7905114 8010797 8114040 46 15 7688418 7798845 7906895 8012538 8115740 45 16 7690278 7800665 7908676 8014278 8117439 44 17 7692137 7802485 7910456 8016017 8119137 43 18 7693995 7804304 7912235 8017756 8120835 42 19 7695853 7806123 7914014 8019494 8122532 41 20 7697710 7807941 7915792 8021232 8124229 40 21 7699566 7809758 7917569 8022969 8125925 39 22 7701422 7811574 7919345 8024705 8127620 38 23 7703277 7813390 7921121 8026440 8129314 37 24 7705132 7815205 7922896 8028175 8131008 36 25 7706986 7817020 7924671 8029909 8132701 35 26 7708839 7818834 7926445 8021642 8134393 34 27 7710692 7820647 7928218 8033375 8136084 33 28 7712544 7822459 7929990 8035107 8137775 32 29 7714395 7824271 7931762 8036838 8139465 31 30 7716246 7826082 7933533 3038569 8141155 30 31 7718096 7827892 7935303 3040299 8142844 29 32 7719945 7829702 7937073 8042028 8144532 28 33 7721794 7831511 7938842 8043757 8146220 27 34 7723642 7833320 7940611 8045485 8147907 26 35 7725490 7835128 7942379 8047212 8149593 25 36 7727337 7836935 7944146 8048938 8151278 24 37 7729183 7838741 7945912 8050664 8152963 23 38 7731028 7840547 7947678 8052389 8154647 22 39 7732872 7842352 7949443 8054114 8156330 21 40 7734716 7844157 7951208 8055838 8158013 20 41 7736559 7845961 7952972 8057561 8159695 19 42 7738402 7847764 7954735 8059283 8161376 18 43 7740244 7849566 7956497 8061005 8163057 17 44 7742085 7851368 7958259 8062726 8164737 16 45 7743926 7853169 7960020 8064446 8166416 15 46 7745766 7854970 7961780 8066166 8168094 14 47 7747606 7856770 7963540 8067885 8169772 13 48 7749445 7858569 7965299 8069603 8171449 12 49 7751283 7860368 7967057 8071321 8173126 11 50 7753121 7862166 7968815 8073038 8174802 10 51 7754958 7863963 7970572 8074754 8176477 9 52 7756794 7865759 7972328 8076470 8178151 8 53 7758630 7867555 7974084 8078185 8179825 7 54 7760465 7869350 7975839 8079899 8181498 6 55 7762299 7871145 7977593 8081613 8183170 5 56 7764132 7872939 7979347 8083326 8184841 4 57 7765965 7874732 7981100 8085038 8186512 3 58 7767797 7876525 7982852 8086749 8188182 2 59 7769629 7878317 7984604 8088460 8189851 1 06 7771460 7880108 7986355 8090170 8191520 0  39 38 37 36 35   The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Quadrant. 55 56 57 58 59 The minutes of the degrees of the Quadrant for right Sins of the compliments of the same Quadrant. 0 8191520 8290376 8386706 8480481 8571673 60 1 8193188 8292002 8388290 8482022 8573171 59 2 8194855 8293628 8389873 8483562 8574668 58 3 8196522 8295253 8391456 8485102 8576164 57 4 8198188 8296877 8393038 8486641 8577660 56 5 8199854 8298501 8394619 8488180 8579155 55 6 8201519 8300124 8396199 8489718 8580649 54 7 8203183 8301746 8397778 8491255 8582142 53 8 8204846 8303367 8399357 8492791 8583635 52 9 8206508 8304987 8400935 8494326 8585127 51 10 8208170 8306607 8402513 8495860 8586619 50 11 8209831 8308226 8404090 8497394 8588110 49 12 8211491 8309844 8405666 8498927 8589600 48 13 8213151 831146● 8407241 8500459 8591089 47 14 8214810 8313079 8408816 8501991 8592577 46 15 8216469 8314696 8410390 8503522 8594064 45 16 8218127 8316312 8411963 8505052 8595551 44 17 8219784 8317927 8413536 8506582 8597037 43 18 8221440 8319541 8415108 8508111 8598523 42 19 8223096 8321155 8416679 8509639 8600008 41 20 8224751 8322768 8418250 8511167 8601492 40 21 8226405 8324380 8419820 8512694 8602975 39 22 8228058 8325991 8421389 8514220 8604457 38 23 8229711 8327602 8422957 8515745 8605939 37 24 8231363 8329212 8424525 8517270 8607420 36 25 8233015 8330822 8426092 8518794 8608901 35 26 8234666 8332431 8427658 8520317 8610381 34 27 8236316 8334039 8429223 8521839 8611860 33 28 8237965 8335646 8430788 8523361 8613338 32 29 8239614 8337252 8432352 8524882 8614815 31 30 8241262 8338858 8433915 8526402 8616292 30 31 8242909 8340463 8435477 8527921 8617768 29 32 8244556 8342067 8437039 8529440 8619243 28 33 8246202 8343671 8438600 8530958 8620718 27 34 8247847 8345247 8440161 8532476 8622192 26 35 8249492 8346877 8441721 8533993 8623665 25 36 8251136 8348479 8443280 8535509 8625137 24 37 8252779 8350080 8444838 8537024 8626608 23 38 8254421 8351680 8446396 8538538 8628079 22 39 8256062 8353279 8447953 8540052 8629549 21 40 8257703 8354878 8449509 8541565 8631019 20 41 8259343 8356476 8451064 8543077 8632488 19 42 8260982 8358073 8452618 8544588 8633956 18 43 8262621 8359670 8454172 8546099 8635423 17 44 8264259 8361266 8455725 8547609 8636889 16 45 8265897 8362862 8457278 8549119 8638355 15 46 8267534 8364457 8458830 8550628 8639820 14 47 8269170 8366051 8460381 8552136 8641284 13 48 8270806 8367644 8461932 8553643 8642748 12 49 8272441 8369236 8463482 8555149 8644211 11 50 8274075 8370828 8465031 8556655 8645673 10 51 8275708 8372419 8466579 8558160 8647134 9 52 8277340 8374009 8468126 8559664 8648595 8 53 8278972 8375599 8469673 9561168 8650055 7 54 8280603 8377188 8471219 8562671 8651514 6 55 8282234 8378776 8472765 8564173 8652973 5 56 8283864 8380363 8474310 8565675 8654431 4 57 8285493 8381950 8475854 8567176 8655888 3 58 8287121 8383536 8477397 8568676 8657344 2 59 8288749 8385121 8478939 8570175 8658799 1 60 8290376 8386706 8480481 8571673 8660254 0  34 33 32 31 30   The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Quadrant. 60 61 62 63 64 The minutes of the degrees of the Quadrant for right Sins of the compliments of the same Quadrant. 0 8660254 8746197 8829476 8910065 8987940 60 1 8661708 8747607 8830841 8911385 8989215 59 2 8663162 8749016 8832205 8912704 8990489 58 3 8664615 8750425 8833569 8914023 8991762 57 4 8666067 8751833 8834932 8915341 8993035 56 5 8667518 8753240 8836295 8916659 8994307 55 6 8668968 8754646 8837657 8917976 8995578 54 7 8670417 8756051 8839018 8919292 8996848 53 8 8671866 8757456 8840378 8920607 8998117 52 9 8673314 8758860 8841737 8921921 8999386 51 10 8674762 8760263 8843095 8923234 9000654 50 11 8676209 8761665 8844452 8924546 9001921 49 12 8677655 8763067 8845809 8925858 9003187 48 13 8679100 8764468 8847165 8927169 9004453 47 14 8680544 8765868 8848521 8928479 9005718 46 15 8681988 8767268 8849876 8929789 9006982 45 16 8683431 8768667 8851230 8931098 9008245 44 17 8684874 8770065 8852583 8932406 9009508 43 18 8686316 8771462 8853936 8933714 9010770 42 19 8687757 8772859 8855288 8935021 9012031 41 20 8689197 8774255 8856639 8936327 9013292 40 21 8690636 8775650 8857989 8937632 9014552 39 22 8692074 8777044 8859338 8938936 9015811 38 23 8693512 8778437 8860687 8940240 9017069 37 24 8694949 8779830 8862035 8941543 9018326 36 25 8696386 8781222 8863383 8942845 9019582 35 26 8697822 8782613 8864730 8944146 9020838 34 27 8699257 8784003 8866076 8945446 9022093 33 28 8700691 8785393 8867421 8946746 9023347 32 29 8702124 8786782 8868765 8948045 9024600 31 30 8703557 8788171 8870108 8949344 9025853 30 31 8704989 8789559 8871451 8950642 9027105 29 32 8706420 8790946 8872793 895193● 9028356 28 33 8707851 8792332 8874134 8953235 9029606 27 34 8709281 8793717 8875475 8954530 9030856 26 35 8710710 8795102 8876815 8955824 9032105 25 36 8712138 8796486 8878154 8957117 9033353 24 37 8713565 8797869 8879492 8958410 9034600 23 38 8714992 8799251 8880830 8959702 9035847 22 39 8716418 8800633 8882167 8960994 9037093 21 40 8717844 8802014 8883503 8962285 9038338 20 41 8719269 8803394 8884838 8963575 9039582 19 42 8720693 8804773 8886172 8964864 9040825 18 43 8722116 8806152 8887506 8966152 9042068 17 44 8723538 8807530 8888839 8967440 9043310 16 45 8724960 8808907 8890171 8968727 9044551 15 46 8726●81 8810283 8891502 8970013 9045791 14 47 8727801 8811659 8892833 8971299 9047031 13 48 8729221 8813034 8894163 8972584 9048270 12 49 8730640 8814408 8895492 8973868 9049508 11 50 8732058 8815782 8896821 8975151 9050746 10 51 8733475 8817155 8898149 8976433 9051983 9 52 8734891 8818527 8899476 8977715 9053219 8 53 8736307 8819898 8900802 8978996 9054454 7 54 8737722 8821268 8902127 8980276 9055688 6 55 8739137 8822638 8903452 8981555 9056922 5 56 8740551 8824007 8904776 8982833 9058155 4 57 8741964 8825375 8906099 8984111 9059387 3 58 8743376 8826743 8907422 8985388 9060618 2 59 8744787 8828110 8908744 8986664 9061848 1 60 8746197 8829476 8910065 8987940 9063078 0  25 26 27 28 29   The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Quadrant. 65 66 67 68 69 The minutes of the degrees of the Quadrant for right Sins of the compliments of the same Quadrant. 0 9063078 9135455 9205049 9271839 9335804 60 1 9064307 9136638 9206185 9272928 9336846 59 2 9065535 9137820 9207321 9274017 9337887 58 3 9066763 9139001 9208456 9275105 9338928 57 4 9067990 9140181 9209590 9276192 9339968 56 5 9069216 9141361 9210723 9277278 9341007 55 6 9070441 9142540 9211855 9278363 9342045 54 7 9071665 9143718 9212986 9279448 9343082 53 8 9072889 9144895 9214117 9280532 9344119 52 9 9074112 9146072 9215247 9281615 9345155 51 10 9075334 9147248 9216376 9282697 9346190 50 11 9076555 9148423 9217504 9283778 9347224 49 12 9077775 9149597 9218631 9284859 9348257 48 13 9078995 9150770 9219758 9285939 9349289 47 14 9080214 9151943 9220884 9287018 9350321 46 15 9081432 9153115 9222010 9288096 9351352 45 16 9082649 9154286 9223135 9289173 9352382 44 17 9083866 9155457 9224259 9290250 9353411 43 18 9085082 9156627 9225382 9291326 9354440 42 19 9086297 9157796 9226504 9292401 9355468 41 20 9087512 9158964 9227625 9293476 9356495 40 21 9088726 9160131 9228746 9294550 9357521 39 22 9089939 9161297 9229866 9295623 9358546 38 23 9091151 9162463 9230985 9296695 9359571 37 24 9092362 9163628 9232103 9297766 9360595 36 25 9093572 9165792 9233220 9298836 9361618 35 26 9094781 9165955 9234337 9299905 9362640 34 27 9095990 9167117 9235453 9300974 9363662 33 28 9097198 9168279 9236568 9302042 9364683 32 29 9098406 9169440 9237682 9303109 9365703 31 30 9099613 9170601 9238795 9304176 9366722 30 31 9100819 9171761 9239908 9305242 9367740 29 32 9102024 9172920 9241020 9306307 9368758 28 33 9103228 9174078 9242131 9307371 9369775 27 34 9104432 9175235 9243242 9308434 9370791 26 35 9105636 9176391 9244352 9309497 9371806 25 36 9106837 9177547 9245461 9310559 9372820 24 37 9108038 9178702 9246569 9311620 9373834 23 38 9109238 9179856 9247676 9312680 9374847 22 39 9110438 9181009 9248782 9313739 9375859 21 40 9111637 9182161 9249888 9314798 9376870 20 41 9112835 9183313 9250993 9315856 9377880 19 42 9114032 9184464 9252097 9316913 9378889 18 43 9115229 9185614 9253200 6317969 9379898 17 44 9116425 9186763 9254303 9319024 9380906 16 45 9117620 9187912 9255405 9320079 9381913 15 46 9118814 9189060 9256506 9321133 9382919 14 47 9120007 9190207 9257606 9322186 9383925 13 48 9121200 9191353 9258706 9323238 9384930 12 49 9122392 9192499 9259805 9324290 9385934 11 50 9123584 9193644 9260963 9325241 9386937 10 51 9124775 9194788 9262000 9326391 9387939 9 52 9125965 9195931 9263096 9327440 9388941 8 53 9127154 9197073 9264192 9328488 9389942 7 54 9128342 9198215 9265287 9329535 9390942 6 55 9129529 9199356 9266381 9330582 9391941 5 56 9130716 9200496 9267474 9331628 9392940 4 57 9131902 9201635 9268566 9332673 9393938 3 58 9133087 9202774 9269658 9333717 9394935 2 59 9134271 9203912 9270749 9334761 9395931 1 60 9135455 9205049 9271839 9335804 9396926 0  24 23 22 21 20   The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Quadrant. 70 71 72 73 74 The minutes of the degrees of the Quadrant for right Sins of the compliments of the same Quadrant. 0 9396926 9455186 9510565 9563048 9612617 60 1 9397921 9456133 9511464 9563898 9613418 59 2 9398915 9457079 9512362 956474● 9614219 58 3 9399908 9458024 9513259 9565596 9615019 57 4 9400900 9458968 9514155 9566444 9615818 56 5 9401891 9459911 9515050 9567291 9616616 55 6 9402882 9460854 9515944 9568137 9617413 54 7 9403872 9461796 9516838 9568982 9618209 53 8 9404861 9462737 9517731 9569826 9619005 52 9 9405849 9463677 9518623 9570670 9619800 51 10 9406836 9464616 9519514 9571513 9620594 50 11 9407822 9465555 9520404 9572355 9621387 49 12 9408808 9466493 9521294 9573196 9622179 48 13 9409793 9467430 9522183 9574036 9622971 47 14 9410777 9468366 9523071 9574875 9623762 46 15 9411760 9469301 9523958 9575714 9624552 45 16 9412742 9470236 9524844 9576552 9625341 44 17 9413724 9471170 9525730 9577389 9626129 43 18 9414705 9472103 9526615 9578225 9626917 42 19 9415685 9473035 9527499 9579061 9627704 41 20 9416665 9473967 9528382 9579896 9628490 40 21 9417644 9474898 9529264 9580730 9629275 39 22 9418622 9475828 9530146 9581563 9630059 38 23 9419599 9476757 9531027 9582395 9630843 37 24 9420575 9477685 9531907 9583226 9631626 36 25 9421550 9478612 9532786 9584057 9632408 35 26 9422525 9479539 9533664 9584887 9633189 34 27 9423499 9480465 9534541 9585716 9633969 33 28 9425472 9481390 9535418 9586544 9634748 32 29 9425444 9482314 9536294 9587371 9635527 31 30 9426415 9483237 9537169 9588197 9636305 30 31 9427380 9484160 9538043 9589023 9637082 29 32 9428356 9485082 9538917 9589848 9637858 28 33 9429325 9486003 9539790 9590672 9638633 27 34 9430293 9486923 9540662 9591495 9639408 26 35 9431260 9487842 9541533 9592318 9640182 25 36 9432227 9488761 9542403 9593140 9640955 24 37 9433193 9489679 9543272 9593961 9641727 23 38 9434158 9490596 9544141 9594781 9642498 22 39 9435122 9491512 9545009 9595600 9643268 21 40 9436085 9492427 9545876 9596419 9644038 20 41 9437048 9493341 9546742 9597237 9644807 19 42 9438010 9494255 9547607 9598054 9645575 18 43 9438971 9495168 9548472 9598870 9646342 17 44 9439931 9496080 9549336 9599685 9647108 16 45 9440890 9496991 9550199 9600499 9647873 15 46 9441849 9497902 9551061 9601313 9648638 14 47 9442807 9498812 9551922 9602126 9649402 13 48 9443764 9499721 9552783 9602938 9650165 12 49 9444720 9500629 9553643 9603749 9650927 11 50 9445676 9501536 9554502 9604559 9651689 10 51 9446631 9502443 9555360 9605368 9652450 9 52 9447585 9503349 9556217 9606177 9653210 8 53 9448538 9504254 9557074 9606985 9653969 7 54 9449490 9505158 9557930 9607792 9654727 6 55 9450441 9506061 9558785 9608598 9655484 5 56 9451392 9506963 9559639 9609403 9656240 4 57 9452342 9507865 9560492 9610208 9656996 3 58 9453291 9508766 9561345 9611012 9657751 2 59 9454239 9509666 9562197 9611815 9658505 1 60 9455186 9510565 9563048 9612617 9659258 0  19 18 17 16 15   The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Quadrant. 75 76 77 78 79 The minutes of the degrees of the Quadrant for right Sins of the compliments of the same Quadrant. 0 9659258 9702957 9743600 9781476 9816272 60 1 9660011 9703660 9744355 9782080 9816827 59 2 9660163 9704363 9745008 9782684 9817381 58 3 9661514 9705065 9745660 9783281 9817934 57 4 9662264 9705766 9740312 9783889 9818486 56 5 9663013 9706466 9746963 9784490 9819037 55 6 9663761 9707165 9747013 9785090 9819587 54 7 9664508 9707863 9748262 9785689 9820137 53 8 9665255 9708561 9748910 9786288 9820686 52 9 9666001 9709258 9749557 9786886 9821234 51 10 9666746 9709954 9750203 9787483 9821781 50 11 9667490 9710649 9750849 9788079 9822327 49 12 9668233 9711343 9751494 9788674 9822872 48 13 9668976 9712036 9752138 9789268 9823417 47 14 9669718 9712729 9752781 9789862 9823961 46 15 9670459 9713421 9753424 9790455 9824504 45 16 9671199 9714112 9754065 9791047 9825046 44 17 9671938 9714802 9754706 9791638 9825587 43 18 9672677 9715491 9755346 9792228 9826128 42 19 9673415 9716180 9755985 9792818 9826668 41 20 9674152 9716868 9756623 9793407 9827207 40 21 9674888 9717555 9757260 9793995 9827745 39 22 9675623 9718241 9757897 9794582 9828282 38 23 9676357 9718926 9758533 9795168 9828818 37 24 9677091 9719610 9759168 9795753 9829354 36 25 9677824 9720294 9759802 6796337 9829889 35 26 9678556 9720977 9760435 9796921 9830423 34 27 9679287 9721659 9761067 9797504 9830956 33 28 9680017 9722340 9761699 9798086 9831488 32 29 9680747 9723020 9762330 9798667 9832019 31 30 9681476 9723699 9762960 9799247 9832549 30 31 9682804 9724378 9763589 9799827 9833079 29 32 9682931 9725056 9764217 9800406 9833608 28 33 9683657 9725733 9764845 9800984 9834136 27 34 9684383 9726409 9765472 9801561 9834663 26 35 9685108 9727085 9766098 9802137 9835189 25 36 9685832 9727760 9766723 9802712 9835714 24 37 9686555 9728434 9767347 9803287 9836239 23 38 9687277 9729107 9767970 9803861 9836763 22 39 9687998 9729779 9768593 9804434 9837286 21 40 9688719 9730450 9769215 9805006 9837808 20 41 9689439 9731120 9769836 9805577 9838329 19 42 9690158 9731789 9770456 9806147 9838850 18 43 9690879 9732458 9771075 9806716 9839370 17 44 9691593 9733126 9771693 9807285 9839889 16 45 9692309 9733793 9772311 9807853 9840407 15 46 9693025 9734459 9772928 9808420 9840924 14 47 9693740 9735124 9773544 9808986 9841440 13 48 9694454 9735789 9774159 9809551 9841956 12 49 9695167 9736453 9774773 9810116 9842471 11 50 9695879 9737116 9775387 9810680 9842985 10 51 9696590 9737778 9776000 9811243 9843498 9 52 9697301 9738439 9776612 9811805 9844010 8 53 9698011 9739099 9777223 9812366 9844521 7 54 9698720 9739759 9777833 9812926 9845032 6 55 9699428 9740418 9778442 9813486 9845542 5 56 9700135 9741076 9779050 9814045 9846051 4 57 9700842 9741733 9779658 9814603 9846559 3 58 9701548 9742389 9780265 9815160 9847066 2 59 9702253 9743045 9780871 9815716 9847572 1 60 9702957 9743700 9781476 9816272 9848078 0  14 13 12 11 10   The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Quadrant. 80 81 82 83 84 The minutes of the degrees of the Quadrant for right Sins of the compliments of the same Quadrant. 0 9848078 9876883 9902681 9925461 9945219 60 1 9848583 9877338 9903085 9925816 9945523 59 2 9849087 9877792 9903489 9926109 9945826 58 3 9849590 9878245 990389● 9926521 9946128 57 4 9850092 9878697 9904294 9926873 9946429 56 5 9850593 9879148 9904695 9927224 9946729 55 6 9851093 9879598 9905095 9927574 9927028 54 7 9851593 9880048 9905494 9927923 9947337 53 8 9852092 9880497 9905893 9928271 9947625 52 9 9852590 9880945 9906291 99●8618 9947622 51 10 9853087 9881392 9906688 9928965 9948218 50 11 9853583 9881838 9907084 9929311 9948513 49 12 9854079 9882283 9907479 9929656 9948807 48 13 9854574 9882728 9907873 9930000 9949100 47 14 9855068 9883172 9908266 9930343 9949393 46 15 9855561 9883615 9908659 9930685 9949685 45 16 9856053 9884057 9909051 9931026 9949976 44 17 9856544 9884498 9909442 9931367 9950266 43 18 9857035 9884938 9909832 9931707 9950555 42 19 9857525 9885378 9910221 9932046 9950844 41 20 9858014 9885817 9910610 9932384 9951132 40 21 9858502 9886255 9910998 9932721 9951419 39 22 9858989 9886692 9911385 9933057 9951705 38 23 9859475 9887128 9911771 9933393 9951990 37 24 9859961 9887564 9912156 9933728 9952274 36 25 9860446 9887999 9912540 9934062 9952557 35 26 9860930 9888433 9912923 9934395 9952840 34 27 9861413 9888866 9913306 9934727 9953122 33 28 9861895 9889298 9913688 9935058 9953403 32 29 9862376 9889729 9914069 9935389 9953683 31 30 9862856 9890159 9914449 9935719 9953962 30 31 9863336 9890588 9914828 9936048 9954240 29 32 9863815 9891017 9915206 9936376 9954518 28 33 9864293 9891445 9915584 9936703 9954795 27 34 9864770 9891872 9915961 9937029 9955071 26 35 9865246 9892298 9916337 9937355 9955346 25 36 9865722 9892723 9916712 9937680 9955620 24 37 9866197 9893147 9917086 9938004 9955893 23 38 9866671 9893571 9917459 9938327 9956165 22 39 9867144 9893994 9917832 9938649 9956437 21 40 9867616 9894416 9918204 9938970 9956708 20 41 9868087 9894837 9918575 9939290 9956978 19 42 9868557 9895257 9918945 9939609 9957247 18 43 9869027 9895677 9919314 9939928 9957515 17 44 9869496 9896096 9919682 9940246 9957782 16 45 9869964 9896514 9920049 9940563 9958049 15 46 9870431 9896931 9920416 9940879 9958315 14 47 9880897 9897347 9920782 9941194 9958580 13 48 9871362 9897762 9921147 9941509 9958844 12 49 9871827 9898177 9921511 9941823 9959307 11 50 9872291 9898591 9921874 9942136 9959370 10 51 9872754 9899004 9922236 9942448 9959632 9 52 9873216 9899416 9922598 9942759 9959893 8 53 9873677 9899827 9922959 9943069 9960153 7 54 9874137 9900237 9923319 9943379 9960412 6 55 9874597 9900646 9923678 9943688 9960670 5 56 9875056 9901055 9924036 9943996 9960927 4 57 9875514 9901463 9924393 9944303 9961183 3 58 9875514 9901463 9924393 9944303 9961183 3 58 9875971 4901870 9924750 9944609 9961438 2 59 9876427 9902276 9925106 9944914 9961693 1 60 9876883 9902681 9925461 9945219 9961947 0  9 8 7 6 5  The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for right Sins of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for right Sins of the Arches of the same Quadrant. 85 86 87 88 89 The minutes of the degrees of the Quadrant for right Sins of the compliments of the same Quadrant. 0 9961947 9975640 9986295 9993908 9998477 60 1 9962200 9975843 9986447 9994009 9998527 59 2 9962452 9976045 9986598 9994109 9998576 58 3 9962703 9976246 9986748 9994208 9998625 57 4 9962954 9976446 9986897 9994307 9998673 56 5 9963204 9976645 9987045 9994405 9998720 55 6 9963453 9976843 9987193 9994502 9998766 54 7 9963701 9977040 9987340 9994598 9998811 53 8 9963948 9977237 9987486 9994693 9998855 52 9 9964194 9977433 9987631 9994787 9998899 51 10 9964440 9977628 9987775 9994881 9998942 50 11 9964685 9977822 9987918 9994974 9998984 49 12 9964929 9978015 9988061 9995066 9999025 48 13 9965172 9978207 9988203 9995157 9999065 47 14 9965414 9978398 9988344 9995247 9999104 46 15 9965655 9978589 9988484 9995336 9999143 45 16 9965895 9978779 9988623 9995424 9999181 44 17 9966235 9978968 9988761 9995512 9999218 43 18 9966374 9979156 9988899 9995599 9999254 42 19 9966612 9979343 9989036 9995685 9999289 41 20 9966849 9979530 9989172 9995770 9999323 40 21 9967085 9979716 9989307 9995854 9999356 39 22 9967320 9979901 9989441 9995937 9999389 38 23 9967555 9980085 9989574 9996019 9999421 37 24 9967789 9980268 9989706 9996101 9999452 36 25 9968022 9980450 9989837 9996182 9999482 35 26 9968254 9980631 9989968 9996262 9999511 34 27 9968485 9980811 9990098 9996341 9999539 33 28 9968715 9980992 9990227 9996419 9999560 32 29 9968944 9981170 9990355 9996496 9999593 31 30 9969173 9981348 9990482 9996573 9999616 30 31 9969401 9981525 9990608 9996649 9999644 29 32 9969628 9981701 9990734 9996724 9999668 28 33 9969854 9981877 9990859 9996798 9999691 27 34 9970079 9982052 9990983 9996871 9999713 26 35 9970304 9982226 9991106 9996943 9999735 25 36 9970528 9982399 9991228 9997014 9999756 24 37 9970751 9982571 9991349 9997085 9999776 23 38 9970973 9982742 9991470 9997155 9999795 22 39 9971194 9982912 9991590 9997224 9999813 21 40 9971414 9983082 9991709 9997292 9999830 20 41 9971633 9983251 9991827 9997359 9999846 19 42 9971851 9983419 9991944 9997425 9999862 18 43 9972069 9983586 9992060 9997491 9999877 17 44 9972286 9983752 9992175 9997556 9999891 16 45 9972502 9983917 9992290 9997620 9999904 15 46 9972717 9984081 9992404 9997683 9999916 14 47 9972931 9984245 9992517 9997745 99999●7 13 48 9973145 9984408 9992629 9997806 9999938 12 49 9973358 9984570 9992740 9997867 9999948 11 50 9973570 9984731 9992850 9997927 9999957 10 51 9973781 9984891 9992960 9997986 9999965 9 52 9973991 9985050 9993069 9998044 9999972 8 53 9974200 9985209 9993177 9998101 9999978 7 54 9974408 9985367 9993284 9998157 9999984 6 55 9974615 9985524 9993390 9998212 9999989 5 56 9974822 9985680 9993495 9998267 9999993 4 57 9975028 9985835 9993599 9998321 9999996 3 58 9975233 9985989 9993703 9998374 9999998 2 59 9975437 9986143 9993806 9998426 9999999 1 60 9975640 9986295 9993908 9998477 1000000 0  4 3 2 1 0   The degrees of the Quadrant for right Sins of the compliments of the Arches of the same Quadrant.    

THE TABLE OF TANGENTS, OTHERWISE CALLED THE FRVITFULL TABLE.  
The Table of Tangents.  
 The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadarnt. 0 1 2 3 4 The minutes of the degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant. 0 0000 174550 349207 524078 699269 60 1 2909 177459 352120 526995 702193 59 2 5818 180369 355033 529911 705116 58 3 8727 183279 357945 532828 708039 57 4 11636 186189 360858 535745 710962 56 5 14544 189100 363770 538663 713886 55 6 17452 192010 366683 541580 716809 54 7 20361 194920 369596 544498 719733 53 8 22270 197820 372508 547415 722657 52 9 2●17● ●00●●● 37●421 550●33 725580 51 10 2●08● ●03●50 37●334 553251 728504 50 11 31906 206561 381247 556169 731428 49 12 34905 302471 384100 559087 734353 48 13 37814 212381 387073 562005 737277 47 14 40723 215291 389987 764923 74020● 46 15 43632 218201 392900 567841 743127 45 16 46541 221111 39●814 570759 746052 44 17 49450 224022 398727 573678 748978 43 18 52359 226932 401641 576596 751903 42 19 55268 229842 404554 579514 754829 41 20 58177 232752 407468 582433 757754 40 21 61086 235663 410382 585352 760680 39 22 63995 238574 413295 588270 763606 38 23 66904 241485 416209 591189 766532 37 24 69813 244395 419123 594108 769459 36 25 72722 247306 422037 597028 772385 35 26 75631 250217 424951 599947 775311 34 27 78540 253128 427866 602866 778238 33 28 81450 256038 430780 605786 781164 32 29 84359 258949 433694 608705 784091 31 30 87268 261859 436609 611625 787017 30 31 90177 264770 439523 614544 789944 29 32 93086 267681 442438 617464 792871 28 33 95995 270592 445353 620384 795799 27 34 98904 273503 448267 623304 798726 26 35 101814 276414 451182 626225 801653 25 36 104723 279325 454097 629145 804581 24 37 107632 282237 457012 632066 807509 23 38 110541 285148 459927 634986 810437 22 39 113450 288059 462842 637907 813365 21 40 116360 290970 465757 640828 816293 20 41 119269 293882 468672 643749 819221 19 42 122178 296794 471588 646671 822150 18 43 125088 299705 474503 649592 825079 17 44 127997 302617 477419 652514 828008 16 45 130906 305528 480335 655435 830937 15 46 133816 308439 483251 658357 833866 14 47 136725 311351 486166 661278 836795 13 48 139635 314262 489082 664200 839724 12 49 142544 317174 491997 667121 842653 11 50 145454 320085 494913 670043 845583 10 51 148363 322997 497829 672965 848513 9 52 151273 325909 500745 675888 851443 8 53 154182 328821 503662 678810 854374 7 54 159092 331733 506578 681733 857304 6 55 160001 334645 509495 684656 860234 5 56 162911 337558 512411 687578 863164 4 57 165820 340470 515328 690501 866095 3 58 168730 343382 518244 693423 869025 2 59 171640 346295 521161 696346 871956 1 60 174550 349207 524078 699269 874886 0  89 88 87 86 85   The degrees of the Quadrant for the Tangent of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant. 5 6 7 8 9 The minutes of the degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant. 0 874886 1051042 1227846 1405408 1583844 60 1 877817 1053983 1230798 1408374 1586826 59 2 880748 1056924 1233751 1411341 1589808 58 3 883680 1059866 1236704 1414308 1592791 57 4 886611 1062808 1239658 1417275 1595774 56 5 889543 1065750 1242612 1420242 1598757 55 6 892475 1068692 1245566 1423210 1601740 54 7 895407 1071634 1248520 1426178 1604723 53 8 898339 1074576 1251474 1429146 1607707 52 9 901271 1077518 1254428 1432115 1610691 51 10 904204 1080461 1257383 1435084 1613675 50 11 907137 1083404 1260338 1438053 1616660 49 12 910070 1086347 1263293 1441022 1619645 48 13 913003 1089291 1266249 1443992 1622630 47 14 915936 1092234 1269205 1446961 1625615 46 15 918870 1095178 1272161 1449931 1627601 45 16 921804 1098122 1275117 1452901 1631587 44 17 924738 1101066 1278073 1455871 1634573 43 18 927771 1104010 1281029 1458842 1637560 42 19 930605 1106954 1283986 1461813 1640547 41 20 933539 1109899 1286943 1464784 1643534 40 21 936473 1112844 1289900 1467755 1646522 39 22 939407 1115789 1292857 1470727 1649510 38 23 942342 1118734 1295815 1473699 1652499 37 24 945277 1121680 1298773 1476671 1655488 36 25 948212 1124625 1301731 1479644 1658477 35 26 951147 1127571 1304689 1482617 1661466 34 27 954083 1130517 1307648 1485590 1664456 33 28 957019 1133463 1310607 1488563 1667446 32 29 959954 1136409 1313566 1491536 1670436 31 30 962890 1139355 1316525 1494510 1673426 30 31 965826 1142302 1319485 1497484 1676417 29 32 968763 1145249 1322445 1500458 1679408 28 33 971699 1148196 1325405 1503433 1682399 27 34 974636 1151144 1328365 1506408 1685390 26 35 977573 1154092 1331325 1509383 1688382 25 36 980509 1157040 1334285 1512358 1691374 24 37 983446 1159988 1337246 1515334 1694366 23 38 986383 1162936 1340207 1518310 1697358 22 39 989320 1165884 1343168 1521286 1700351 21 40 992257 1168832 1346129 1524262 1703344 20 41 995195 1171781 1349091 1527239 1706337 19 42 998133 1174730 1352053 1530216 1709331 18 43 1001072 1177679 1355015 1533193 1712325 17 44 1004010 1180628 1357977 1536170 1715319 16 45 1006949 1183577 1360940 1539148 1718313 15 46 1009887 1186527 1363903 1542126 1721308 14 47 1012825 1189477 1366866 1545104 1724304 13 48 1015763 1192427 1369830 1548082 1727300 12 49 1018702 1195377 1372793 1551061 1730296 11 50 1021641 1198328 1375757 1554040 1733292 10 51 1024580 1201279 1378721 1557019 1736287 9 52 1027519 1204230 1381686 1559999 1739284 8 53 1030459 1207181 1384650 1562979 1742281 7 54 1033399 1210132 1387615 1565959 1745278 6 55 1036339 1213084 1390580 1568939 1748275 5 56 1039279 1216036 1393545 1571920 1751273 4 57 1042219 1218988 1396510 1574901 1754271 3 58 1045160 1221940 1399476 1577882 1757270 2 59 1048101 1224892 1402442 1580863 1760269 1 60 1051042 1227845 1405408 1583844 1763268 0  84 83 82 81 80   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant. 10 11 12 13 14 The minutes of the degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant. 0 1763268 1943803 2125565 2308682 2493280 60 1 1766268 1946822 2128605 2311746 2496370 59 2 1769268 1949841 2131646 2314810 2499411 58 3 1772268 1952861 2134687 2317875 2502552 57 4 1775269 1955881 2137729 2320940 2505643 56 5 1778270 1958901 2140771 2324006 2508735 55 6 1781271 1961922 2143814 2327072 2511827 54 7 1784272 1964943 2146857 2330139 2514920 53 8 1787274 1967964 2149900 2333206 2518013 52 9 1790276 1970985 2152944 2336273 2521106 51 10 1793278 1974007 2155988 2339341 2524200 50 11 1796281 1977029 2159032 2342419 2527294 49 12 1799284 1980052 2162077 2345478 2530389 48 13 1802287 1983075 2165122 2348547 2533484 47 14 1805291 1986098 2168167 2351616 2536580 46 15 1808295 1989122 2171213 2354686 2539676 45 16 1811299 1992146 2174259 2357757 2542773 44 17 1814303 1995171 2177306 2360828 2545870 43 18 1817308 1998196 2180352 2363899 2548968 42 19 1820313 2001221 2183400 2366971 2552066 41 20 1823318 2004247 2186448 2370043 2555165 40 21 1826324 2007273 2189496 2373116 2558264 39 22 1829329 2010299 2192544 2376189 2561364 38 23 1832335 2013326 2192544 2379263 2564464 37 24 1835342 2016353 2195593 2382337 2567564 36 25 1838349 2019380 2201692 2385411 2570665 35 26 1841357 2022408 2204742 2388486 2573766 34 27 1844365 2025436 2207792 2391561 2576868 33 28 1847373 2028464 2210843 2394636 2579970 32 29 1850382 2031493 2213894 2397712 2583073 31 30 1853301 2034522 2216946 2400788 2586176 30 31 1856400 2037552 2219998 2403865 2589280 29 32 1859409 2040582 2223051 2406942 2592384 28 33 1862419 2043612 2226104 2410020 2595489 27 34 1865429 2046643 2229157 2413098 2598594 26 35 1868439 2049674 2232211 2416176 2601700 25 36 1871449 2052705 2235265 2419255 2604806 24 37 1874460 2055737 2238319 2422334 2607912 23 38 1877471 2058769 2241374 2425414 2611019 22 39 1880482 2061801 2244429 2428494 2614126 21 40 1883494 2064834 2247485 2431574 2617234 20 41 1886506 2067867 2250541 2434655 2620342 19 42 1889516 2070900 2253597 2437736 2623451 18 43 1892531 2073934 2256654 2440818 2626560 17 44 1895544 2076968 2259711 2443900 2629670 16 45 1898558 2080002 2262769 2446983 2632780 15 46 1901572 2083037 2265827 2450066 2635891 14 47 1904586 2086073 2268885 2453150 2639002 13 48 1907601 2089109 2271944 2456234 2642114 12 49 1910616 2092145 2275003 2459319 2645226 11 50 1913632 2095182 2278063 2462404 2648339 10 51 1916648 2098219 2281123 2465490 2651452 9 52 1999664 2101256 2284183 2468576 2654566 8 53 1922680 2104293 2287244 2471662 2657680 7 54 1925697 2107331 2290305 2474749 2660795 6 55 1928714 2110369 2293367 2477836 2663910 5 56 1931731 2113407 2296429 2480924 2667026 4 57 1934749 2116446 2299492 2484012 2670142 3 58 1937767 2119485 2302555 2487101 2673258 2 59 1940785 2122525 2305618 2490191 2676375 1 60 1943803 2125565 2308682 2493280 2679492 0  79 78 77 76 75   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant. 15 16 17 18 19 The minutes of the degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant. 0 2679492 2867453 3057307 3249197 3443276 60 1 2662610 2870601 3060487 3252413 3446530 59 2 2685728 2873749 3063669 3255630 3449785 58 3 2688847 2876898 3066851 3258848 3453040 57 4 2691966 2880048 3070034 3262066 3456296 56 5 2695086 2883198 3073218 3265285 3459553 55 6 2698206 2886349 3076402 3268504 3462810 54 7 2701327 2889501 3079587 3271724 3466068 53 8 ●704448 2892653 3082772 3274944 3469326 52 9 2707570 2895806 3085958 3278165 3472585 51 10 2710693 2898960 3085144 3281387 3475845 50 11 2713816 2902114 3092331 3284609 3479105 49 12 2716940 2905268 3095518 3287832 3482366 48 13 2720064 2908423 3198706 3291055 3485628 47 14 2723189 2911578 3101895 3294280 3488891 46 15 2726314 2914734 3105084 3297505 3492154 45 16 2729439 2917890 3108274 3300731 3495418 44 17 2732565 2921047 3111464 3303957 3498683 43 18 2735691 2924204 3114655 3307184 3501949 42 19 2738818 2927362 3117846 3310411 3505215 41 20 2741945 2930520 3121038 3313639 3508482 40 21 2745073 2933679 3124230 3316868 3511749 39 22 2748201 2936839 3127423 3320097 3515017 38 23 2751330 2939999 3130617 3323327 3518286 37 24 2754459 2943160 3133811 3326558 3521555 36 25 2757589 2946321 3137006 3329789 3524825 35 26 2760729 2946483 3140201 3333020 3528096 34 27 2763850 2952645 3143397 3336252 3531368 33 28 2766981 2955808 3146594 3339485 3534640 32 29 2770113 2958971 3149791 3342719 3537913 31 30 2773245 2962135 3152989 3345953 3541186 30 31 2776378 2965299 3156187 3349188 3544460 29 32 2779511 2968464 3159386 3352423 3547735 28 33 2782645 2971629 3162585 3355659 3515010 27 34 2785779 2974795 3165785 3358896 3554286 26 35 2788914 2977962 3168986 3362133 3557563 25 36 2792050 2981129 3172187 3365371 3560840 24 37 2795186 2984297 3175389 3368610 3564118 23 38 2798323 2987465 3178591 3371850 3567397 22 39 2801460 2990634 3181794 3375090 3570676 21 40 2804597 2993804 3184998 3378331 3573956 20 41 2807735 2996973 3188202 3381572 3577237 19 42 2810873 3000143 3191407 3384814 3580519 18 43 2814012 3003314 3194613 3388057 3583801 17 44 2817151 3006486 3197819 3391300 3587084 16 45 2820291 3009658 3201026 3394544 3590367 15 46 2823432 3012831 3204233 3397798 3593651 14 47 2826573 3016004 3207441 3401033 3596936 13 48 2829714 3019178 3210649 3404279 3600221 12 49 2832856 3022353 3213858 3407525 3603507 11 50 2835999 3025528 3217067 3410772 3606794 10 51 2839142 3028703 3220277 3414020 3610082 9 52 2842286 3031879 3223488 3417268 3613370 8 53 2845430 3035055 3226699 3420517 3616659 7 54 2848575 3038232 3229911 3423766 3619949 6 55 2851720 3041410 3233124 3427016 3623239 5 56 2854866 3044588 3236337 3430267 3626530 4 57 2858012 3047767 3239551 3433518 3629822 3 58 2861159 3050946 3242766 3436770 3633115 2 59 2864306 3054126 3245981 3440023 3636408 1 60 2867453 3057307 3249197 3443276 3639702 0  74 73 72 71 70   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant. 20 21 22 23 24 The minutes of the degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant. 0 3639702 3838640 4040262 4244748 4452286 60 1 3642997 3841978 4043647 4248182 4455772 59 2 3646293 3845316 4047031 4251617 4459259 58 3 3649589 3848655 4050416 4255052 4462747 57 4 3652886 3851995 4053802 4258488 4466236 56 5 3656183 3855336 4057189 4261925 4469726 55 6 3659481 3858678 4060577 4265363 4473216 54 7 3662780 3862020 4063966 4268801 4476707 53 8 3666079 3865363 4067356 4265363 4480199 52 9 3669379 3868707 4070747 4268801 4483692 51 10 3672680 3872052 4074139 4272240 4487186 50 11 3675982 3875397 4077531 4275680 4490681 49 12 3679284 3878743 4080924 4279121 4494177 48 13 3682587 3882090 4084318 4282563 4497674 47 14 3685891 3885438 4087713 4286006 4501172 46 15 3689195 3888787 4091109 4289450 4504671 45 16 3692500 3892136 4094506 4292895 4508171 44 17 3695806 3895486 4097903 4296340 4511672 43 18 3699113 3898837 4101301 4299786 4515173 42 19 3702420 3902188 4104699 4303233 4518675 41 20 3705728 3905540 4108097 4306681 4522178 40 21 3709037 3908893 4111497 4310130 4525682 39 22 3712347 3912247 4114898 4313580 4529187 38 23 3715657 3915601 4118300 4317031 4532693 37 24 3718968 3918956 4121703 4327387 4536200 36 25 3722279 3922312 4125107 4330841 4539708 35 26 3725591 3925669 4128511 4334296 4543217 34 27 3728904 3929027 4131916 4337752 4546727 33 28 3732218 3932385 4135322 4341209 4550238 32 29 3735533 3935744 4138728 4344666 4553750 31 30 3738848 3939104 4142135 4348124 4557264 30 31 3742164 3942465 4145544 4351583 4560778 29 32 3745480 3945826 4148953 4355043 4564293 28 33 3748797 3949188 4152363 4358504 4567809 27 34 3752115 3952551 4155773 4361966 4571326 26 35 3755434 3955915 4159184 4365429 4574843 25 36 3758753 3959280 4162596 4368893 4578361 24 37 3762073 3962646 4166009 4372357 4581880 23 38 3765394 3966012 4169423 4375822 4585400 22 39 3768716 3969379 4172838 4379288 4588921 21 40 3772038 3972746 4176255 4382755 4592443 20 41 3775361 3976114 4179672 4386223 4595966 19 42 3778685 3979483 4183090 4389692 4599490 18 43 3782010 3982853 4186509 4393162 4603015 17 44 3785335 3986224 4189928 4396633 4606541 16 45 3788661 3989596 4193348 4400105 4610068 15 46 3791988 3992969 4196769 4403578 4613596 14 47 3795315 3996342 4200191 4407051 4617125 13 48 3798643 3999716 4203613 4410525 4620654 12 49 3801972 4003090 4207036 4414000 4624184 11 50 3805302 4006465 4210460 4417476 4627715 10 51 3808632 4009841 4213885 4420953 4631247 9 52 3811962 4013217 4217311 4424432 4634780 8 53 3815295 4016594 4220738 4427910 4638314 7 54 3818628 4019972 4224165 4431390 4641849 6 55 3821961 4023351 422759● 4434871 4645385 5 56 3825295 4026731 4231022 4438352 4648922 4 57 3828630 4030112 4234452 4441834 4652460 3 58 3831966 4033494 4237883 4445317 4655999 2 59 3835303 4036877 4241315 4448801 4659540 1 60 3838640 4040262 4244748 4452286 4663081 0  69 68 67 66 65   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant. 25 26 27 28 29 The minutes of the degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant. 0 4663081 4877328 5095254 5317094 5543090 60 1 4666623 4880930 5098919 5320826 5546893 59 2 4670166 4884533 5102585 5324559 5550697 58 3 4673710 4888137 5106252 5328293 5554503 57 4 4677255 4891742 5109920 5332028 5558310 56 5 4680801 4895347 5113589 5335765 5562118 55 6 4684348 4898953 5117259 5339503 5565927 54 7 4687896 4902560 5120930 5343242 5569738 53 8 4691444 4906168 5124602 5346982 5573550 52 9 4694993 4909777 5128275 5350723 5577363 51 10 4698543 4913387 5131949 5354465 5581177 50 11 4702094 4916998 5135625 5358209 5584993 49 12 4705646 4920610 5139302 5361954 5588810 48 13 4709199 4924223 5142980 5365700 5592628 47 14 4712753 4927838 5146659 5369447 5596447 46 15 4716308 4931454 5150339 5373195 5600268 45 16 4719864 4935071 5154020 5376944 5604090 44 17 4723422 4938689 5157702 5380694 5607913 43 18 4726981 4942308 5161385 5384445 5611737 42 19 4730541 4945928 5165069 5388198 5615562 41 20 4734102 4949549 5168755 5391952 5619388 40 21 4737664 4953171 5172442 5395707 5623216 39 22 4741227 4956794 5176130 5399463 5627045 38 23 4744790 4960418 5179819 5403221 5630875 37 24 4748354 4964043 5183509 5406980 5634707 36 25 4751919 4967669 5187200 5410740 5638540 35 26 4755485 4971296 5190892 5414501 5642374 34 27 4759052 4974924 5194585 5418263 5646210 33 28 4762620 4978553 5198279 5422026 5650047 32 29 4766189 4982184 5201974 5425791 5653885 31 30 4769759 4985816 5205670 5429557 5657725 30 31 4773330 4989448 5209368 5433324 5661566 29 32 4776902 4993081 5213067 5437092 5665408 28 33 4780475 4996716 5216767 5440861 5669251 27 34 4784049 5000352 5220468 5444632 5673096 26 35 4787624 5003989 5224170 5448404 5676942 25 36 4791200 5007627 5227873 5452177 5680789 24 37 4794777 5011266 5231577 5455951 5684637 23 38 4798355 5014906 5235283 5459726 5688486 22 39 4801934 5018547 5238990 5463503 5692337 21 40 4805515 5022189 5242698 5467281 5696189 20 41 4809096 5025832 5246407 5471060 5700043 19 42 4812678 5029476 5250117 5474840 5703898 18 43 4816261 5033121 5253828 5478621 5707754 17 44 4819845 5036767 5257540 5482404 5711611 16 45 4823430 5040414 5261254 5486188 5715469 15 46 4827016 5044062 5264969 5489973 5719329 14 47 4830603 5047712 5268685 5493759 5723190 13 48 4834191 5051363 5272402 5497546 5727052 12 49 4837780 5055015 5276120 5501335 5730916 11 50 4841371 5058668 5279839 5505125 5734781 10 51 4844962 5062322 5283959 5508916 5738647 9 52 4848554 5065977 5287280 5512708 5742515 8 53 4852147 5069633 5291003 5516501 5746384 7 54 4855741 5073290 5294727 5520296 5750254 6 55 4859336 5076948 5298452 5524092 5754125 5 56 4862932 5080607 5302178 5527889 5757998 4 57 4866529 5084267 5305905 5531687 5761872 3 58 4870127 5087928 5309633 5535487 5765747 2 59 4873727 5091590 5313363 5539288 5769624 1 60 4877328 5095254 5317094 5543090 5773502 0  64 63 62 61 60   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant. 30 31 32 33 34 The minutes of the degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant. 0 5773502 6008606 6248693 6494076 6745085 60 1 5777381 6012566 6252738 6498212 6749318 59 2 5781262 6016528 6256785 6502350 6753553 58 3 5785144 6020491 6260834 6506489 6757789 57 4 5789027 6024455 6264884 6510630 6762027 56 5 5792911 6028420 6268935 6514773 9766267 55 6 5796797 6032387 6272988 6518917 6770508 54 7 5800684 6036355 6277042 6523063 6774751 53 8 5804572 6040324 6281098 6527200 6778996 52 9 5808462 6044295 6285155 6531359 6783243 51 10 5812353 6048267 6289214 6535510 6787491 50 11 5816245 6052241 6293274 6539662 6791741 49 12 5820139 6056216 6297336 6543816 6795993 48 13 5824034 6060193 6301399 6547971 6800246 47 14 5827930 6064171 6305464 6552128 6804501 46 15 5831828 6068150 6309530 6556287 6808758 45 16 5835727 6072131 6313598 6560447 6813016 44 17 5839627 6076113 6317667 6564609 6817276 43 18 5843528 6080096 6321738 6568772 6821538 42 19 5847431 6084081 6325810 6572937 6825801 41 20 5851335 6088067 6329883 6577103 6830066 40 21 5855241 6092055 6333958 6581271 6834333 39 22 5859148 6096044 6338034 6585440 6838602 38 23 5863056 6100035 6342112 6589611 6842872 37 24 5866966 6104027 6346191 6593784 6847144 36 25 5870877 6108020 6350272 6597958 6851417 35 26 5874489 6112015 6354355 6602134 6855692 34 27 5878702 6116011 6358439 6606312 6859969 33 28 5882617 6120009 6362525 6610491 6864247 32 29 5886533 6124008 6366613 6614672 6868527 31 30 5890450 6128008 6370702 6618855 6872809 30 31 5894369 6132010 6374792 6623039 6877093 29 32 5898289 6136013 6378884 6627225 6881379 28 33 5902211 6140018 6382977 6631413 6885666 27 34 5906134 6144024 6387072 6635603 6889955 26 35 5910058 6148032 6391169 6639792 6894246 25 36 5913984 6152041 6395267 6643984 6898539 24 37 5917911 6156052 6399366 6648178 6902833 23 38 5921839 6160064 6403467 6652373 6907129 22 39 5925769 6164077 6407569 6656570 6911426 21 40 5929700 6168092 6411673 6660768 6915725 20 41 5933633 6172108 6415779 6664968 6920026 19 42 5937567 6176126 6419886 6669170 6924329 18 43 5941502 6180147 6423995 6673373 6928634 17 44 5945438 6184168 6428105 6677578 6932940 16 45 5949376 6188190 6432216 6681785 6937248 15 46 5955315 6192213 6436329 6685994 6941558 14 47 5957255 6196237 6440444 6690204 6945869 13 48 5961197 6200263 6444560 6694416 6950182 12 49 5965140 6204290 6458678 6698630 6954497 11 50 5969084 6208319 6452798 6702845 6958813 10 51 5973030 6212350 6456919 6707062 6963131 9 52 5976776 6216382 6461042 6711281 6967451 8 53 5980926 6220416 6465166 6715501 6971773 7 54 5984876 6224451 6469292 6719723 6976097 6 55 5988827 6228488 6473419 6723946 6980423 5 56 5992780 6232526 6477548 6728171 6984750 4 57 5996734 6246566 6481678 6732397 6989079 3 58 6000690 6240607 6485809 6736625 6993409 2 59 6004647 6244649 6489942 6740854 6997741 1 60 6008606 6248693 6494076 6745085 7002075 0  95 58 57 56 55   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadarnt. 35 36 37 38 39 The minutes of the degrees of the Quadrant for the Tangents of the compliment of the Arches of the same Quadrant. 0 7002075 7265424 7535541 7812856 8097840 60 1 7006411 7269869 7540103 7817542 8102658 59 2 7010749 7274316 7544667 7822230 8107478 58 3 7015088 7278765 7549233 7826920 8112300 57 4 7019429 7283216 7553801 7831612 8117124 56 5 7023772 7287669 7558371 7836306 8121951 55 6 7028117 7292124 7562943 7841002 8126780 54 7 7032463 7296581 7567517 7845700 8131611 53 8 7036811 7301040 7572093 7850400 8136444 52 9 7041161 7305501 7576670 7855102 8141280 51 10 7045513 7309563 7581249 7859807 8146118 50 11 7049867 7314427 7585830 7864514 8150958 49 12 7054223 7318893 7590413 7869223 8155801 48 13 7058581 7323361 7594999 7873934 8160646 47 14 7062940 7327831 7599587 7872647 8165493 46 15 7067301 7332303 7604177 7883363 8170343 45 16 7071664 7336777 7608769 7888081 8175195 44 17 7076029 7341253 7613363 7892801 8180049 43 18 7070395 7345731 7617959 7897523 8184905 42 19 7084763 7350210 7622557 7902247 8189764 41 20 7089133 7354691 7627157 7906973 8194625 40 21 7093505 7359174 7631759 7911702 8199488 39 22 7097879 7363659 7636363 7916433 8204354 38 23 7102254 7368146 7640969 7921166 8209222 37 24 7106631 7372635 7645577 7925901 8214092 36 25 7111010 7377126 7650187 7930638 8218965 35 26 7115391 7381619 7654799 7935378 8223840 34 27 7119773 7386114 7659413 7940120 8228717 33 28 7124167 7390611 7664030 7944864 8233597 32 29 7128543 7395110 7668649 7949610 8238479 31 30 7132931 7399610 7673270 7954358 8243363 30 31 7137321 7404112 7677893 7959109 8248250 29 32 7141713 7408616 7682518 7963862 8253139 28 33 7146106 7413122 7687145 7968617 8258031 27 34 7150501 7417630 7691774 7973374 8262925 26 35 7154898 7422140 7696405 7978133 8267821 25 36 7159297 7426652 7701038 7982895 8272720 24 37 7163698 7431167 7705673 7987659 8277621 23 38 7168100 7435684 7710310 7992425 8282524 22 39 7172504 7440203 7714949 7997193 8287429 21 40 7176910 7444724 7719590 8001963 8292337 20 41 7181318 7449246 7724233 8006736 8297247 19 42 7185728 7453770 7728878 8011511 8302160 18 43 7190140 7458296 7733525 8016288 8307075 17 44 7194554 7462824 7738175 8021067 8311992 16 45 7198970 7476354 7742827 8025849 8316912 15 46 7203387 74718●6 7747481 8030633 8321834 14 47 7207806 7476420 7752137 8035419 8326759 13 48 7212227 7480956 7756795 8040207 8331686 12 49 7216650 7485494 7761455 8044997 8336615 11 50 7221075 7490033 7766117 8049790 8341547 10 51 7225502 7494574 7770781 8054585 8346481 9 52 7229931 7499117 7775447 8059382 8351418 8 53 7234362 7503663 7780116 8064181 8356357 7 54 7238794 7508211 7784787 8068983 8361298 6 55 7243228 7512761 7789460 8073787 8366242 5 56 7247664 7517313 7794135 8078593 8371188 4 57 7252102 7521867 7798812 8083401 8376136 3 58 7256541 7526423 7803491 8088212 8381087 2 59 7260982 7530981 7808172 8093025 8386040 1 60 7265424 7535541 7812856 8097840 8390996 0  54 53 52 51 50   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for Tangents of the Arches of the same Quadrant. 40 41 42 43 44 The minutes of the degrees of the Quadrant for the Tangents of the compliment of the Arches of the same Quadrant. 0 8390996 8692867 9004040 9325151 9656888 60 1 8395954 8697975 9009308 9330591 9662511 59 2 8400915 8703085 9014579 9336034 9668137 58 3 8405878 8708198 901985● 9341480 9673766 57 4 8410844 8713344 9025130 9346929 9679398 56 5 8415812 8718433 9030410 9352381 9685034 55 6 8420782 8723555 9035693 9357835 9690674 54 7 8425754 8728679 9040978 9363292 9696315 53 8 8430729 8733806 9046266 9368752 9701960 52 9 8435706 8738935 9051557 9374215 9707609 51 10 8440686 8744067 9056850 9379682 9713261 50 11 8445668 8749201 9062146 9385152 9718916 49 12 8450653 8754338 9067445 9390625 9724574 48 13 8455640 8759478 9072747 9396101 9730235 47 14 8460630 8764620 9078052 9401580 9735900 46 15 8465622 8769764 9083360 9407062 9741568 45 16 8470617 8774911 9088670 9412547 9747239 44 17 8475614 8780061 9093983 9418034 9752913 43 18 8480614 8785214 9099299 9423524 9758591 42 19 8485617 8790369 9104618 9429017 9764272 41 20 8490622 8795527 9109940 9434513 9769956 40 21 8495629 8800688 9115265 9440012 9775643 39 22 8500639 8805851 9120593 9445514 9781334 38 23 8505651 8811017 9125923 9451019 9787028 37 24 8510666 8816186 9131256 9456528 9792725 36 25 8515683 8821357 9136592 9462040 9798425 35 26 8520703 8826531 9141930 9467555 9804128 34 27 8525725 8831708 9147271 9473072 9809835 33 28 8530750 8836887 9152615 9478594 9815545 32 29 8535777 8842069 9157962 9484118 9821258 31 30 8540806 8847253 9163312 9489645 9826974 30 31 8545838 8852440 9168665 9495175 9832694 29 32 8550872 8857630 9174021 9400708 9838417 28 33 8555909 8862822 9179380 9506244 9844143 27 34 8560949 8868017 9184741 9511783 9849872 26 35 8565991 8873015 9190105 9517325 9855605 25 36 8571036 8878415 9195472 9522870 9861341 24 37 8576083 8883628 9200842 9528419 9867180 23 38 8581133 8888824 9206215 9533971 9872922 22 39 8586185 8899033 9211590 9539526 9878668 21 40 8591239 8899244 9216968 9545084 9884317 20 41 8596296 8904458 9222349 9550645 9890070 19 42 8601355 8909675 9227733 9556209 9895826 18 43 8606417 8914894 9233120 9561776 9901585 17 44 8611482 8920116 9238510 9567346 9907347 16 45 8616549 8925341 9243903 9572919 9913113 15 46 8621619 8930568 9249299 9578495 9918882 14 47 8626692 8935798 9254698 9584074 9924654 13 48 8631767 8941031 9260100 9589656 9930430 12 49 8636845 8946267 9265505 9595241 9936209 11 50 8641926 8951506 9270913 9600830 9941991 10 51 8647009 8956747 9276324 9606422 9947777 9 52 8652095 8961991 9281738 9612017 9953566 8 53 8657683 8967238 9287155 9617615 9959359 7 54 8662273 8972487 9292574 9623216 9965155 6 55 8667366 8977739 9297996 9628820 9970954 5 56 8672461 8982994 9303421 9634427 9976756 4 57 8677559 8988252 9308849 9640037 9982562 3 58 8682659 8993512 9314280 9645651 9988371 2 59 8687762 8998775 9319714 9651268 9994184 1 60 8692867 9004040 9325151 9656888 10000000 0  49 48 47 46 45   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for Tangents of the Arches of the same Quadrant. 45 46 47 48 The minutes of the degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant. 0 10000000 10355302 10723686 11106124 60 1 10005820 10367332 10729942 11112623 59 2 10011643 10367365 10736202 11119126 58 3 10017469 10373402 10742466 11125634 57 4 10023299 10379443 10748734 11132146 56 5 10029132 10385487 10755006 11138662 55 6 10034968 10391535 10761282 11145182 54 7 10040808 10397587 10767562 11151706 53 8 10046651 10403643 10773845 11158235 52 9 10052497 10409702 10780132 11164768 51 10 10058347 10415765 10786423 11171305 50 11 10064201 10421832 10792718 11177846 49 12 10070058 10427902 10799017 11184392 48 13 10075918 10433976 10805320 11190942 47 14 10081782 10340054 10811627 11197496 46 15 10087649 10446135 10817938 11204054 45 16 10093520 10452220 10824253 11210617 44 17 10099394 10458309 10830572 11217184 43 18 10105272 10464401 10836895 11223755 42 19 10111153 10470497 10843222 11230330 41 20 10117038 10476597 10849554 11236910 40 21 10122926 10482701 10855889 11243494 39 22 10128818 10488808 10862228 11250082 38 23 10134713 10494919 10868571 11256675 37 24 10140611 10501034 10874918 11263272 36 25 10146513 10507153 10881269 11269873 35 26 10152418 10513275 10887624 11276478 34 27 10158327 10519401 10893983 11283088 33 28 10164239 10525531 10900346 11289702 32 29 10170154 10531664 10906713 11296321 31 30 10176073 10537801 10913084 11302944 30 31 10181996 10543942 10919459 11309571 29 32 10187922 10550087 10925838 11316203 28 33 10193852 10556235 10932221 11322899 27 34 10199785 10562387 10938608 11329480 26 35 10205722 10568543 10945000 11336125 25 36 10211663 10574703 10951396 11342774 24 37 10217607 10580867 10957796 11349428 23 38 10223555 10587034 10964200 11356086 22 39 10229506 10593205 10970608 11362748 21 40 10235460 10599280 10977020 11369415 20 41 10241418 10605559 10983436 11376086 19 42 10247380 10611742 10989856 11382762 18 43 10253345 10617929 10996280 11389442 17 44 10259314 10624119 11002708 11396126 16 45 10265286 10630313 11009140 11402815 15 46 10271262 10636511 11015577 11409508 14 47 10277242 10642713 11022028 11416206 13 48 10283225 10648919 11028463 11422908 12 49 10289212 10655128 11034912 11429615 11 50 10295202 10661341 11041365 11436326 10 51 10301196 10667558 11047832 11443042 9 52 10307193 10673779 11054283 11449762 8 53 10313194 10680004 11060748 11456487 7 54 10319199 10686233 11067218 11463216 6 55 10325207 10692466 11073692 11469950 5 56 10331219 10698702 11080170 11476688 4 57 10337234 10704942 11086652 11483431 3 58 10343253 10711186 11093138 11490178 2 59 10349276 10717434 11099629 11496929 1 60 10355302 10723686 11106124 11503684 0  44 43 42 41   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadarnt. 49 50 51 52 The minutes of the degrees of the Quadrant for the Tangents of the compliment of the Arches of the same Quadrant. 0 11503684 11917537 12348972 12799416 60 1 11510444 11924580 12356320 12807093 59 2 11517208 11931628 12363673 12814776 58 3 11523977 11938680 12371032 12822465 57 4 11530751 11945737 12378394 12830159 56 5 11537529 11952799 12385762 12837859 55 6 11544312 11959866 12393136 12845565 54 7 11551100 11966938 12400515 12853277 53 8 11557893 11974015 12407999 12860994 52 9 11564691 11981097 12415288 12868717 51 10 11571494 11988183 12422683 12876445 50 11 11578301 11995274 12430083 12884179 49 12 11585112 12002370 12437489 12891919 48 13 11591928 12009471 12444900 12899665 47 14 11598748 12016578 12452317 12907417 46 15 11605572 12023690 12459739 12915175 45 16 11612401 12030807 12467167 12922939 44 17 11619234 12037929 12474600 12930709 43 18 11626072 12045056 12482039 12938485 42 19 11632915 12052188 12489484 12946267 41 20 11639763 12059325 12496934 12954055 40 21 11646615 12066467 12504389 12961848 39 22 11653472 12073614 12511850 12969647 38 23 11660334 12080766 12519316 12977457 37 24 11667200 12087923 12526787 12985263 36 25 11674071 12095085 12534264 12993080 35 26 11680947 12102252 12541746 13000903 34 27 11687827 12109424 12549233 13008732 33 28 11694712 12116601 12556725 13016567 32 29 11701602 12123783 12564222 13024407 31 30 11708497 12130970 12571724 13032253 30 31 11715396 12138162 12579232 13040105 29 32 11722300 12145359 12586746 13047963 28 33 11729208 12152561 12594265 13055827 27 34 11736121 12159768 12601790 13063697 26 35 11743039 12166981 12609321 13071573 25 36 11749962 12174199 12616858 13079455 24 37 11756989 12181422 12624400 13087343 23 38 11763821 12188650 12631948 13095237 22 39 11770758 12195883 12639501 13103138 21 40 11777700 12203121 12647060 13111045 20 41 11784646 12210364 12654624 13118958 19 42 11791597 12217613 12662194 13126877 18 43 11798553 12224867 12669769 13134802 17 44 11805514 12232126 12677350 13142732 16 45 11812479 12239390 12684937 13150668 15 46 11819449 12246659 12692530 13158610 14 47 11826424 12253933 12700128 13166558 13 48 11833404 12261212 12707732 13174512 12 49 11840388 12268496 12715341 13182472 11 50 11847377 12275786 12722956 13190438 10 51 11854371 12283081 12730577 13198411 9 52 11861370 12290381 12738203 13206390 8 53 11868374 12297687 12745835 13214375 7 54 11875383 12304998 12753473 13222367 6 55 11882397 12312314 12761116 13230365 5 56 11889417 12319635 12768765 13238369 4 57 11896438 12326961 12776420 13246379 3 58 11903466 12334293 12784080 13254396 2 59 11910499 12341630 12791745 13262419 1 60 11917537 12348972 12799416 13270448 0  40 39 38 38   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadarnt. 53 54 55 56 The minutes of the degrees of the Quadrant for the Tangents of the compliment of the Arches of the same Quadrant. 0 13270448 13763820 14281480 14825610 60 1 13278483 13772243 14290325 14834916 59 2 13286524 13780673 14299177 14844230 58 3 13294571 13789109 14308037 14853553 57 4 13302624 13797552 14316905 14862884 56 5 13310683 13806002 14325780 14872223 55 6 13318749 13814459 14334662 14881570 54 7 13326821 13822922 14343552 14890925 53 8 13334899 13831392 14352451 14909288 52 9 13342984 13839869 14361354 14909659 51 10 13351075 13848352 14370266 14919038 50 11 13359172 13856842 14379186 14928426 49 12 13367276 13865339 14388113 14937822 48 13 13375386 13873843 14397048 14947226 47 14 13383502 13882354 14405990 14956638 46 15 13391624 13890872 14414939 14966058 45 16 13399753 13899397 14423896 14975486 44 17 13407888 13907930 14432861 14984923 43 18 13416029 13916470 14441833 14994368 42 19 13424177 13925017 14450812 15003821 41 20 13432331 13933571 14459799 15013283 40 21 13440492 13942131 14468794 15022753 39 22 13448659 13950698 14477797 15032231 38 23 13456832 13959272 14486807 15041717 37 24 13465011 13967853 14495825 15051211 36 25 13473197 13976441 14504850 15060714 35 26 13481390 13985035 14513883 15070225 34 27 13489589 13993636 14522924 15079744 33 28 13497794 14002244 14531972 15089271 32 29 13506006 14010859 14541028 15078807 31 30 13514224 14019481 14550091 15108351 30 31 13522449 14028110 14559162 15117903 29 32 13530680 14036746 14568241 15127464 28 33 13538918 14045389 14577327 15137034 27 34 13547162 14054040 14586421 15146612 26 35 13555413 14062698 14595523 15156199 25 36 13563670 14071363 14604633 15165794 24 37 13571834 14080035 14613750 15175398 23 38 13580104 14088715 14622875 15185011 22 39 13588381 14097402 14632007 15194632 21 40 13596764 14106097 14641146 15204261 20 41 13605054 14114798 14650293 15213899 19 42 13613350 14123506 14659449 15223545 18 43 13621653 14132221 14668613 15233200 17 44 13629963 14140923 14677785 15242863 16 45 13638279 14149672 14686965 15252535 15 46 13646602 14158409 14696153 15262216 14 47 13654932 14167153 14705349 15271905 13 48 13663268 14175904 14714553 15281603 12 49 13671610 14184663 14723765 15291309 11 50 13679959 14193429 14732985 15301024 10 51 13688315 14202202 14742212 15310748 9 52 13696677 14210982 14751447 15320481 8 53 13705046 14219769 14760690 15330222 7 54 13713422 14228563 14769941 15339972 6 55 13721805 14237365 14779200 15349730 5 56 13730194 14246174 14788466 15349497 4 57 13738590 14254990 14797740 15369273 3 58 13746993 14263813 14807022 15379057 2 59 13755403 14272643 14816312 15388850 1 60 13763820 14281480 14825610 15398651 0  36 35 34 33   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant. 57 58 59 60 The minutes of the degrees of the Quadrant for the Tangents of the compliment of the Arches of the same Quadrant. 0 15398651 16003347 16642794 17320508 60 1 15408461 16013710 16653766 17332150 59 2 15418280 16024083 16664749 17343804 58 3 15428108 16034466 16675742 17355469 57 4 15437945 16044859 16686746 17367146 56 5 15447791 16055261 16697760 17378834 55 6 15457646 16065673 16708785 17390534 54 7 15467510 16076095 16719820 17402246 53 8 15477382 16086527 16730866 17413969 52 9 15487263 16096968 16741922 17425704 51 10 15497153 16107419 16752989 17437451 50 11 15507052 16117880 16764067 17449210 49 12 15516960 16128351 16775156 17460981 48 13 15526877 16138832 16786256 17472764 47 14 15536803 16149322 16797367 17484559 46 15 15546738 16159822 16808489 17496366 45 16 15556682 16170332 16819621 17508185 44 17 15566636 16180852 16830764 17520026 43 18 15576599 16191381 16841918 17531869 42 19 15586571 16201920 16853083 17543724 41 20 15596552 16212469 16864259 17555591 40 21 15606542 16224028 16875446 17567470 39 22 15616541 16233597 16886644 17579362 38 23 15626549 16244176 16897853 17591266 37 24 15636566 16254766 16909074 17603182 36 25 15646592 16265366 16920306 17615111 35 26 15656627 16275976 16931549 17627052 34 27 15666671 16286596 16942803 17639006 33 28 15676724 16297226 16954068 17650972 32 29 15686786 16307866 16965344 17662951 31 30 15696857 16318516 16976631 17674941 30 31 ●5706938 16329176 16987929 17686945 29 32 15717028 16339847 16999239 17698960 28 33 1572127 16350528 17010560 17710987 27 34 15737235 16361219 17021892 17723027 26 35 15747353 16371920 17033236 17735079 25 36 15757480 16382631 17044591 17747143 24 37 15767616 16393352 17055957 17759220 23 38 15777761 16404083 17067325 17771309 22 39 15787915 16414824 17078714 17783410 21 40 15798078 16425575 17090115 17795524 20 41 15808251 16436337 17101527 17808651 19 42 15818433 16447109 17112950 17819790 18 43 15828625 16457892 17124384 17831942 17 44 15838827 16468685 17135829 17844107 16 45 15849038 16479488 17147285 17856285 15 46 15859259 16490302 17158752 17868475 14 47 15869489 16501126 17170231 17880678 13 48 15879729 16511960 17181721 17892894 12 49 15889979 16522805 17193222 17905123 11 50 15900238 16533660 17204734 17917364 10 51 15910507 16544526 17216258 17929618 9 52 15920785 16555402 17227794 17941885 8 53 15931073 16566289 17239342 17954164 7 54 15941370 16577186 17250902 17966456 6 55 15951676 16588094 17262473 17978761 5 56 15961992 16599013 17274056 17991079 4 57 15972317 16609942 17285651 18003410 3 58 15982651 1662088● 17297258 18015753 2 59 15992994 16631833 17308877 18028109 1 60 16003347 16642794 17320508 18040478 0  32 31 30 29   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant. 61 62 63 64 The minutes of the degrees of the Quadrant for the Tangents of the compliment of the Arches of the same Quadrant. 0 18040478 18807265 19626104 20503034 60 1 18052860 18820471 19640225 20518180 59 2 18065255 18833691 19654362 20533344 58 3 18077663 18846925 19668516 20548526 57 4 18090084 18860174 19682686 20563726 56 5 18102518 18873437 19696872 20578945 55 6 18114966 18886715 19711074 20594182 54 7 18127427 18900007 19725293 20609437 53 8 18139901 18913314 19739528 20624711 52 9 18152388 18926636 19753780 20640003 51 10 18164889 18939972 19768048 20655313 50 11 18177403 18953323 19782333 20670642 49 12 18189930 18966689 19796634 20685989 48 13 18202470 18980070 19810951 20701355 47 14 18215024 18993466 19825285 20716739 46 15 18227591 19006876 19839635 20732142 45 16 18240171 19020301 19854002 20747564 44 17 18252765 19033741 19868386 20763004 43 18 18265372 19047196 19882786 20778463 42 19 18277992 19060665 19897203 20793941 41 20 18290626 19074149 19911637 20809438 40 21 18303273 19087648 19926088 20824953 39 22 18315934 19101162 19940555 20840487 38 23 18328608 19114691 19955039 20856040 37 24 18341296 19128235 19669540 26871612 36 25 18353997 19141795 19984057 20887202 35 26 18366712 19155370 19998591 20902811 34 27 18379440 19168960 20013142 20918439 33 28 18392182 19182565 20027709 20934086 32 29 18404938 19196185 20042297 20949752 31 30 18417707 19209821 20056898 20965436 30 31 18430490 19223472 20071516 20981140 29 32 18443287 19237138 20086152 20996863 28 33 18456098 19250819 20100805 21012605 27 34 18468922 19264516 20115475 21028367 26 35 18481760 19278228 20130163 21044148 25 36 18494612 19291955 20144868 21059949 24 37 18507478 19305698 20159590 21075769 23 38 18520357 19319456 20174329 21091609 22 39 18533250 19333230 20189086 21107468 21 40 18546157 19347019 20203860 21123347 20 41 18559078 19360824 20218651 21139246 19 42 18572013 19374644 20233460 21155164 18 43 18584962 19388480 20248286 21171102 17 44 18597925 19402331 2026●130 21187059 16 45 18610902 19416198 20277991 21203036 15 46 18623894 19430081 20292870 21219032 14 47 18636900 19443980 20307767 21235048 13 48 18649920 19457894 20322681 21251083 12 49 18662954 19471824 20337613 21267138 11 50 18676002 19485770 20352563 21283213 10 51 18689064 19499732 20367531 21299308 9 52 18702140 19513710 20382516 21315423 8 53 18715231 19527704 20397519 21331558 7 54 18728335 19541714 20412539 21347713 6 55 18741454 19555739 20427577 21363888 5 56 18754587 19569780 20442633 21380083 4 57 18767735 19583837 20457706 21396298 3 58 18780897 19597910 20472797 21412534 2 59 18794074 19611999 20487906 21428790 1 60 18807265 19626104 20503034 21445067 0  28 27 26 25   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant. 65 66 67 68 The minutes of the degrees of the Quadrant for the Tangents of the compliment of the Arches of the same Quadrant. 0 21445067 22460371 23558529 24750869 60 1 21461364 22477965 23577595 24771613 59 2 21477681 22495582 23596687 24792387 58 3 21494019 22513222 23615805 24813191 57 4 21510377 22530885 23634950 24834024 56 5 21526756 22548571 23654121 24854887 55 6 21543155 22566281 23673318 24875780 54 7 21559575 22584014 23692542 24896704 53 8 21576015 22601771 23711793 24917659 52 9 21592475 22619951 23731071 24938644 51 10 21608956 22637355 23750375 24959659 50 11 21625458 22655183 23769706 24980705 49 12 21641981 22673034 23789064 25001782 48 13 21658525 22690909 23808448 25022890 47 14 21675090 22708808 23827859 25044029 46 15 21691676 22726730 23847297 25065198 45 16 21708283 22744676 23866762 25086398 44 17 21724911 22762646 23886254 25107629 43 18 21741559 22780639 23905773 25128991 42 19 21758228 22798656 23925320 25150183 41 20 21774918 22816696 23944895 25171506 40 21 21791629 22834760 23964496 25192861 39 22 21808362 22852848 23984124 25214248 38 23 21825116 22870960 24003779 25235666 37 24 21841892 22889096 24023462 25257116 36 25 21858689 22907256 24043172 25278597 35 26 21875508 22925441 24062910 25300115 34 27 21892348 22943650 24082675 25321655 33 28 21909210 22961883 24102468 25343232 32 29 21926094 22980141 24122289 25364841 31 30 21944000 22998424 24142137 25386482 30 31 21959926 23016731 24162013 25408154 29 32 21976874 23035062 24181917 25429858 28 33 21993843 23053418 24201849 25451594 27 34 22010834 23071798 24221809 25473362 26 35 22027846 23090203 24241798 25495162 25 36 22044879 23108632 24261815 25516995 24 37 22061934 23127086 24281860 25538860 23 38 22079011 23145565 4301934 25560758 22 39 22096109 23164068 14322037 25582688 21 40 22113229 23182597 24342169 25604651 20 41 22130372 23201151 24362329 25626647 19 42 22147537 23219730 24382518 25648675 18 43 22164725 23238335 24402735 25670736 17 44 22181935 23256965 24422981 25692830 16 45 22199168 23275621 24443256 25714957 15 46 22216424 23294302 24463559 25737218 14 47 22233703 23313008 24483891 25759312 13 48 22251004 23331740 24504252 25781540 12 49 22268328 23350498 24524641 25803801 11 50 22285675 2336928● 24545061 25826096 10 51 22303044 23388092 24565509 25848424 9 52 22320435 23406927 24585986 25870786 8 53 22337848 23425788 24606492 25893181 7 54 22355284 23444674 24627028 25915610 6 55 22372742 23463586 24647594 25938073 5 56 22390223 23482523 24668189 25960569 4 57 22407726 23501486 24688814 25983099 3 58 22425252 23520475 24709469 26005663 2 59 22442800 23539489 24730154 26028261 1 60 22460371 23558529 24750869 26050893 0  24 23 22 21   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant. 69 70 71 72 The minutes of the degrees of the Quadrant for the Tangents of the compliment of the Arches of the same Quadrant. 0 26050893 27474777 29042105 30776834 60 1 26073559 27499665 29069569 30807323 59 2 26096260 2752459● 29097080 30837866 58 3 26118996 27549559 29124638 30868465 57 4 26141766 27574565 29152243 30890119 56 5 2616457● 27599612 29179895 30929828 55 6 26187411 27624699 29207595 30960593 54 7 26210286 27649827 ●9235343 30991413 53 8 26233196 27674995 29263139 31022289 52 9 26256141 27700204 29290382 31053221 51 10 26279120 27725453 29318873 31084208 50 11 26302135 27750742 29346811 31115252 49 12 26325185 27776072 29374797 31146352 48 13 26348270 27802443 29402831 31177508 47 14 26371390 27826855 29430913 31208720 46 15 26394546 27852308 29459043 31239989 45 16 26417738 27877803 29487221 31271315 44 17 26440966 27903339 29515446 31302698 43 18 26464229 27928917 29543719 31334138 42 19 26487528 27954536 29572041 31365636 41 20 26510863 27980196 29600411 31397191 40 21 26534234 28005898 29628831 31428805 39 22 26557641 28031642 29657301 31460470 38 23 26581084 28057429 29685820 31492205 37 24 26604563 28083258 29714388 31523992 36 25 26628079 28109129 29743006 31555838 35 26 26651631 28135043 29771674 31587742 34 27 26675220 28160999 29800392 31619705 33 28 26698845 28186998 29829160 31651727 32 29 26722507 28213040 29857978 31683807 31 30 26746206 28239125 29886847 31715946 30 31 26769942 28265253 29915765 31748144 29 32 26793716 28291424 29944734 31780401 28 33 26816527 28317638 29973753 31812717 27 34 26841375 28343895 30002823 31845093 26 35 26865260 28379195 30031943 31877528 25 36 26889183 28396539 30061113 31910 24 24 37 26913143 28422926 30090334 31942580 23 38 26937141 28449357 30119605 31975197 22 39 26961177 28475832 30148927 32007875 21 40 26985251 28502350 30178299 32040613 20 41 27009362 28528913 30207723 32073413 19 42 27033511 28555520 30237200 32106275 18 43 27057698 28582172 30266730 32139200 17 44 27081922 28608868 30296312 32172187 16 45 27106184 28635608 30325947 32205237 15 46 27130484 28662393 30355635 32238349 14 47 27154823 28689222 30385375 32271524 13 48 27179200 28716096 30415169 32304762 12 49 27203616 28743015 30445015 32338064 11 50 27228070 28769979 30474915 32371430 10 51 27252563 28796987 30504867 32404858 9 52 27272095 28824040 30534872 32438348 8 53 27301667 28851139 30564930 32471901 7 54 27326278 28878283 30595041 32505517 6 55 27350929 28905472 30625205 32539196 5 56 27375620 28932707 30655423 32572937 4 57 27400350 28959988 30685695 32606741 3 58 27425120 28987315 30716020 32640907 2 59 27449929 29014687 30746400 32674536 1 60 27474777 29042105 30776834 32708528 0  20 19 18 17   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant. 73 74 75 76 The minutes of the degrees of the Quadrant for the Tangents of the compliment of the Arches of the same Quadrant. 0 32708528 34874151 37320517 40107808 60 1 32745286 34912477 37363987 40157569 59 2 32776709 34950881 37407551 40207446 58 3 32810898 34989364 37451210 40257440 57 4 32845153 35027925 37494964 40307552 56 5 32879747 35066565 37538814 40357781 55 6 32913862 35105283 37582760 40408129 54 7 32948317 35144080 37626803 40458596 53 8 32982839 35182956 37670943 40509183 52 9 33017427 35221911 37715180 40559890 51 10 33052082 35260945 37759515 40610718 50 11 33086802 35300059 37803948 40661665 49 12 33121588 35339253 37848479 40712731 48 13 33156441 35378528 37893109 40763917 47 14 33191362 35417883 37937838 40815224 46 15 33226351 35457320 37982666 40866652 45 16 33261408 35496838 38027592 40918201 44 17 33296534 35536438 38072616 40969871 43 18 33331728 35576121 38117740 41021663 42 19 33366990 35615888 38162963 41073577 41 20 33402321 35655739 38208285 41125614 40 21 33437720 35695672 38253708 41177775 39 22 33473188 35735689 38299232 41230062 38 23 33508725 35775789 38344857 41282475 37 24 33544330 35815973 38390584 41335015 36 25 33580005 35856241 38436414 41387683 35 26 33615750 35896593 38482347 41440480 34 27 33651566 35937029 38528384 41493407 33 28 33687453 35977550 38574525 41546464 32 29 33723410 36018156 38620772 41599653 31 30 33759438 36058848 38667125 41652974 30 31 33795535 36099623 38713580 41706424 29 32 33831703 36140483 38760139 41760003 28 33 33867942 36181427 38806801 41813712 27 34 33904252 36222456 38853567 41867550 26 35 33940634 36263570 38900438 41921518 25 36 33977088 36304771 38947416 41975617 24 37 34013615 36346060 38994501 42029848 23 38 34050215 36387437 39041695 42084211 22 39 34086888 36428903 39088998 42138706 21 40 34123634 36470459 39136409 42193334 20 41 34160453 36512103 39183929 42248096 19 42 34197345 36553836 39231557 42302993 18 43 34234310 36595659 39279294 42358025 17 44 34271348 36637572 39327139 42413193 16 45 34308459 36679574 39375094 42468497 15 46 34345644 36721666 39423158 42523937 14 47 34382903 36763849 39471331 42579514 13 48 34420237 36806121 39519614 42635228 12 49 34457647 36848483 39568006 42691080 11 50 34495132 36890936 39616509 42747070 10 51 34532692 36933479 39665124 42803199 9 52 34570327 36976114 39713852 42859468 8 53 34608038 37018840 39762695 42915878 7 54 34645824 37061659 39811654 42972429 6 55 34683686 37104570 39860729 43029122 5 56 34721625 37147574 39909917 43085958 4 57 34759640 37190670 39959218 43142937 3 58 34797733 37233859 40008633 43200060 2 59 34835903 37277141 40058103 43257328 1 60 34874151 37320517 40107808 43314742 0  16 15 14 13   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadrant. 77 78 79 80 The minutes of degrees of the Quadrant for the Tangents of the compliment of the Arches of the same Quadrant 0 43314742 47046295 51445543 56712854 60 1 43372301 47113680 51525561 56809480 59 2 43430006 47181249 51605820 56906425 58 3 43487857 47249003 51686321 57003690 57 4 43545855 47316942 51767065 57101277 56 5 43604000 47385067 51848053 57199188 55 6 43662293 47453380 51929285 57297425 54 7 43720733 47521882 52010762 57395990 53 8 43779321 47590575 52092485 57494885 52 9 43838057 47659460 52174455 57594111 51 10 43896942 47728538 52256673 57693670 50 11 43955977 47797809 52339140 57793564 49 12 44015163 47867274 52421857 57893795 48 13 44074501 47936934 52504826 57994366 47 14 44133992 48006790 52588048 58095279 46 15 44193637 48076841 52671525 58196536 45 16 44253435 48147088 52755259 58298138 44 17 44313387 48217531 52839251 58400087 43 18 44373494 48288171 52923503 58502385 42 19 44433756 48359008 53008016 58605934 41 20 44494174 48430043 53092792 58708035 40 21 44554749 48501278 53177831 58811388 39 22 44615481 48572714 53263134 58915095 38 23 44676371 48644352 53348702 59019157 37 24 44737419 48716193 53434536 59123576 36 25 44798626 48788238 53520637 59228353 35 26 44859993 48860488 53607006 59333490 34 27 44921521 48932945 53693644 59438989 33 28 44983211 49005610 53780552 59544852 32 29 45045065 49078483 53867731 59651081 31 30 45107083 49151565 53955183 59757678 30 31 45169263 49224856 54042909 59864646 29 32 45231607 49298357 54130911 59971987 28 33 45294114 49372069 54219190 60079703 27 34 45356785 49445993 54307748 60187796 26 35 45419621 49520130 54396586 60296268 25 36 45482623 49594481 54485705 60405112 24 37 45545790 49669047 54575107 60514358 23 38 45609123 49743829 54664793 60623981 22 39 45672623 49818827 54754764 60733992 21 40 45736291 49894042 54845022 60844392 20 41 45800128 49969475 54935569 60955184 19 42 45864135 50045127 55029406 61066370 18 43 45928314 50120999 55117535 61177952 17 44 45992666 50197092 55208958 61289930 16 45 46057192 50273407 55300676 61402307 15 46 46121892 50349935 55392692 61515085 14 47 46186767 50246707 55485007 61628267 13 48 46251817 50503605 55577622 61741856 12 49 46318043 50580910 55670539 61855854 11 50 46382445 50658353 55763759 61970263 10 51 46448023 50736025 55857283 62085085 9 52 46513778 50813927 55951112 62200323 8 53 46579711 50892060 56045247 62315979 7 54 46645823 50970425 56139689 62432056 6 55 46712115 51049023 56234439 62548556 5 56 46778587 51127855 56329498 62665481 4 57 46845240 51206922 56424868 62782833 3 58 46912075 51286225 56520550 62900615 2 59 46979093 51365765 56616545 63018829 1 60 47046295 51445543 56712854 63137478 0  12 11 10 9   The degrees of the Quadrant for the Tangent of the compliment of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadarnt. 81 82 83 84 The minutes of the degrees of the Quadrant for the Tangents of the compliment of the Arches of the same Quadrant. 0 63137478 71153707 81443502 95143611 60 1 63256564 71304198 81639821 95410585 59 2 63376089 71455313 81837074 95679034 58 3 63496056 71607058 82035268 95948971 57 4 63616468 71759440 82234410 96220411 56 5 63737327 71912459 82434508 96493467 55 6 63858635 72066117 82635570 96767939 54 7 63980394 72220422 82837603 97044063 53 8 64102607 72375376 83040614 97321646 52 9 64225276 72530983 83244610 97600890 51 10 64348404 72687247 83449598 97881716 50 11 64471994 72844173 83655585 98164135 49 12 64596049 73001766 83862572 98448162 48 13 64720571 73160031 84070565 98733810 47 14 64845563 73318972 84279571 99021104 46 15 64971028 73478593 84489598 99310047 45 16 65096969 73638898 84700687 99600655 44 17 65223388 73799892 84912817 99893042 43 18 65350287 73961579 85125995 100187022 42 19 65477669 74123964 85340229 100482822 41 20 65605537 74287052 85555525 100780346 40 21 65733894 74450847 85771891 101079507 39 22 65862743 74615354 85989335 101380525 38 23 65992087 74780577 86207866 101683314 37 24 66121928 74946521 86427493 101987889 36 25 66252268 75113189 86648225 102294266 35 26 66383110 75280586 86870072 102602473 34 27 66514457 75448716 87093043 102912514 33 28 66646313 75617584 87317150 103224405 32 29 66778681 75787195 87542404 103538166 31 30 66911564 75957554 87768816 103853919 30 31 67044965 76128666 87996394 104171468 29 32 67178887 76300536 88225146 104491055 28 33 67313334 76473170 88455079 104812581 27 34 67448309 76646573 88686196 105136063 26 35 67583815 76820751 88918508 105461519 25 36 67719855 76995710 89152021 105788969 24 37 67856423 77171455 89386745 106118428 23 38 67993549 77347991 89622688 106449917 22 39 68131209 77525324 89859858 106783466 21 40 68269416 77703459 90098268 107119198 20 41 68408173 77882402 90337927 107456902 19 42 68547438 78062159 90578848 107796712 18 43 68687350 78242737 90821043 108138767 17 44 68827777 78424142 91064526 108482852 16 45 68968769 78606379 91309309 108829233 15 46 69110326 78789454 91555401 109177805 14 47 69252455 78973371 91802810 109528589 13 48 69395158 79158136 92051546 109881598 12 49 69538439 76343754 92301618 110236864 11 50 69682302 79530231 92553036 110594415 10 51 69826751 79717572 92805759 110954264 9 52 69971789 79905783 93059875 111316432 8 53 70117419 80094869 93315361 111680940 7 54 70263645 80284835 93572238 112047814 6 55 70410470 80475688 93830595 112417202 5 56 70557898 80667435 94090270 112788878 4 57 70705932 80860083 94351448 113163656 3 58 70854576 81053639 94614055 113539681 2 59 71003833 81248110 94878103 113918875 1 60 71153706 81443502 95143611 114300579 0  8 7 6 5   The degrees of the Quadrant for the Tangent of the compliment of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadarnt. 85 86 87 The minutes of the degrees of the Quadrant for the Tangents of the compliment of the Arches of the same Quadrant. 0 114300579 143006601 190811200 60 1 114684819 143606943 191879163 59 2 115071619 144212307 192959095 58 3 115461005 144822757 194051200 57 4 115853017 145438358 195155685 56 5 116247668 146059175 196273146 55 6 116644985 146685275 197403054 54 7 117044995 147316726 198545993 53 8 117447864 147953611 199702191 52 9 117853346 148595987 200871878 51 10 118261757 149244148 202055705 50 11 118672834 149897753 203253093 49 12 119086890 150557233 204464726 48 13 119503669 151222301 205691260 47 14 119923488 151893462 206932111 46 15 120346233 152570581 208188402 45 16 120771937 153253487 209459545 44 17 121200643 153942729 210746693 43 18 121632370 154638158 212049271 42 19 122067151 155339855 213368214 41 20 122505017 156047923 214704085 40 21 122946003 156762433 216056022 39 22 123390142 157483474 217425507 38 23 123837634 158211136 218812405 37 24 124288195 158945509 220217049 36 25 124742169 159686753 221639784 35 26 125199280 160434770 223080983 34 27 125659878 161189849 224540987 33 28 126123842 161952305 226020167 32 29 126591211 162721698 227518902 31 30 127062036 163498660 229037584 30 31 127536341 164282764 230576614 29 32 128014165 165074651 232136427 28 33 128495548 165873906 233717425 27 34 128980531 166681172 235320041 26 35 129469305 167496287 236945285 25 36 129961652 168319085 238592501 24 37 130457692 169150247 240262714 23 38 130957670 169989613 241957021 22 39 131461286 170837304 243674732 21 40 131968930 171693461 245417543 20 41 132480297 172558198 247184785 19 42 132995769 173431641 248978216 18 43 133515636 174313925 250797165 17 44 134038804 175205183 252643455 16 45 134566419 176105555 254517088 15 46 135098153 177015180 256417991 14 47 135634096 177934219 258348100 13 48 136174272 178862806 260307416 12 49 136718731 179801085 262296605 11 50 137267523 180749537 264316358 10 51 137820702 181707670 266366704 9 52 138378319 182676299 268449755 8 53 138940429 183654941 270565570 7 54 139507087 184644417 272714927 6 55 140078545 185644562 274898633 5 56 140654481 186655202 277117516 4 57 141235334 187677207 279372435 3 58 141820765 188710414 281664304 2 59 142411234 189755028 283994009 1 60 143006601 190811200 286362498 0  4 3 2   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Tangents of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Tangents of the Arches of the same Quadarnt. 88 89 The minutes of the degrees of the Quadrant for the Tangents of the compliment of the Arches of the same Quadrant. 0 286362498 572899830 60 1 288770746 582610421 59 2 291219764 592655713 58 3 293710598 603057015 57 4 296244357 613825994 56 5 298823024 624990311 55 6 301445987 636564040 54 7 304115322 648591509 53 8 306833212 661050728 52 9 309599077 674016435 51 10 312416191 687500725 50 11 315283945 701531474 49 12 318204757 716149676 48 13 321181137 731385593 47 14 324212583 747289264 46 15 327302782 763899813 45 16 330451272 781259259 44 17 333661982 799432199 43 18 336934467 818463792 42 19 340272744 838430438 41 20 343677949 859395374 40 21 347150587 881427652 39 22 350695255 904627361 38 23 354312962 929081086 37 24 358006024 954893332 36 25 361776788 982180553 35 26 365626388 1011062679 34 27 369560062 1041705454 33 28 373579199 1074263399 32 29 377686614 1108922084 31 30 381885288 1145891136 30 31 386178258 1185395877 29 32 390568737 1227736470 28 33 395060088 1273213435 27 34 399655828 1322188681 26 35 404359642 1375082163 25 36 409175388 1432363027 24 37 414111295 1494645462 23 38 419159137 1562590046 22 39 424335793 1637005697 21 40 429641796 1718863124 20 41 435082056 1809337410 19 42 440661780 1909864971 18 43 446386310 2022219818 17 44 452261453 2148619711 16 45 458293185 2291873854 15 46 464487853 2455533838 14 47 470852152 2644433955 13 48 477393195 2864819229 12 49 484118353 3125276745 11 50 491038024 3437829002 10 51 498155754 3819696333 9 52 505482730 4297181900 8 53 513030946 4911098124 7 54 520805157 5729633839 6 55 528821258 6875680006 5 56 537085003 8594003953 4 57 545610968 11457529506 3 58 554414914 17188033688 2 59 563504309 34376070815 1 60 572899830 Infinita. 0  1 0   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.    

THE TABLE OF SECANTS OTHERWISE CALLED THE BENEFICIAL TABLE.  
The Table of Secants.  
 The degrees of the Quadrant for Secants. of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 0 1 2 3 The minutes of the Quadrant for the Scants of the Compliments of the Arches of the same Quadrant. 0 10000000 10001524 10006095 10013723 60 1 10000001 10001574 10006198 10013875 59 2 10000002 10001626 10006301 10014029 58 3 10000004 10001679 10006405 10014184 57 4 10000008 10001733 10006509 10014339 56 5 10000010 10001788 10006615 10014495 55 6 10000014 10001844 10006721 10014653 54 7 10000020 10001900 10006828 10014811 53 8 10000027 10001957 10006936 10014970 52 9 10000034 10002015 10007045 10015130 51 10 10000042 10002074 10007155 10015291 50 11 10000051 10002134 10007265 10015453 49 12 10000060 10002195 10007376 10015615 48 13 10000071 10002256 10007488 10015778 47 14 10000083 10002318 10007601 10015942 46 15 10000095 10002381 10007716 10016107 45 16 10000108 10002445 10007831 10016273 44 17 10000122 10002510 10007946 10016440 43 18 10000137 10002576 10008062 10016608 42 19 10000152 10002642 10008179 10016777 41 20 10000168 10002709 10008298 10016946 40 21 10000186 10002777 10008417 10017116 39 22 10000204 10002846 10008537 10017287 38 23 10000223 10002916 10008658 10017459 37 24 10000243 10002987 10008779 10017632 36 25 10000264 10003058 10008902 10017806 35 26 10000285 10003130 10009025 10017981 34 27 10000308 10003203 10009149 10018157 33 28 10000332 10003277 10009274 10018333 32 29 10000357 10003352 10009400 10018510 31 30 10000381 10003428 10009527 10018687 30 31 10000407 10003505 10009655 10018865 29 32 10000433 10003582 10009783 10019044 28 33 10000461 10003660 10009912 10019224 27 34 10000489 10003739 10010043 10019405 26 35 10000518 10003819 10010174 10019587 25 36 10000548 10003900 10010306 10019770 24 37 10000579 10003982 10010439 10019954 23 38 10000611 10004060 10010572 10020138 22 39 10000643 10004148 10010706 10020324 21 40 10000677 10004232 10010841 10020510 20 41 10000711 10004317 10010977 10020698 19 42 10000746 10004403 10011114 10020886 18 43 10000782 10004490 10011252 10021086 17 44 10000819 10004578 10011390 10021266 16 45 10000857 10004666 10011529 10021456 15 46 10000895 10004755 10011670 10021649 14 47 10000934 10004845 10011811 10021842 13 48 10000975 10004936 10011952 10022035 12 49 10001016 10005028 10012098 10022239 11 50 10001058 10005122 10012238 10022424 10 51 10001100 10005216 10012383 10022620 9 52 10001144 10005310 10012528 10022817 8 53 10001188 10005405 10012674 10023015 7 54 10001233 10005501 10012822 10023213 6 55 10001280 10005598 10012970 10023412 5 56 10001327 10005696 10013119 10023612 4 57 10001375 10005795 10013269 10023813 3 58 10001423 10005894 10013419 10024014 2 59 10001473 10005994 10013570 10024217 1 60 10001524 10006095 10013723 10024420 0  89 88 87 86   The degrees of the Quadrant for the Tangents of the compliments of the Arches of the same Quadrant.   
 The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The degrees of the Quadrant for Secants. of the Arches of the same Quadrant. 4 5 6 7 The minutes of degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant. 0 10024420 10038198 10055082 10075098 60 1 10024625 10038454 10055390 10075459 59 2 10024830 10038710 10055699 10075820 58 3 10025036 10038968 10056009 10076182 57 4 10025242 10039226 10056320 10076545 56 5 10025450 10039486 10056632 10076909 55 6 10025658 10039747 10056944 10077274 54 7 10025868 10040008 10057256 10077639 53 8 10026078 10040269 10057570 10078005 52 9 10026289 10040532 10057884 10078372 51 10 10026500 10040796 10058200 10078740 50 11 10026713 10041061 10058517 10079009 49 12 10026927 10041326 10058834 10079479 48 13 10027141 10041592 10059153 10079850 47 14 10027357 10041859 10059472 10080222 46 15 10027573 10042128 10059792 10080595 45 16 10027790 10042397 10060113 10080968 44 17 10028009 10042667 10060435 10081332 43 18 10028227 10642936 10060757 10081717 42 19 10028447 10043207 10061080 10082093 41 20 10028667 10043479 10061405 10082470 40 21 10028889 10043752 10061730 10082848 39 22 10029111 10044025 10062056 10083226 38 23 10029334 10044300 10062383 10083606 37 24 10029559 10044576 10062711 10083987 36 25 10029784 10044853 10063039 10084368 35 26 10030009 10045130 10063369 10084750 34 27 10030236 10045409 10063700 10085134 33 28 10030463 10045689 10064031 10085518 32 29 10030692 10045969 10064364 10085903 31 30 10030920 10046250 10064690 10086289 30 31 10031150 10046532 10065035 10086677 29 32 10031381 10046815 10065365 10087065 28 33 10031614 10047098 10065701 10087454 27 34 10031846 10047383 10066038 10087843 26 35 10032079 10047669 10066376 10088243 25 36 10032314 10047954 10066715 10088623 24 37 10032550 10048241 10067054 10089015 23 38 10032786 10048529 10067394 10089408 22 39 10033023 10048818 10067735 10089802 21 40 10033261 10049107 10068076 10090196 20 41 10033500 10049398 10068419 10090592 19 42 10033740 10049690 10068763 10090988 18 43 10033981 10049983 10069107 10091385 17 44 10034223 10050276 10069452 10091783 16 45 10034465 10050571 10069808 10092182 15 46 10034708 10050865 10070155 10092582 14 47 10034952 10051160 10070493 10092983 13 48 10035196 10051456 10070842 10093385 12 49 10035441 10051753 10071192 10093787 11 50 10035688 10052051 10071543 10094190 10 51 10035936 10052350 10071895 10094624 9 52 10036184 10052649 10072247 10095030 8 53 10036434 10052951 10072600 10095406 7 54 10036684 10053252 10072954 10095813 6 55 10036934 10053555 10073310 10096221 5 56 100371●5 10053858 10073666 10096630 4 57 10037438 10054162 10074023 10097040 3 58 10037690 10054468 10074380 10097451 2 59 10037944 10054775 10074737 10097863 1 60 10038198 10055082 10075098 10098275 0  85 84 83 82   The minutes of degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The degrees of the degrees of the Quadrant for the Secants. of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 8 9 10 11 The minutes of the Quadrant for the Secants of the Compliments of the Arches of the same Quadrant. 0 10098275 10124650 10154264 10187166 60 1 10098698 10125117 10154786 10187743 59 2 10099103 10125585 10155308 10188320 58 3 10099518 10126054 10155831 10188899 57 4 10099934 10126524 10156356 10189478 56 5 10100351 10126994 10156881 10190058 55 6 10100769 10127465 10157407 10190639 54 7 10101188 10127947 10157934 10191221 53 8 10101607 10128410 10158462 10191804 52 9 10102028 10128684 10158991 10192387 51 10 10102450 10129358 10159520 10192972 50 11 10102872 10129634 10160051 10193557 49 12 10103295 10130311 10160582 10194144 48 13 10103720 10130788 10161114 10194732 47 14 10104144 10131266 10161648 10195320 46 15 10104570 10131746 10162182 10195910 45 16 10104996 10132226 10162707 10196500 44 17 10105423 10132707 10163252 10197092 43 18 10105851 19133189 10163789 10197684 42 19 10106286 10133672 10164327 10198277 41 20 10106710 10134156 10165865 10198872 40 21 10107140 10134641 10165495 10199467 39 22 10107572 10135127 10165944 10200063 38 23 10108005 10135614 10166485 10200060 37 24 10108438 10136102 10167028 10201258 36 25 10108873 10136591 10167571 10201857 35 26 10109309 10137080 10168116 10202457 34 27 10109755 10137571 10168661 10203058 33 28 10110182 10138163 10169207 10203659 32 29 10110620 10138555 10169765 10204262 31 30 10111059 10139048 10170303 10204867 30 31 10111509 10139543 10170852 10205470 29 32 10111940 10140038 10171401 10206075 28 33 10112482 10140534 10171952 10206681 27 34 10112825 10141036 10172504 10207289 26 35 10113279 10141528 10173056 10207897 25 36 10113713 10142027 10173609 10208506 24 37 10114159 10142526 10174163 10209116 23 38 10114606 10143026 10174718 10209727 22 39 10115053 10143528 10175274 10210339 21 40 10115501 10144030 10175831 10210952 20 41 10115951 10144533 10176389 10211566 19 42 10116401 10145037 10176947 10211180 18 43 10116852 10145542 10177507 10212796 17 44 10117303 10146048 10178068 10213412 16 45 10117754 10146554 10178630 10214030 15 46 10118209 10147062 10179193 10214668 14 47 10118663 10147572 10179756 10215268 13 48 10119118 10148082 10180321 10215889 12 49 10119574 10148593 10180886 10216510 11 50 10120031 10149104 10181453 19217113 10 51 10120489 10149615 10182021 10217756 9 52 10120948 10150128 10182589 10218380 8 53 10121408 10150642 10183158 10219015 7 54 10121868 10151156 10183728 10219631 6 55 10122330 10151672 10184299 10220258 5 56 10122792 10152188 10184870 10220885 4 57 10123256 10152705 10185443 10221514 3 58 10123720 10153224 10186017 10222143 2 59 10124275 10153744 10186591 10222774 1 60 10124650 10154264 10187166 10223405 0  81 80 79 78   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Secants. of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 12 13 14 15 The minutes of degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant. 0 10223405 10263040 10306136 10352762 60 1 10224037 10263730 10306884 10353569 59 2 10224671 10264420 10307633 10354377 58 3 10225305 10265112 10308383 10355186 57 4 10225941 10265804 10309134 10355996 56 5 10226577 10266498 10309886 10356807 55 6 10227215 10267192 10310639 10357619 54 7 10227854 10267888 10311393 10358433 53 8 10228493 10268584 10312148 10359247 52 9 10229134 10269281 10312903 10360063 51 10 10229775 10269979 10313660 10360880 50 11 10230417 10270688 10314417 10361698 49 12 10231060 10271379 12315176 10362517 48 13 10231644 10272080 10315935 10363337 47 14 10232288 10272782 10316696 10364158 46 15 10232994 10273485 10317457 10364980 45 16 10233641 10274190 10318220 10365802 44 17 10234289 10274895 10318984 10366626 43 18 10234938 10275601 10319749 10367450 42 19 10235587 10276318 10320525 10368276 41 20 10236238 10277016 10321282 10369102 40 21 10236889 10277726 10322050 10369930 39 22 10237541 10278436 10322819 10370758 38 23 10238195 10279148 10323589 10371588 37 24 10238849 10279860 10324359 10372418 36 25 10239505 10280573 10325131 10373250 35 26 10240161 10281287 15325903 10374092 34 27 10240818 10282002 10326677 10374916 33 28 10241476 10282717 10327451 10375750 32 29 10242135 10283434 10328127 10376586 31 30 10242795 10284151 10329003 10377422 30 31 10243456 10284870 10329781 10378260 29 32 10244118 10285589 10330559 10379098 28 33 10245782 10286310 10331339 10379938 27 34 10245445 10287032 10332119 10380778 26 35 10246110 10287754 10332902 10381620 25 36 10246776 10288478 10333684 10382463 24 37 10247442 10289202 10334467 10383307 23 38 10248110 10289928 10335252 10384153 22 39 10248778 10290654 10336037 10384999 21 40 10249448 10291381 10336824 10385846 20 41 10250119 10292119 10337612 10386694 19 42 10250790 10292838 10338400 10387543 18 43 10251461 10293569 10339189 10388393 17 44 10252136 10294300 10339980 10389244 16 45 10252811 10295043 10340771 10390096 15 46 10253482 10295766 10341564 10390949 14 47 10254162 10296501 10342347 10391803 13 48 10254839 10297237 10343152 10392657 12 49 10255517 10297973 10343947 10393513 11 50 10256196 10298710 10344743 10394370 10 51 10256876 10299449 10345541 10395228 9 52 10257557 10300188 10346340 10396087 8 53 10258239 10300928 10347139 10396947 7 54 10258922 10301669 10347940 10397808 6 55 10259606 10302411 10348741 10398670 5 56 10260291 10303154 10349544 10899533 4 57 10260977 10303898 10350347 10400397 3 58 10261661 10304643 10351151 10401262 2 59 10262351 10305390 10351956 10402128 1 60 10263040 10306136 10352762 10402994 0  77 76 75 74   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Secants. of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 16 17 18 19 The minutes of degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant. 0 10402994 10456917 10514621 10576207 60 1 10403862 10457847 10515616 10577267 59 2 10404730 10458779 10516612 10578328 58 3 10405590 10459711 10517609 10579400 57 4 10406471 10460645 10518607 10580463 56 5 10407343 10461580 10519606 10581518 55 6 10408216 10462516 10520606 10582583 54 7 10409091 10463453 10521607 10583650 53 8 10409966 10464391 10522608 10584717 52 9 10410843 10465330 10523611 10585795 51 10 10411721 10466270 10524615 10586855 50 11 10412600 10467211 10525620 10587925 49 12 10413479 10468153 10526626 10588997 48 13 10414360 10469096 10527633 10590070 47 14 10415241 10470041 10528642 10591145 46 15 10416124 10470986 10529651 10592220 45 16 10417007 10471933 10530662 10593297 44 17 10417892 10472880 10531673 10594375 43 18 10418778 10473829 10532686 10595455 42 19 10419665 10474778 10533699 10596534 41 20 10420553 10475729 10534714 10597615 40 21 10421442 10476680 10535730 10598697 39 22 10422333 10477633 10536747 10599780 38 23 10423224 10478587 10537765 10600865 37 24 10424116 10479542 10538785 10601950 36 25 10425009 10480498 10539805 10603037 35 26 10425903 10481454 10540826 10604125 34 27 10426798 10482412 10541848 10605214 33 28 10427694 10483371 10542872 10606304 32 29 10428591 10484331 10543897 10607395 31 30 10429489 10485292 10544923 10608487 30 31 10430388 10486254 10545950 10609580 29 32 10431288 10487217 10546977 10610675 28 33 10432189 10488181 10548006 10611770 27 34 10433091 10489146 10549036 10612867 26 35 10433995 10490113 10550067 10613964 25 36 10434899 10491080 10551099 10615063 24 37 10435805 10492049 10552133 10616163 23 38 10436711 10493018 10553168 10617264 22 39 10437619 10493989 10554204 10618366 21 40 10438528 10494961 10555241 10619469 20 41 10439436 10494934 10556279 10620574 19 42 10440346 10496908 10557318 10621680 18 43 10441257 10497883 10558359 10622787 17 44 10442170 10498059 10559400 10623895 16 45 10443083 10499836 10560443 10625004 15 46 10443998 10500814 10561496 10626114 14 47 10444913 10501793 10562531 10627226 13 48 10445830 10502773 10563577 10628338 12 49 10446749 10503754 10564623 10629451 11 50 10447668 10504736 10565670 10630566 10 51 10448588 10505719 10566719 10631682 9 52 10449509 10506704 10567769 10632799 8 53 10450431 10507689 10568820 10633917 7 54 10451354 10508676 10569872 10635037 6 55 10452279 10509664 10570925 10636157 5 56 10453204 10510653 10571980 10637279 4 57 10454131 10511643 10573034 10638402 3 58 10455058 10512635 10574091 10639526 2 59 10455987 10513627 10575149 10640651 1 60 10456917 10514621 10576207 10641777 0  73 72 71 70   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 20 21 22 23 The minutes of the Quadrant for the Secants of the Compliments of the Arches of the same Quadrant. 0 10641777 10711449 10785347 10863603 60 1 10642905 10712646 10786616 10864945 59 2 10644034 10713888 10787885 10866289 58 3 10645164 10715042 10789155 10867633 57 4 10646295 10716242 10790427 10868979 56 5 10647427 10717444 10791700 10870326 55 6 10648560 10718647 10792974 10871675 54 7 10649694 10719850 10794250 10873024 53 8 10650829 10721056 10795527 10874374 52 9 10651965 10722261 10796805 10875626 51 10 10653103 10723469 10798085 10877079 50 11 10654242 10724677 10799365 10878434 49 12 10655381 10725887 10800647 10879790 48 13 10656522 10727098 10803214 10881147 47 14 10657664 10728310 10803214 10882506 46 15 10658807 10729524 10804500 10883865 45 16 10659951 10730738 10805787 10885226 44 17 10661097 10731953 10807074 10886588 43 18 10662244 10733170 10808363 10887952 42 19 10663392 10734387 10809652 10889317 41 20 10664541 10735606 10810942 10890683 40 21 10665692 10736826 10812234 15892051 39 22 10666844 10738048 10813528 10893417 38 23 10667996 10739270 10814823 10894788 37 24 10669150 10740494 10816119 10896159 36 25 10670304 10741719 10817417 10897531 35 26 10671460 10742945 10818715 10898905 34 27 10672617 10744173 10820015 10900280 33 28 10673776 10745401 10821316 10901656 32 29 10674936 10746631 10822617 10903033 31 30 10676096 10747864 10823920 10904413 30 31 10677258 10749094 10825225 10905790 29 32 10678420 10750327 10826531 10907171 28 33 10679584 10751561 10827838 10908553 27 34 10680749 10752797 10829146 10909936 26 35 10681915 10754034 10830455 10911322 25 36 10683082 10755273 10831766 10912709 24 37 10684250 10756513 10833078 10914096 23 38 10685420 10757753 10834391 10915484 22 39 10686591 10758995 10835706 10916874 21 40 10687763 10760237 10837023 10918265 20 41 10688936 10761481 10838341 10919657 19 42 10690111 10762726 10839660 10921051 18 43 10691287 10763972 10840980 10922436 17 44 10692464 10765220 10842301 10923833 16 45 10693642 10766469 10843623 10925241 15 46 10694821 10767720 10844947 10926641 14 47 10696001 10768971 10846272 10928041 13 48 10697182 10770224 10847597 10929442 12 49 10698364 10771477 10848924 10930846 11 50 10699548 10772732 10850252 10932249 10 51 10700732 10773988 10851583 10933654 9 52 10701918 10775244 10852914 10935061 8 53 10703105 10776502 10854246 10936469 7 54 10704294 10777761 10855578 10937879 6 55 10705483 10779022 10856912 10939290 5 56 10706674 10780284 10858247 10940702 4 57 10707866 10781547 10859584 10942115 3 58 10709059 10782802 10860922 10943527 2 59 10710254 10784078 10862262 10944945 1 60 10711449 10785347 10863603 10946362 0  69 68 67 66   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 24 25 26 27 The minutes of degrees of the Quadrant for the Secants of the Compliments of the Arches of the same Quadrant. 0 10946362 11033783 11126021 11223262 60 1 10947781 11035280 11127601 11224927 59 2 10949201 11036779 11129182 11226593 58 3 10950622 11038279 11130765 11228260 57 4 10952045 11039780 11132349 11229929 56 5 10953469 11041283 11133933 11231599 55 6 10954898 11042787 11135519 11233270 54 7 10956320 11044293 11137106 11234943 53 8 10957747 11045799 11138694 11236617 52 9 10959175 11047306 11140284 11238292 51 10 10960605 11048815 11141875 11239969 50 11 10962036 11050325 11143467 11241648 49 12 10963469 11051837 11145061 11243329 48 13 10964903 11053350 11146656 11245011 47 14 10966338 11054865 11148254 11246694 46 15 10967775 11056381 11149853 11248378 45 16 10969213 11057898 11151453 11250064 44 17 10970652 11059420 11153055 11251751 43 18 10972092 11060939 11154658 11253440 42 19 10973533 11062461 11156262 11255130 41 20 10974976 11063985 11157868 11256822 40 21 10976420 11065510 11159475 11258516 39 22 10977865 11067037 11161084 11260211 38 23 10979312 11068564 11162694 11261907 37 24 10980760 11070092 1116430● 11263605 36 25 10982210 11071621 11165919 11265304 35 26 10983661 11073152 11167533 11267005 34 27 10985113 11074684 11169149 11268707 33 28 10986567 11076218 11170766 11270410 32 29 10988022 11077753 11172385 11272114 31 30 10989480 11079289 11174006 11273820 30 31 10990938 11080827 11175627 11275528 29 32 10992398 11082366 11177249 11277238 28 33 10993859 11083906 11178873 11278949 27 34 10995321 11085448 11180499 11280661 26 35 10996783 11086990 11182125 11282374 25 36 10998247 11088536 11183753 11284089 24 37 10999712 11090082 11185383 11285805 23 38 11001179 11091629 11187014 11287524 22 39 11002647 11093178 11188647 11289244 21 40 11004116 11094729 11190281 11290965 20 41 11005587 11096280 11191916 11292688 19 42 11007059 11097833 11193553 11294412 18 43 11008533 11099387 11195191 11296132 17 44 11010008 11100943 11196831 11297864 16 45 11011484 1110250● 11198472 11299593 15 46 11019262 11104058 11200114 11301324 14 47 11014441 11105618 11201758 11303056 13 48 11015921 11107179 11203404 11304789 12 49 11017402 11108741 11205051 11306523 11 50 11018884 11110306 11206700 11308259 10 51 11020367 11111871 11208550 11309996 9 52 11021852 11113438 11210001 11311735 8 53 11023338 11115006 11211654 11313476 7 54 11024826 11116575 11213308 11315218 6 55 11026315 11118145 11214963 11316961 5 56 11027806 11119717 11216620 11318706 4 57 11029298 11121290 11218278 11319452 3 58 11030791 11122865 11219938 11322199 2 59 11032287 11124442 11221599 11323949 1 60 11033783 11126021 11223262 11325700 0  65 64 63 62   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 28 29 30 31 The minutes of the Quadrant for the Scants of the Compliments of the Arches of the same Quadrant. 0 11325700 11433540 11547004 11666331 60 1 11327452 11435384 11548944 11668371 59 2 11329206 11437230 11550886 11670413 58 3 11330961 11439078 11552829 11672457 57 4 11332718 11440927 11554774 11674502 56 5 11334479 11442777 11556720 11676548 55 6 11336237 11444629 11558669 11678597 54 7 11337999 11446483 11560619 11680647 53 8 11339762 11448339 11562570 11682698 52 9 11341526 11450196 11564523 11684752 51 10 11343292 11452054 11566480 11686807 50 11 11345060 11453915 11568434 11688864 49 12 11346830 11455776 11570393 11690923 48 13 11348601 11457639 11572353 11692984 47 14 11350373 11459503 11574314 11695046 46 15 11352149 11461370 11576277 11697110 45 16 11353923 11463238 11578242 11699176 44 17 11355698 11465107 11580208 11701243 43 18 11357475 11466978 11582175 11703312 42 19 11359255 11468850 11584145 11705383 41 20 11361036 11470723 11586116 11707455 40 21 11362819 11472599 11588089 11709530 39 22 11364603 11474483 11590064 11711606 38 23 11366389 11476354 11592040 11713684 37 24 11368177 11478235 11594018 11715764 36 25 11369966 11480117 11595998 11717845 35 26 11371756 11482001 11597979 11719928 34 27 11373548 11483887 11599961 11722012 33 28 11375341 11485774 11601946 11724099 32 29 11377136 11487662 11603932 11726187 31 30 11378933 11489353 11605919 11728276 30 31 11380731 11491445 11607909 11730367 29 32 11382530 11493338 11609900 11732460 28 33 11384331 11495233 11611893 11734555 27 34 11386134 11497140 11613888 11736652 26 35 11387938 11499028 11615876 11738751 25 36 11389744 11500928 11617882 11740851 24 37 11391551 11502829 11619881 11742953 23 38 11393359 15504731 11621882 11745057 22 39 11395169 11506626 11623885 11747162 21 40 11396981 11508532 11625889 11749269 20 41 11398794 11510450 11627996 11751378 19 42 11400609 11512360 11629904 11753489 18 43 11402425 11514271 11631913 11755603 17 44 11404243 11516183 11633924 11757718 16 45 11406063 11518097 11635937 11759834 15 46 11407884 11520013 11637952 11761951 14 47 11409706 11521930 11639968 11764069 13 48 11411530 11523849 11641986 11766190 12 49 11413356 11525770 11644005 11768312 11 50 11415183 11527692 11646026 11770437 10 51 11417012 11529616 11648049 11772564 9 52 11418842 11531542 11650075 11774696 8 53 11420673 11533469 11652099 11776822 7 54 11422507 11535398 11654127 11778954 6 55 11424342 11537328 11656156 11781088 5 56 11426178 11539260 11658188 11783223 4 57 11428016 11541193 11660221 11785361 3 58 11429856 11543128 11662256 11787500 2 59 11431689 11545065 11664292 11789640 1 60 11433540 11547004 11666331 11791783 0  61 60 59 58   The degrees of the Quadrant for the Secants of the compliment of the Arches of the same Quadrant.   
 The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 32 33 34 35 The minutes of the Quadrant for the Scants of the Compliments of the Arches of the same Quadrant. 0 11791783 11923633 12062179 12207745 60 1 11793927 11925886 12064546 12210233 59 2 11796073 11928141 12066916 12212723 58 3 11798221 11930397 12069286 12215214 57 4 11800371 11932656 12071660 12217708 56 5 11802522 11934917 12074036 12220204 55 6 11804675 11937180 12076413 12222702 54 7 11806830 11939445 12078792 12225201 53 8 11808987 11941701 12081174 12227703 52 9 11811145 11943979 12083558 12230207 51 10 11813306 11946250 12085943 12232713 50 11 11815468 11948522 12088330 12235221 49 12 11817632 11950796 12090720 12237732 48 13 11819797 11953071 12093111 12240245 47 14 11821965 11955349 12095504 12242759 46 15 11824134 11957629 12097899 12245275 45 16 11826306 11959910 12100296 12247794 44 17 11828479 11962194 12102696 12250315 43 18 11830654 11964479 12105097 12252837 42 19 11832830 11966766 12107500 12255361 41 20 11835008 11969055 12109905 12257888 40 21 11837188 11971346 12112312 12260417 39 22 11839369 11973638 12114722 12262948 38 23 11841552 11975932 12117133 12265481 37 24 11843737 11978229 12119546 12268016 36 25 11845924 11980527 12121960 12270553 35 26 11848114 11982828 12124377 12273093 34 27 11850305 11985131 12126796 12275634 33 28 11852498 11987435 12129216 12278187 32 29 11854693 11989741 12131638 12280722 31 30 11856890 11992050 12134063 12283270 30 31 11859088 11994360 12136490 12285820 29 32 11861288 11996672 12138919 12288372 28 33 11863489 11998986 12141350 12290925 27 34 11865693 12001303 12143783 12293481 26 35 11867899 12003619 12146218 12296039 25 36 11870107 12005938 12148656 12298599 24 37 11872316 12008259 12150095 12301161 23 38 11874527 12010582 12153536 12303725 22 39 11876739 12012907 12155978 12306291 21 40 11878954 12015233 12158423 12308859 20 41 11881171 12017562 12160870 12311430 19 42 11883389 12019893 12163319 12314003 18 43 11885609 12022226 12165770 12316578 17 44 11887831 12024560 12168223 12319156 16 45 11890054 12026897 12170677 12321736 15 46 11892280 12029236 12173135 12324317 14 47 11894508 12031576 12175594 12326900 13 48 11896737 12033919 12178055 12329486 12 49 11898968 12036264 12180518 12332074 11 50 11901202 12038610 12182983 12334664 10 51 11903437 12040958 12185450 12337256 9 52 11905674 12043309 12187919 12339851 8 53 11907912 12045661 12190390 12342448 7 54 11910153 12048016 12192864 12345046 6 55 11912395 12050372 12195340 12347646 5 56 11914640 12052730 12197817 12350●49 4 57 11916886 12055089 12200296 12352854 3 58 11919133 12057451 12202777 12355460 2 59 11921382 12059814 12205260 12358068 1 60 11923633 12063179 12207745 12360678 0  57 56 55 54   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 36 37 38 39 The minutes of the Quadrant for the Scants of the Compliments of the Arches of the same Quadrant. 0 12360678 12521357 12690184 12867599 60 1 12363290 12524103 12693070 12870632 59 2 12365906 12526851 12695957 12873667 58 3 12368524 12529601 12698847 12876704 57 4 12371144 12532354 12701739 12879744 56 5 12373766 12535110 12704634 12882787 55 6 12376391 12537867 12707531 12885832 54 7 12379018 12540627 12710430 12888879 53 8 12381647 12543389 12713332 12891929 52 9 12384278 12546152 12716236 12894982 51 10 12386911 12548918 12719143 12898035 50 11 12389546 12551686 12722052 12901094 49 12 12●92183 12554456 12724964 12904155 48 13 12394822 12557229 12727878 12907218 47 14 12397464 12560005 12730794 12910283 46 15 12400108 12562783 12733713 12913351 45 16 12402754 12565563 12736635 12916422 44 17 12405402 12568345 12739559 12919494 43 18 12408053 12571130 12742485 12922569 42 19 12410705 12573917 12745413 12925647 41 20 12413359 12576706 12748344 12928727 40 21 12416015 12579597 12751277 12931809 39 22 12418674 12582912 12754213 12934895 38 23 12421335 12585087 12757151 12937983 37 24 12423998 12587885 12760092 12941073 36 25 12426663 12590685 12763035 12944166 35 26 12429331 12593488 12765981 12947262 34 27 12432001 12596293 12768929 12950360 33 28 12434673 12599101 12771880 12953461 32 29 12437348 12601911 12774833 12956565 31 30 12440024 12604724 12777788 12959671 30 31 12442702 12607539 12780746 12962780 29 32 12445383 12610356 12783707 12965892 28 33 12448066 12613175 12786670 12969007 27 34 12450751 12615997 12789635 12972124 26 35 12453438 12618821 12792602 12975243 25 36 12456128 12621648 12795573 12978366 24 37 12458821 12624477 12798546 12981491 23 38 12461516 12627308 12801521 12984618 22 39 12464213 12630141 12804498 12987747 21 40 12466913 12632977 12807478 12990880 20 41 12469614 12635815 12810460 12994015 19 42 12472317 12638655 12813445 12997153 18 43 12475022 12641597 12816432 13000293 17 44 12477730 12644343 12819422 13003436 16 45 12480440 12646191 12822415 13006582 15 46 12483152 12650041 12825410 13009730 14 47 12485866 12652893 12828407 13012881 13 48 12488583 12655748 12831407 13016034 12 49 12491302 12658605 12834409 13019189 11 50 12494022 12661464 12837414 13022348 10 51 12496743 12664325 12840421 13025509 9 52 12499469 12667189 12843431 13028673 8 53 12502197 12670055 12846443 13031839 7 54 12504927 12672924 12849458 13035008 6 55 12507659 12675795 12852475 13038180 5 56 12510394 12678668 12855495 13041354 4 57 12513132 12681543 12858517 13044530 3 58 12515871 12684421 12861542 13047710 2 59 12518613 12687301 12864569 13050892 1 60 12521357 12690184 12867599 13054077 0  53 52 51 50   The degrees of the Quadrant for the Secants of the compliment of the Arches of the same Quadrant.   
 The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 40 41 42 43 The minutes of degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant. 0 13054077 13250131 13456326 13673275 60 1 13057264 13253482 13459851 13676986 59 2 13060455 13256835 13463380 13680700 58 3 13063646 13260192 13466912 13684417 57 4 13066843 13263582 13470447 13688138 56 5 13070041 13266915 13473985 13691861 55 6 13073242 13270282 13477527 13695587 54 7 13076445 13273651 13481071 13699316 53 8 13079651 13277023 13484618 13703048 52 9 13082859 13280397 13488168 13706783 51 10 13086071 13283775 13491721 13710523 50 11 13089285 13287155 13495276 13714266 49 12 13092502 13290538 13498835 13718012 48 13 13095721 13293924 13502397 13721761 47 14 13098944 13297313 13505962 13725514 46 15 13102169 13300704 13509530 13729270 45 16 13105397 13304098 13513101 13733029 44 17 13108627 13307495 13516675 13736790 43 18 13111861 13310896 13520252 13740555 42 19 13114098 13314299 13523832 13744322 41 20 13118337 13317705 13527416 13748092 40 21 13121578 13321114 13531003 13751867 39 22 13124823 13324526 13534593 13755644 38 23 13128070 13327941 13538185 13759424 37 24 13131320 13331359 13541781 13763209 36 25 13134572 13334779 13545380 13766997 35 26 13134828 13338203 13548981 13770788 34 27 13141085 13341629 13552585 13774582 33 28 13144346 13345058 13556193 13778380 32 29 13147509 13348490 13559803 13782181 31 30 13150874 13351924 13563417 13785985 30 31 13154142 13355361 13567034 13789792 29 32 13157413 13358802 13570654 13793603 28 33 13160687 13362245 13574277 13797416 27 34 13163964 13365691 13577903 13801233 26 35 13167243 13369140 13581532 13805053 25 36 13170526 13372592 13585164 13808876 24 37 13173811 13376057 13588799 13812703 23 38 13177099 13379505 13592438 13816534 22 39 13180389 13382966 13596079 13820368 21 40 13183682 13386430 13599723 13824205 20 41 13186978 13389897 13603370 13828045 19 42 13190276 13393367 13607021 13831889 18 43 13193577 13396839 13610975 13835736 17 44 13196882 13400315 13614332 13839586 16 45 13200189 13403794 13617992 13843439 15 46 13203499 13407275 13621656 13847296 14 47 13206812 13410759 13625323 13851156 13 48 13210128 13414247 13628993 13855019 12 49 13213447 13417738 13632666 13858885 11 50 13216769 13421232 13636342 13862755 10 51 13220093 13424728 13640021 13866628 9 52 13223421 13428227 13643704 13870505 8 53 13226750 13431729 13647390 13874385 7 54 13230082 13435234 13651078 13878268 6 55 13233417 13438742 13654769 13882154 5 56 13236754 13442253 13658464 13886044 4 57 13240094 13445767 13662162 13889636 3 58 13243437 13449284 13665863 13893833 2 59 13246783 13452804 13669567 13897733 1 60 13250131 13456326 13673275 13901636 0  49 48 47 46   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 44 45 46 47 The minutes of degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant. 0 13901636 14142135 14395564 14662790 60 1 13905542 14146251 14399901 14667366 59 2 13909451 14150371 14404242 14671946 58 3 13913365 14154494 14408587 14676530 57 4 13917281 14158621 14412937 14681119 56 5 13921201 14162751 14417290 14685712 55 6 13925126 14166884 14421647 14690309 54 7 13929052 14171021 14426008 14694910 53 8 13932982 14175162 14430374 14699514 52 9 13936916 14179306 14434743 14704122 51 10 13940854 14183454 14439116 14708735 50 11 13944795 14187606 14443493 14713352 49 12 13948739 14191761 14447874 14717973 48 13 13952686 14195919 14452259 14722598 47 14 13956638 14200082 14456648 14727228 46 15 13960592 14204248 14461040 14731862 45 16 13964550 14208418 14465437 14736500 44 17 13968511 14212591 14469838 14741142 43 18 13972476 14216769 14474242 14745788 42 19 13976444 14220950 14478650 14750438 41 20 13980416 14225135 14483062 14755094 40 21 13984391 14229324 14487478 14759753 39 22 13988370 14233517 14491898 14764416 38 23 13992352 14237713 14496322 14769083 37 24 13996338 14241913 14500750 14773755 36 25 14000327 14246115 14505182 14778430 35 26 14004319 14250321 14509617 14783110 34 27 14008315 14254531 14514056 14787794 33 28 14012314 14258745 14518500 14792482 32 29 14016316 14262961 14522946 14797174 31 30 14020322 14267182 14527397 14801871 30 31 14024332 14271407 14531852 14806571 29 32 14028345 14275635 14536311 14811276 28 33 14032361 14279867 14540773 14815985 27 34 14036381 14284103 14545250 14820698 26 35 14040404 14288343 14549711 14825416 25 36 14044431 14292587 14554186 14830139 24 37 14048461 14296834 14558665 14834866 23 38 14052494 14301086 14563148 14839597 22 39 14056531 14305331 14567635 14844332 21 40 14060572 14309599 14572126 14849072 20 41 14064616 14313861 14576621 14853815 19 42 14068664 14318127 14581120 14858563 18 43 14072715 14322396 14585624 14863315 17 44 14076770 14326670 14590131 14868071 16 45 14080829 14330947 14594642 14872831 15 46 14084891 14335228 14599157 14877597 14 47 14088956 14339513 14603676 14882377 13 48 14093026 14343802 14608199 14887141 12 49 14097099 14348095 14612725 14891919 11 50 14101175 14352391 14617256 14896701 10 51 14105255 14356691 14621791 14901487 9 52 14109339 14360995 14626330 14906278 8 53 14113427 14365303 14630873 14911073 7 54 14117518 14369615 14635421 14915873 6 55 14121612 14373930 14639973 14920677 5 56 14125709 14378350 14644528 14925486 4 57 14129810 14382573 14649087 14930299 3 58 14133915 14386900 14653651 14935116 2 59 14138023 14391230 14658218 14939938 1 60 14142135 14395564 14662790 14944764 0  45 44 43 42   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 48 49 50 51 The minutes of degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant. 0 14944764 15242532 15557239 15890158 60 1 14949594 15247634 15562635 15895869 59 2 14954429 15252741 15568036 15091586 58 3 14959268 15257852 15573441 15907307 57 4 14964112 15262969 15578852 15913034 56 5 14968960 15268090 15584267 15918766 55 6 14973812 15273216 15589688 15924504 54 7 14978668 15278347 15595114 15930247 53 8 14983530 15283484 15600545 15936095 52 9 14988396 15288626 15605981 15941748 51 10 14993266 15293773 15611422 15947508 50 11 14998104 15298924 15616868 15953273 49 12 15003020 15304080 15622319 15959044 48 13 15007903 15309240 15627775 15964820 47 14 15012791 15314405 15633237 15970603 46 15 15017683 15319574 15639704 15976390 45 16 15022580 15324748 15644177 15982184 44 17 15027481 15329926 15649655 15987983 43 18 15032387 15335109 15655138 15993788 42 19 15037297 15340297 15660626 15999599 41 20 15042212 15345491 15666119 16005416 40 21 15047131 15350689 15671617 16011237 39 22 15052054 15355892 15677121 16017065 38 23 15056982 15361100 15682630 16022898 37 24 15061915 15366313 15688144 16028736 36 25 15066852 15371530 15693663 16034579 35 26 15071791 15376753 15699188 16040429 34 27 15076739 15381980 15704717 16046283 33 28 15081690 15387212 15710252 16052143 32 29 15086645 15392449 15715791 16058008 31 30 15091605 15397692 15721337 16063878 30 31 15096569 15402939 15726887 16069754 29 32 15101538 15408191 15732443 16075637 28 33 15106571 15413447 15738003 16081524 27 34 15111490 15418708 15743569 16087418 26 35 15116472 15423974 15749141 16093318 25 36 15121459 15429246 15754718 16099224 24 37 15126451 15434522 15760300 16105135 23 38 15131447 15439803 15765887 16111053 22 39 15136447 15445089 15771479 16116976 21 40 15141453 15450380 15777077 16122905 20 41 15146463 15455675 15782680 16128839 19 42 15151478 15460976 15788289 16134779 18 43 15156497 15466182 15793903 16140424 17 44 15161520 15471593 15799523 16146676 16 45 15166548 15476908 15805147 16152634 15 46 15171581 15482229 15810777 16158598 14 47 15176619 15487554 15816412 16164567 13 48 15181661 15492885 15822052 16170542 12 49 15186708 15498220 15827697 16176522 11 50 15191760 15503560 15833349 16182509 10 51 15196816 15508905 15839005 16188501 9 52 15201877 15514256 15844667 16194499 8 53 15206943 15519611 15850335 16200503 7 54 15212013 15524972 15856008 16206513 6 55 15217088 15530338 15861676 16212528 5 56 15222168 15535710 15867370 16218550 4 57 15227253 15541083 15873058 16224577 3 58 15232342 15546463 15878753 16230610 2 59 15237435 15551848 15884453 16236648 1 60 15242532 15557239 15890158 16242692 0  41 40 39 38   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 52 53 54 55 The minutes of degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant. 0 16242692 16616401 17013017 17434469 30 1 16248742 16622819 17019832 17441715 29 2 16254799 16629243 17026654 17448968 28 3 16260861 16635673 17033482 17456229 27 4 16266929 16642109 17040318 17463499 26 5 16273003 16648551 17047160 17470775 25 6 16279083 16655001 17054010 17478059 24 7 16285169 16661457 17060866 17485351 23 8 16291261 16667919 17067729 17492650 22 9 16297358 16674408 17074599 17499957 21 10 16303461 16680864 17081476 17507272 20 11 16309570 16687345 17088359 17514594 19 12 16315685 16693834 17095250 17521924 18 13 16321806 16700328 17102148 17529262 17 14 16327934 16706829 17109053 17536607 16 15 16334067 16713336 17115965 17543959 15 16 16340197 16719850 171●2885 17551319 14 17 16346353 16726362 17129812 17558687 13 18 16352505 16732877 17136747 17566063 12 19 16358663 16739430 17143689 17573446 11 20 16364827 16745970 17150638 17580837 10 21 16370996 16752517 17157593 17588236 9 22 16377172 16759070 17164556 17595643 8 23 16383359 16765629 17171525 17603057 7 24 16389542 16772195 17178502 17610480 6 25 16395736 16778767 17185485 17617909 5 26 16401936 16785347 17192476 17625347 4 27 16408152 16791933 17199472 17632793 3 28 16414365 16798525 17206477 17640246 2 29 16420573 16805124 17213488 17647707 1 30 16426798 16811729 17220507 17655175 60 31 16433027 16818341 17227532 17662651 59 32 16439263 16824960 17234565 17670136 58 33 16445505 16831585 17241605 17677627 57 34 16451754 16838217 17248653 17685127 56 35 16458008 16844856 17255708 17692635 55 36 16464269 16851502 17262770 17700151 54 37 16470536 16858154 17269839 17707674 53 38 16476809 16864813 17276917 17715206 52 39 16483089 16871479 17284002 17722744 51 40 16489385 16878151 17291095 17730290 50 41 16495668 16884830 17298194 17737844 49 42 16501967 16891515 17305300 17745407 48 43 16508272 16898207 17312413 17752978 47 44 16514582 16904907 17319514 17760555 46 45 16520898 16911613 17326662 17768142 45 46 15527220 16918326 17333798 17775740 44 47 16533548 16925046 17340941 17783343 43 48 16539883 16931772 17348091 17790955 42 49 16546224 16948504 17355249 17798575 41 50 16552571 16945244 17362415 17806203 40 51 16558920 16951990 17369587 17813838 39 52 16565286 16958743 17376767 17821481 38 53 16571642 16965498 17383954 17829132 37 54 16578026 16972270 17391148 17836792 36 55 16584406 16979044 17398350 17844460 35 56 16590792 16985824 17405560 17852135 34 57 16597184 16992611 17412776 17859818 33 58 16603584 16999406 17420000 17867509 32 59 16609989 17006208 17427231 17875209 31 60 16616401 17013017 17434469 17882917 30  37 36 35 34   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 56 57 58 59 The minutes of degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant. 0 17882917 18360816 18870800 19416039 60 1 17890632 18369014 18879589 19425445 59 2 17898356 18377251 18888389 19434862 58 3 17906089 18385497 18897196 19444290 57 4 17913830 18393753 18906018 19453727 56 5 17921579 18402017 18914846 19463175 55 6 17929337 18410291 18923685 19472635 54 7 17937102 18418574 18932534 19482114 53 8 17944876 18426865 18941393 19491595 52 9 17952658 18435165 18950261 19501076 51 10 17960448 18443454 18959139 19510578 50 11 17968247 18451792 18968027 19520091 49 12 17976054 18460120 18976926 19529615 48 13 17983869 18468456 18985834 19539150 47 14 17991693 18476802 18994752 19548697 46 15 17999525 18485157 19003680 19558254 45 16 18007365 18493521 19012618 19567822 44 17 18015214 18501895 19021516 19577401 43 18 18023071 18510278 19030523 19586991 42 19 18030936 18518670 19039491 19596592 41 20 18038811 18527072 19048468 19606204 40 21 18046693 18535483 19057455 19615827 39 22 18054584 18543903 19066453 19625462 38 23 18062482 18552332 19075461 19635107 37 24 18070389 18560770 19084480 19644765 36 25 18078305 18569217 19093509 19654434 35 26 18086229 18577674 19102549 19664114 34 27 18094161 18586139 19111598 19673805 33 28 18102102 18594614 19120658 19683507 32 29 18110051 18603098 19129727 19693220 31 30 18118009 18611591 19138807 19702945 30 31 18125975 18620094 19147897 19712680 29 32 18133950 18629606 19156998 19722428 28 33 18141934 18637127 19166109 19732186 27 34 18149926 18645658 19175231 19741956 26 35 18157927 18654198 19184362 19751738 25 36 18165937 18662748 19193504 19761531 24 37 18173956 18671507 19202656 19771335 23 38 18181984 18679875 19211818 19781141 22 39 18190021 18688452 19220990 19790968 21 40 18198065 18697038 19230172 19800808 20 41 18206118 18705634 19239365 19810658 19 42 18214179 18714239 19248569 19820320 18 43 18222249 18722854 19257783 19830393 17 44 18230328 18731480 19267008 19840277 16 45 18238416 18740115 19276242 19850172 15 46 18246513 18748760 19285488 19860079 14 47 18254618 18757414 19294744 19869997 13 48 18262732 18766078 19304010 19879927 12 49 18270854 18774752 19313287 19889068 11 50 18278986 18783436 19322574 19899820 10 51 18287126 18792130 19331872 19909784 9 52 18295276 18800833 19341181 19919760 8 53 18303434 18809546 19350501 19929748 7 54 18211601 ●8818268 19359831 19939749 6 55 18319776 18826999 19369172 19949760 5 56 18327961 18835741 19378524 19959784 4 57 18337154 18844492 19387886 19966820 3 58 18344356 18853252 19397260 19979868 2 59 18352567 18862021 19406644 19989928 1 60 18360816 18870800 19416939 20000000 0  33 32 31 30   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 60 61 62 63 The minutes of the Quadrant for the Scants of the Compliments of the Arches of the same Quadrant. 0 20000000 20626654 21300545 22026892 60 1 20010083 20637484 21312206 22039475 59 2 20020179 20648338 21323882 22052074 58 3 20030285 20659184 21335570 22064690 57 4 20040404 20670054 21347275 22077322 56 5 20050534 20680937 21358993 22089970 55 6 20060676 20691834 21370727 22102635 54 7 20070832 20702744 21382475 22115316 53 8 20080995 20713667 21394238 22128014 52 9 20091172 20724603 21407016 22140728 51 10 20101361 20735554 21417808 22153459 50 11 20111562 20746517 21429615 22166204 49 12 20121776 20757494 21441438 22178971 48 13 20132001 20768484 21453275 22191751 47 14 20142239 20779488 21465128 22204548 46 15 20152489 20790505 21476995 22217361 45 16 20162751 20801535 21488877 22230191 44 17 20173035 20812579 21500774 22243038 43 18 20183321 20823636 21512686 22255902 42 19 20193619 20834706 21524612 22268782 41 20 20203930 20845791 21536553 22281680 40 21 20214252 20856888 21548509 22294595 39 22 20224588 20868000 21560481 22307526 38 23 20234936 20879125 21572467 22320474 37 24 20245296 20890264 21584469 22333439 36 25 20255669 20901416 21596487 22346420 35 26 20266054 20912582 21608520 22359419 34 27 20276452 20923761 21620568 22372434 33 28 20286863 20934955 21632631 22385466 32 29 20297286 20946162 21644710 22398418 31 30 20307721 20957383 21656804 22411584 30 31 20318170 20968618 21668913 22424667 29 32 20328630 20979867 21681038 22437768 28 33 20339102 20991130 21693178 22450886 27 34 20349587 21002406 21705334 22464022 26 35 20360084 21013696 21717505 22477175 25 36 20370594 21025001 21729691 22490346 24 37 20381116 21036319 21741893 22503543 23 38 20391751 21047651 21754111 22516748 22 39 20402198 21058997 21766344 22529965 21 40 20412758 21070357 21778593 22543201 20 41 20423331 21081731 21790858 22556358 19 42 20433916 21093119 21803138 22569723 18 43 20444514 21104522 21815434 22583025 17 44 20455126 21115938 21827745 22596336 16 45 20465750 21127368 21840072 22609663 15 46 20476387 21138814 21852415 22623009 14 47 20487037 21150273 21864774 22636372 13 48 20497700 21161747 21877149 22649754 12 49 20508376 21173235 21889539 22663152 11 50 20519064 21184737 21901946 22676569 10 51 20529765 21196253 21914369 22690004 9 52 20540479 21207783 21926808 22703456 8 53 20551205 21219328 21939263 22716924 7 54 20561945 21230887 21951734 22730414 6 55 20572697 21242460 21964220 22743919 5 56 20583463 21254048 21976722 22757443 4 57 20594242 21265650 21989240 22770984 3 58 20605033 21277267 22001775 22784543 2 59 20615837 21288899 22014325 22798120 1 60 20626654 21300545 22026892 22811726 0  29 28 27 26   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 64 65 66 67 The minutes of the Quadrant for the Scants of the Compliments of the Arches of the same Quadrant. 0 22811726 23662013 24585936 25593051 60 1 22825329 23676784 24602010 25610602 59 2 22838962 23691575 24618107 25628180 58 3 22852612 23706387 24634227 25645783 57 4 22866281 23721220 24650370 25663414 56 5 22879968 23736073 24666536 25681071 55 6 22893674 23750947 24682727 25698754 54 7 22907387 23765842 24698940 25716464 53 8 22921140 23780757 24715178 25734201 52 9 22934901 23795692 24731439 25751965 51 10 22948680 23810648 24747724 25769755 50 11 22962478 23825625 24764033 25787582 49 12 22976294 23840623 24780365 25805417 48 13 22990129 23805642 24796721 25823287 47 14 23003983 23870683 24813101 25841185 46 15 23017855 23885844 24829504 25859104 45 16 23031747 23900827 24845932 25877061 44 17 23045657 23915931 24862383 25895040 43 18 23059586 23931055 24879958 25913046 42 19 23073534 23946200 24895356 25931080 41 20 23087501 23961366 24911878 25949142 40 21 23101486 23976553 24928423 25967230 39 22 23115490 23991762 24944993 25985345 38 23 23129513 24006992 24961587 26003487 37 24 23143556 24022245 24978●05 26021658 36 25 23157616 24037518 24994847 26039855 35 26 23171696 24052814 25011514 26058081 34 27 23185795 24068130 25028205 26076333 33 28 23199913 24083469 25044920 26094614 32 29 23214050 24098850 25061660 26112923 31 30 23228205 24114213 25078426 26131259 30 31 23242380 24129616 25095216 26149623 29 32 23256574 24145041 25112030 26168015 28 33 23270797 24160487 25128869 26186436 27 34 23285021 24175956 25145732 26204884 26 35 23299273 24191445 25162620 26223361 25 36 23313546 24206956 25179532 26241867 24 37 23327838 24222488 25196469 26260400 23 38 23342150 24238043 25213432 26278963 22 39 23356481 24253619 25230418 26297555 21 40 23370832 24269217 25247431 26316176 20 41 23385203 24284838 25264468 26334825 19 42 23399593 24300481 25281531 26353503 18 43 23414003 24316147 25298620 26372209 17 44 23428433 24331835 25315734 26390945 16 45 23442882 24347546 25332874 26409709 15 46 23457351 24363281 25350039 26428502 14 47 23471840 24379038 25367229 26447323 13 48 23486348 24394818 25384445 26366174 12 49 23500876 24410620 25401687 26485053 11 50 23515424 24426446 25418956 26503962 10 51 23529992 24442294 25436250 26522890 9 52 23544580 24458164 25453570 26541867 8 53 23559188 24474056 25470915 26560863 7 54 23573817 24489973 25488286 26579889 6 55 23588565 24505908 25505683 26598945 5 56 23603134 24521869 25523005 26618030 4 57 23617822 24537851 25540553 26637145 3 58 23632532 24553857 255580●7 26656251 2 59 23647262 24569885 25575526 26675466 1 60 23662013 24585936 25593051 26694672 0  25 24 23 22   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 68 69 70 71 The minutes of the Quadrant for the Scants of the Compliments of the Arches of the same Quadrant. 0 26694672 27904284 29238045 30715531 60 1 26713907 27925445 29261433 30741500 59 2 26733172 27946642 29284861 30767516 58 3 26752467 27967873 29308328 30793579 57 4 26771791 27989139 29331835 30819689 56 5 26791145 28010440 29355382 30845846 55 6 26810529 28031776 29378970 30872051 54 7 26829942 28053147 29402599 30898304 53 8 26849390 28074553 29426268 30924605 52 9 26868867 28095994 29449978 30950953 51 10 26888373 28117469 29473728 30977350 50 11 26907910 28138980 29497519 31003793 49 12 26927479 28160527 29521350 31030285 48 13 26947078 28182108 29545222 31056824 47 14 26966709 28203725 29569136 31083412 46 15 29986370 28225378 29593090 31110047 45 16 27006062 28247067 29617087 31136731 44 17 27025785 28268793 29641124 31163462 43 18 27045539 28290553 29665204 31190241 42 19 27065323 28312349 29689326 31217019 41 20 27085138 28334181 29713488 31243945 40 21 27104985 28356049 29737692 31270871 39 22 27124864 28377954 29761938 31297848 38 23 27144774 28399894 29786227 31324873 37 24 27164717 28421871 29810558 31351948 36 25 27184690 28443884 29834931 31379072 35 26 27204686 28465934 29859347 31406247 34 27 27224734 28488021 29883705 31433472 33 28 27244804 28510144 29908306 31460747 32 29 27264906 28532304 29932850 31488072 31 30 27285040 28554501 29957438 31515448 30 31 27305205 28576735 29982069 31542873 29 32 27325402 28599007 30006743 31570349 28 33 27345631 28621316 30031460 31597875 27 34 27365893 28643662 30056220 31625453 26 35 27386186 28666045 30081023 31653080 25 36 17406513 28688467 30105870 31680758 24 37 27426872 28710925 30130760 31708486 23 38 27447264 28733422 30155714 31736265 22 39 27467688 28755956 30180672 31764094 21 40 27488145 28778549 30205694 31791974 20 41 27508635 28801139 30230760 31819906 19 42 27529157 28823787 30255871 31847891 18 43 27549722 28846473 30281026 31875929 17 44 27570301 28869196 30306226 31904019 16 45 27590922 28891957 30331460 31932164 15 46 27611578 28914756 30356759 31960358 14 47 27632266 28937594 30382092 31988606 13 48 27952989 28960471 30407470 32016909 12 49 27673745 28983386 30432893 32045263 11 50 27694535 29006340 30458361 32073672 10 51 27715358 29029332 30483873 32102132 9 52 27736215 29052363 30509430 32130649 8 53 27757105 29075435 30535033 32159212 7 54 27778029 29098546 30560682 32187832 6 55 27798987 29121697 30586375 32216504 5 56 27819978 29144888 30612115 32245231 4 57 27841003 29168●18 30637890 32274012 3 58 27862060 29191388 30663732 32302846 2 59 27883156 29214697 30689608 32331735 1 60 27904284 29238045 30715531 32360678 0  21 20 19 18   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 72 73 74 75 The minutes of the Quadrant for the Scants of the Compliments of the Arches of the same Quadrant. 0 32360678 34203038 36279559 38637042 60 1 32389676 34235609 36316402 38679033 59 2 32418726 34268245 36353333 38721117 58 3 32447837 34300947 36390323 38763296 57 4 32477001 34333716 36427401 38805571 56 5 32506219 34366553 36464558 38847941 55 6 32535494 34399452 36501793 ●8890408 54 7 32564823 34432420 36539107 38932971 53 8 32594209 34465456 36570511 38975632 52 9 32623651 34498557 36613973 39018390 51 10 32653148 34531726 36651525 39061246 50 11 32682701 34564959 36689156 39104200 49 12 32712311 34598259 36726868 39147252 48 13 32741977 34631626 36764660 39190423 47 14 32771699 34665061 36802533 39233653 46 15 32801478 34698564 36840488 39277002 45 16 32831314 34732135 36878524 39320449 44 17 32861207 34765775 36916641 39363994 43 18 32891157 34799483 36954842 39407640 42 19 32921165 34833259 36993127 39451384 41 20 32951231 34867105 37031496 39495228 40 21 32981355 34901024 37069947 39539172 39 22 33011537 34935005 37108482 39583218 38 23 33041776 34966052 37147101 39627364 37 24 33072074 35003172 37185803 39671613 36 25 33102431 35037361 37224589 39715965 35 26 33131846 35071621 37263459 39760420 34 27 33163320 35105952 37302413 39804979 33 28 33193853 35140354 37341453 39849642 32 29 33224444 35174826 37380577 39894411 31 30 33255094 35209369 37419788 39939226 30 31 33285803 35243981 37459081 39984263 29 32 33316571 35278664 37498460 40029344 28 33 33347398 35313418 37537923 40074528 27 34 33378286 35348244 37577471 40119816 26 35 33409132 35383140 37617104 40165289 25 36 33440240 35418110 37656824 40210709 24 37 33471307 35453152 37696632 40256316 23 38 33502436 35488268 37736518 40302033 22 39 33533625 35523456 37776513 40347858 21 40 33564875 35558718 37816588 40393792 20 41 33596187 35594052 37856751 40439834 19 42 33627561 35629460 37897004 40485985 18 43 33658998 35664940 37937146 40532245 17 44 33690497 35700494 37977779 40578613 16 45 33722059 35736121 38018300 40625091 15 46 33753683 35771822 38058912 40671678 14 47 33785370 35807597 38099614 40718374 13 48 33817120 35843447 38140406 40765180 12 49 33848934 35879373 38181288 40812093 11 50 33880813 35915374 38222261 40859121 10 51 33912753 35951451 38263324 40906259 9 52 33944756 35987602 38304479 40953510 8 53 33976821 36023829 38345725 41004876 7 54 34008950 36060132 38387064 41048358 6 55 34041141 36096510 38428495 41095957 5 56 34073395 36132966 38470019 41143668 4 57 34105712 36169497 38511635 41191492 3 58 34138091 36206107 38553344 41239431 2 59 34170523 36242794 38595146 41287425 1 60 34203038 36279559 38637042 41335654 0  17 16 15 14   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 76 77 78 79 The minutes of the Quadrant for the Scants of the Compliments of the Arches of the same Quadrant. 0 41335654 44454097 48097335 52408433 60 1 41383937 44510183 48163151 52486983 59 2 41432338 44566415 48229350 52565774 58 3 41480856 44622793 48295633 52644807 57 4 41529492 44679318 48362102 52724084 56 5 41578245 44735990 48428756 52803604 55 6 41627117 44792810 48495599 52883368 54 7 41676108 44849777 48562631 52963377 53 8 41725219 44906892 48629854 53043632 52 9 41774450 44964155 48697269 53124134 51 10 41823802 45021567 48764877 53204885 50 11 41873273 45079129 48832678 53285884 49 12 41922863 45136843 48900673 53367134 48 13 41972573 45194707 48968853 53448635 47 14 42022405 45252726 49037249 53530390 49 15 42072357 45310898 49105830 53612399 45 16 42122431 45369223 49174607 53694666 44 17 42172625 45427703 49243590 53777191 43 18 42222942 45486338 49312751 53859976 42 19 42273380 45545127 49382118 53943022 41 20 42323942 45604073 49451684 54026331 40 21 42374627 45663175 49521449 54109903 39 22 42425439 45722435 49591416 54193739 38 23 42476377 45781853 49661584 54277840 37 24 42527442 45841429 49731956 54362207 36 25 42578635 45901164 49802532 54446842 35 26 42629957 45961059 49873313 54531744 34 27 42681409 46021115 49944301 54616915 33 28 42732991 46081333 50015497 54702356 32 29 42784705 46141715 50086901 54788068 31 30 42836551 46202261 50158514 54874053 30 31 42888527 46262969 50230335 54960312 29 32 42940631 46323841 50302367 55046847 28 33 42992865 46384877 50374610 55133659 27 34 43045229 46446076 50447065 55220751 26 35 43097722 46507440 50519732 55308122 25 36 43150347 46568970 50592614 55395775 24 37 43203103 46630665 50665711 55483710 23 38 43255992 46692527 50739024 55571930 22 39 43309012 46754555 50812553 55660434 21 40 43362166 46816752 50886299 55749226 20 41 43415454 46879117 50960263 55838300 19 42 43468877 46941653 51034447 55927677 18 43 43522435 47004361 51108850 56017340 17 44 43576129 47067242 51183475 56107297 16 45 43629●5● 47130297 51258321 56197549 15 46 43683925 47193526 51333391 56288099 14 47 43738728 47256930 51408684 56378948 13 48 43792268 47320509 51484204 56470097 12 49 43846646 47384264 51559951 56561548 11 50 43901162 47448195 51635936 56653302 10 51 43955817 47512302 51712129 56745360 9 52 44000612 47576586 51788563 56837723 8 53 44065548 47641048 51865227 56930392 7 54 44120625 47705689 51942124 57023369 6 55 44175844 47770510 52019254 57116653 5 56 44231207 47835511 52096618 57210246 4 57 44286712 47900693 52174216 57304150 3 58 44342362 47966058 52252051 57398367 2 59 44398156 48031605 52330123 57492896 1 60 44454097 48097385 52408433 57587740 0  13 12 11 10   The degrees of the Quadrant for the Secants of the compliment of the Arches of the same Quadrant.   
 The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 80 81 82 83 The minutes of the Quadrant for the Scants of the Compliments of the Arches of the same Quadrant. 0 57587740 63924495 71852975 82055127 60 1 57682901 64042118 72002006 82249986 59 2 57778381 64160180 72151659 82445779 58 3 57874180 64278683 72301942 82642513 57 4 57970302 64397632 72452863 82840196 56 5 58066748 64517028 72604421 83038833 55 6 58163520 64636873 72756618 83238436 54 7 58260619 64757168 72909461 83439009 53 8 58358049 64877918 73062954 83640561 52 9 58455●10 64999124 73217100 83843097 51 10 58553904 65120789 73371903 84046626 50 11 58652333 65242916 73527367 84251153 49 12 58751099 65365508 73683499 84456680 48 13 58850205 65488566 73840302 84663213 47 14 58949653 65612095 73997782 84870760 46 15 59049444 65736097 74155942 85079327 45 16 59149581 65859675 74314786 85288957 44 17 59250065 65985531 74474318 85499628 43 18 59350898 66110967 74634544 85711347 42 19 59452082 66246886 74795468 85924121 41 20 59553618 66363291 74957095 86137958 40 21 59655506 66490185 75119429 86352864 39 22 59757728 66617572 75282475 86568849 38 23 59860346 66745453 75446238 86785921 37 24 59963291 66873831 75610721 87004089 36 25 60066612 67002708 75775928 87223362 35 26 60170285 67132088 75941864 87443750 34 27 60274319 67261972 76108533 87665261 33 28 60378718 67392365 76275941 87887909 32 29 60483482 67523270 76444091 88111704 31 30 60588615 67654691 76612989 88336657 30 31 60694118 67786629 76782641 88562776 29 32 60799995 67919089 76953050 88790069 28 33 60906246 68052073 77124223 89018543 27 34 61012875 68185585 77296165 89248201 26 35 61119882 68319630 77468882 89479054 25 36 61227271 68454208 77642381 89711108 24 37 61335043 68589313 77816665 89944373 23 38 61443202 68724977 77991740 90178856 22 39 61551749 68861175 78167612 90414568 21 40 61660686 68997920 78344287 90651519 20 41 61770013 69135315 78521769 90889717 19 42 61879735 69273018 78700066 91129181 18 43 61989853 69411469 78879183 91369917 17 44 62100367 69550434 79059128 91611941 16 45 62211280 69689963 79239905 91855265 15 46 62322594 69830059 79421520 92099899 14 47 62434312 69970726 79603976 92345849 13 48 62546437 70111967 79787381 92593126 12 49 62658971 70253786 79971439 92841739 11 50 62771918 70396188 80156456 93091699 10 51 62885274 70539174 80342336 93342963 9 52 62999049 70682751 80529087 93595620 8 53 63113241 70826919 80716713 93849647 7 54 63227855 70971684 80905219 94105066 6 55 63342890 71117047 81094612 94361964 5 56 63458352 71263014 81284899 94620181 4 57 63574240 71409586 81476087 94879901 3 58 63690559 71556760 81668183 95141050 2 59 63807309 71704564 81861195 95403639 1 60 63924495 71852975 82055127 95667689 0  9 8 7 6   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 84 85 86 The minutes of the Quadrant for the Scants of the Compliments of the Arches of the same Quadrant. 0 95867689 114737188 143355808 60 1 95933204 115●19970 143954694 59 2 96200195 115505313 144558602 58 3 96468673 115893242 145167595 57 4 96738655 116283797 145781740 57 5 97010253 116676991 146401101 55 6 97283267 117072851 147025745 54 7 97557932 117471403 147655740 53 8 97834057 117872815 148291169 52 9 98111843 118276840 148932108 51 10 98391211 118683794 149578791 50 11 98672171 119093414 150230942 49 12 98954738 119506013 150888966 48 13 99236930 119921335 15155●578 47 14 99524766 120339695 152222283 46 15 99812250 120760985 152897946 45 16 100101400 121185232 153579394 44 17 100392329 121612482 154267179 43 18 100684851 122042752 154961155 42 19 100979193 122476076 155661396 41 20 101275259 122912485 156368008 40 21 101572962 123352014 157081063 39 22 101872522 123794696 157800648 38 23 102173854 124240732 158526854 37 24 102476971 124689836 159259771 36 25 102781890 125142353 159999560 35 26 103088639 125598007 160746121 34 27 103397202 126057149 161499724 33 28 103707656 126519656 162260744 32 29 104019959 126985568 163028671 31 30 104334254 127454936 163804188 30 31 104650345 127927785 164586836 29 32 104968474 128404152 165377268 28 33 105288542 128884078 166175067 27 34 105610566 129367604 166980877 26 35 105934564 129854921 167794536 25 36 106260557 130345812 168615879 24 37 106588558 130840395 169445585 23 38 106918589 131338917 170283495 22 39 107250680 131841076 171129820 21 40 107584955 132347264 171984431 20 41 107921201 132857174 172847712 19 42 108259554 133371390 173719700 18 43 108600151 133889600 174600528 17 44 108942779 134411312 175490331 16 45 109287702 134937471 176389247 15 46 109634817 135467749 177297417 14 47 109984143 136002235 178215000 13 48 110335695 136540955 179142131 12 49 110689503 137083887 180078954 11 50 111045597 137631223 181025951 10 51 111403988 138183016 181982628 9 52 111764699 138739177 182949802 8 53 112127750 139299830 183926988 7 54 112493167 139865032 184915009 6 55 112861097 140435034 185913698 5 56 113231316 141009514 186922883 4 57 113604036 141588910 187943432 3 58 113979204 142172885 188975184 2 59 114356941 142761897 190018342 1 60 114737188 143355808 191073059 0  5 4 3   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.   
 The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant.  The minutes of the degrees of the Quadrant for the Secants of the Arches of the same Quadrant. 87 88 89 The minutes of the Quadrant for the Scants of the Compliments of the Arches of the same Quadrant. 0 191073059 286537048 572987098 60 1 192139567 288943841 582696234 59 2 193218044 291391404 592740072 58 3 194308693 293880683 603139919 57 4 195411723 296413087 613907444 56 5 196527729 298990299 625070305 55 6 197656182 301611807 636642580 54 7 198797665 304279687 648655621 53 8 199952408 306996123 661126359 52 9 201120639 309760533 674090521 51 10 202303011 312576192 687573461 50 11 203498943 315442491 701612741 49 12 204709121 318361849 716229489 48 13 205934200 321336774 731453951 47 14 207173596 324366765 747356168 46 15 208428431 327455509 763965262 45 16 209698119 330602545 781323254 44 17 210983811 333811800 799494739 43 18 212284914 337082830 818524878 42 19 213602421 340419652 838490069 41 20 214936837 343823403 859453551 40 21 216287319 347294586 881484374 39 22 217655350 350837799 904682629 38 23 219040792 354454051 929134899 37 24 220443981 358145679 954945691 36 25 221865261 361914968 982231437 35 26 223305005 365763113 1011112129 34 27 224763453 369095332 1041753449 33 28 226241278 373713015 1074309940 32 29 227738558 377818975 1108967170 31 30 229255785 382016194 1145934768 30 31 230793360 386307709 1185438054 29 32 232351718 390696734 1227777193 28 33 233931261 395186630 1273252703 27 34 235532422 399780916 1322226495 26 35 237156211 404483275 1375118522 25 36 238801972 409397566 1432297932 24 37 240470730 414227875 1494678912 23 38 242163582 419278406 1562622042 22 39 243879838 424453607 1637036239 21 40 245621193 429758156 1718892212 20 41 247386980 435196961 1809365043 19 42 249178956 440775230 1909891150 18 43 250996450 446498305 2022234532 17 44 252841285 452371994 2148642981 16 45 254713463 458402271 2291895669 15 46 256612911 464595485 2455554199 14 47 258541565 470958329 2644450861 13 48 260499426 477497828 2864894681 12 49 262487160 484221619 3125282743 11 50 264505458 491139838 3437843546 10 51 266554348 498256113 3819709423 9 52 268635944 505581634 4297193536 8 53 270750304 513128395 4911255640 7 54 272898206 520901152 5729642566 6 55 275080457 528915798 6875687278 5 56 277297985 537178089 8594018365 4 57 279551349 545702599 11458691197 3 58 281841763 554505091 17188036598 2 59 284170013 563593031 34376072269 1 60 286537048 572987098 INFINITA 0  2 1 0   The degrees of the Quadrant for the Secants of the compliments of the Arches of the same Quadrant.    

A plain Treatise of the first principles of cosmography, and specially of the Sphere, representing the shape of the whole world: Together with all the chiefest and most necessary uses thereof, written by M. Blundevill of Newton Flotman, Anno Dom. 1594  The heavens declare the glory of God, and the firmament showeth his handy work. Psal. 19  depiction of astrolabe 
LONDON Printed by John Windet. 1594.   

The exposition of certain terms or principles of Geometry.  
MInding to treat of the principles of cosmography, and especially of the Sphere, I think it good first to expound unto you certain terms of Geometry, without the which the unlearned shall hardly understand the Contents of this Treatise, which terms are certain principles of Geometry, called definitions: For there are but three kinds of principles, whereon the demonstration of all Geometrical conclusions dependeth: that is, definitions, petitions, and maxims, but I mind here to deal only with certain definitions, Whereof I will set down as many as I think needful for this purpose, in such order as followeth.  
1 A point called in Latin punctus, is a thing supposed to be indivisible, having neither length, breadth, nor depth, as the point or prick a.  
2 A line called in Lataine linea, is a supposed length, having neither breadth nor thickness, as the line a. b.  
3. The ends or bounds of a line, are two points, and of lines some be right, and some be crooked.  
4 A right line is that which goeth right from one point to another, and not bowingly as doth the line c. d. but so strait as is possible, as the line a. b. Again the crooked line is either a whole Circle, or portion of a Circle, or else goeth winding in and out, as a serpent called in Latin Linea tortuosa, or else winding about like the shell of a Snail, called of the Latins, linea Spiralis, as these figures here do show.  
A whole Circle  
A Portion of a circle  
Linea tortuosa.  
Linea spiralis.  
5 Superficies or upperface, is that which only hath length and breadth, without depth, which is twofould, that is to say, plain and crooked.  
6 The bounds of superficies are lines.  
7 A plain superficies is that which lieth strait betwixt his lines, as the figure a. & a crooked superficies is that which goeth bowing and lieth not strait betwixt his lines as the figure b. Again superficies being considered in an hollow body, as a barrel, tun, or vault, may be divided into two other kinds, that is Conuexe, and Concave, for the upper part of such vault is said to be convex, and the inward part cancave as the figure b. showeth.  
Conuex. Concave.  
Right. Crooked. Mixt.  
9 Of plain Angles, some are called right line angles, because that both the lines whereof it consisteth are right, and some are called crooked line angles, because that both lines are crooked, & some are said to be mixed, because the one line is crooked and the other right, as you may perceive by the three sundry shapes thereof made in the Margin.  
10 If one right line standing upon another right line do make two equal angles, that is to say, of each side one, then either of those Angles is a right Angle, and the line so standing upon his fellow, is called the perpendicular or plumb line, as the line a. b. standing upon the line c. d.  
11 A blunt Angle, is that which is greater than a right Angle, as the Angle e.  
12 A sharp Angle, is that which is lesser than a right Angle, as f. so as there be in all three Angles, that is right, blunt, and sharp, in Latin rectus, obtusus, & acutus: And besides these Angles, there be also certain Spherical, that is to say, round Angles, which consist of two circular lines, drawn upon a Spherical superficies, which do cross one another in some point, either rightly, or obliquely: if rightly, than they make right Angles, on each side of the point where they cross, as the figure E. showeth, if obliquely then they make sharp Angles of the inside, and blunt Angles on the out side of the point, where they cross as the figure F. doth partly show, for such Angles cannot be so well described in Plano, as upon the superficies of some Spherical body.  
13 A term called in Latin terminus, is the bound or linute of any thing, as points are the bounds of lines, & lines the bounds of superficieses, and superficieses the bounds of a body, which is that which hath imaginatively, but not materially, both length, breadth, and depth, and if such body have many faces or sides, than it is bounded with many superficieses, as the figure g. made like a six square die. But if such body be round as a bowl, Sphere, or Globe, than it is bounded or covered with one superficies only, as the figure H. doth show.  
14 A figure is that which is comprehended within one bound, or many bounds.  
Circumference.  
15 A Circle is a plain figure bounded with one circular line, which is called the circumference, unto the which as many right lines as are drawn from the point standing in the midst, are all equal one to the other, that is to say, one is as long as another, as the figure I. here doth show.  
diameter.  
16 And the middle point of a Circle is called the Centre, as the point A. in figure I  
17 The Diameter of a Circle, is a right line passing through the Centre, from one side of the Circumference to the other, which divideth the Circle into two equal parts, as the line B. C.  
18 A Semicircle is a figure, contained within the Diameter and half the Circumference of a Circle as the figure D. C. E.  
The cord.  
19 The portion of a Circle, is a part of a Circle, greater or lesser than the Semicircle, as the figures F. and G. do show, the right line in either of which figures is called in Latin chorda, in English the string, and the Circular line in Latin is called arcus in English the bow, but the Diameter is always the greatest cord in any Circle.  
The cord.  
20 Right line figures, are those which are bounded with right lines.  
21 Triangles or three cornered figures, are those which are bounded with three right lines as the figure H.  
22 Four square figures, are those which are bounded with four right lines, as the figure I  
23 Many square figures, are those which are bounded with more right lines then four as K.  
24 Of Triangles or three cornered figures, there be six kinds, whereof the first ●s called Isoplurus, having three equal sides and three equal Angles as A.  
25 The second is called Isosceles, ha●ing but two equal sides and angles as B.  
26 The third called scalenos having ●o side equal, one with the other, but one ●horter or longer than another as C.  
27 The fourth is called Ort hogonius, ●auing one right Angle, as D.  
28 The fift is called ampligonius, because it hath one blunt Angle, as E.  
29 The sixth is called oxigonius, because 〈◊〉 hath three sharp Angles, but not equal ●●des as F.  
30 Of four square figures there is one just four square, having four equal sides, and is right angled as G.  
31 Another is called a long Square, which is right angled, but not having four equal sides, as H.  
32 There is another called rombus, th●● is to say a Turbot, in shape like to Diamant, which hath four equal sides, but is not right angled, as I  
33 There is another called romboides, which though it hath sides and Angles one right against another, yet hath it neither four equal sides, nor yet is right angled, as K.  
34 All other kinds of four squares are called trapezia, as L. or such like.  
Right line parallels.  
Circular parallels.  
35 Parallels are two right lines equally distant one from another, which being drawn forth infinitely, would never tou●● or meet one another in any part, as th● two lines M. N. And though the lines be● circular, yet if they be equally distant in a●● places one from another, they are also call●● parallels, as the figure P. showeth, y●● though the lines be winding in and out 〈◊〉 Serpents, yet if they be equally distant in 〈◊〉 places one from another, they are parallel as well as if they were right lines as the ●●gure Q. showeth.  
Serpentine parallels.   

The order and contents of this Treatise, touching the first principles of Cosmography, and specially of the Sphere.  
Having set down the exposition of certain terms of Geometry, for the better understanding of this Treatise, I do first define what Cosmography is, and therewith do briefly show what Sciences it comprehendeth, and who were first inventors thereof: That done, because the Sphere doth represent the shape of the whole world, I do define what a Sphere is, and then I do divide the world into two essential parts, that is, the Celestial and Elemental part, according to which two parts I do also divide this Treatise into two Books, the first whereof treateth of the Celestial parts, and the second of the Elemental part of the world, with what order the Chapters of each Book hereafter following, do plainly show.  

The Chapters and Contents of the first Book.  
OF Cosmography, what it is, and what kind of Sciences it comprehendeth, and who were first inventors thereof, Chap. 1.  
The definition of a Sphere, and of the unity, roundness, and capacity of the world, also of the Poles and axle-tree thereof. Chap. 2.  
Of the division of the world, and of the two Essential parts thereof, and what things each part containeth. Chap. 3.  
A figure of the whole world, wherein are set forth the two Essential parts before mentioned, that is to say, the eleven heavens and the four Elements.  
A demonstration to prove the plurality of the heavens, Chap. 4.  
Of the highest Sphere or heaven, called the Imperial heaven. Chap. 5.  
Of the tenth Sphere or heaven called in Latin Primum mobile, that is, the first movable, & what motion it hath. Chap. 6.  
Of the ninth heaven, what motions & names it hath, & whether there be any waters above the firmament or not. Chap. 7.  
Of the eight heaven what motions it hath, and what circles are imagined by the Astronomers to be in that heaven, and to what use and purpose they serve: Also in what time every one of the seven Planets maketh his revolution, & of what thickness their Spheres be. Chap. 8.  
Of the circles whereof a material Sphere consisteth, and of their divers divisions. Chap. 9  
Of the Equinoctial line, why it is so called, and of the diverse uses thereof. Chap. 10.  
A figure showing the Equinoctial line, the two poles and the axle-tree of the world.  
Of the zodiac, why it is so called, and of the 12. signs therein contained. Also of the Latitude, Longitude and declination thereof. Chap. 11.  
How much the zodiac declineth from the Equinoctial towards either of the Poles, and of the greatest declination of the sun, what it is at this present, and what it hath been● in times past. Chap. 12.  
How to know the quantity of the suns declination, be it Northward or Southward, every day throughout the year aswell by a Table as by help of a material Sphere or globe▪ Chap. 13.  
An instrument to know thereby in what sign or degree the sun is every day throughout the year.  
A Table showing the declination of the sun every day throughout the year lately calculated.  
Upon what poles the zodiac turneth about, also of the Ecliptic line and of the diverse uses thereof. Chap. 14.  
Of the Eclipses both Solar and Lunar, & of the head & tail of the Dragon with certain figures showing the same. cap. 15.  
Of the two Colours, why they are so named, and whereto they serve: also of the four Cardinal points, that is, the tw● Equinoctial, and the two Solsticiall points, and of the entrance of the sun into any of those points, or into any other sign. Chap. 16.  
Of the Horizon both right and obliqne, making thereby three kinds of Spheres, that is, the right, the parallel, and the obliqne Sphere. Chap. 17.  
A figure showing the Latitude of any place to be equal unto the elevation of the Pole.  
Of the Meridian, and of the uses thereof. Chap. 18.  
Of the vertical circle and uses thereof, whereof no mention is commonly made by those that writ of the Sphere. Chap. 19  
Of the four lesser circles, that is to say, the circle Arctic, the circle Antartique, the Tropic of Cancer, and the Tropic of Capricorn, and also of the five Zones, that is to say, two cold, two temperate, and one extremely hot. Chap. 20.  
A figure showing the five foresaid Zones.  
A Table showing how many minutes are requisite to make one degree, in every lesser circle answerable to one degree of the Equinoctial.  
Of the stars and celestial bodies contained in the firmament, and first of their substance. Chap. 21.  
Of the moving and shape of the stars. Chap. 22.  
Of the number of the stars, and of their magnitude and greatness, and into how many Images they are divided, & how many stars every Image containeth. Chap. 23.  
Of the xii. Images or signs of the zodiac. Chap. 24.  
Of the xxj. Northern Images. Chap. 25.  
Of the 15. Southern Images. Chap. 26.  
Of the longitude of the fixed stars, and of the procession of the vernal Equinoctial point, and what it is. Chap. 27.  
Of the Latitude of the fixed stars. Chap. 28.  
Of the Declination of the fixed stars. Chap. 29.  
Of the ascension and dissension, that is the rising and setting of the stars, aswell according to the Astronomers, as according to the Poets. Chap. 30.  
Of the Astronomical ascension and dissension in general both right, mean, and obliqne, & what a given ark is. Cap. 31.  
Of the right, obliqne, and mean ascension in particular, and of the chief causes of such diversity of ascensions. cap. 32.  
How to know the diversities of the ascensions and dissensions, as well in the right as obliqne Sphere. Chap. 33.  
Of the ascentionall difference and uses thereof. Chap. 34.  
Of the threefold Poetical rising and setting of the stars, Chap. 35.  
Of time, what it is, and into what parts it is divided. chap. 36.  
Of the year, and of his diverse kinds, and of the diverse computations had thereof in diverse ages, and amongst divers Nations. Chap. 37.  
Of the suns year called in Latin annus solaris, and of the diverse kinds thereof, and first of the Tropical year, both equal and unequal. Chap. 38.  
Of the Syderall year, & how much it containeth. chap. 39  
Of the Political year, and diverse kinds thereof. chap. 40.  
Of the julian year, why it is so called, and of the 2. kinds thereof, that is, the common year, and the bisextile year, otherwise called the leap year. Chap. 41.  
Of the Egyptian year, and how many days it containeth. Chap. 42.  
How many Moons the jews year, and the Athenian year do contain. Chap. 43.  
Of the year Lunar, and of the kinds thereof. Chap. 44.  
Of the divers kinds of months, and into what parts every Solar month is divided according to the Romans, that is, into Kallends, Nones, and Ideses. Chap. 45.  
Of the diverse kinds of months Lunar. Chap. 46.  
Of a week. Chap. 47.  
Of days and nights both natural and artificial, as well in the right Sphere, as in the obliqne Sphere. Chap. 48.  
Two figures showing the right and obliqne Sphere, together with the arckes of the days and nights in each Sphere. Chap. 49.  
How to found out by the material Sphere or Globe, and by help of the ascentionall difference before defined, the increase & decrease of every day throughout the year in every several latitude, & at what hour the sun riseth & setteth. c. 50  
How to know by the material Sphere or Globe, in what part of the Horizon the sun riseth and setteth every day, and thereby the length of the day. Also how to know the Meridian altitude of the sun every day throughout the year, & being at his Meridian altitude, to know how far distant he is from the Zenith every day, Chap. 51.  
Of hours as well equal as unequal, and into what parts they are divided. Chap. 52.  
How and in what manner the jews do divide the artificial day and night, each of them into four quarters. chap. 53.  
How to know what Planet reigneth in every hour of the day or night artificial, as well by help of a table, as by a rule contained in one verse. chap. 54.  
Here end the Contents of the first Book.   

The Chapters and Contents of the second Book, containing the Elemental part of the world.  
OF the Elemental part of the world. Chap. 1:  
Of the fire, his nature and motion, Chap: 2:  
Of the Air, and into how many Regions it is divided, Chap. 3.  
A figure showing the three several Regions of the air.  
Of the water, and whether it be round or not. Chap. 4.  
A figure showing that the water is round.  
Of the Earth, and whether it be in all parts round or not. Chap. 5.  
Of the compass of the earth, and of the diversity of measures, according to diverse countries. chap. 6.  
Of the Longitude and Latitude of the earth. chap. 7.  
A figure showing the Longitude and Latitude of the world.  
How to know the Latitude of any place, as well in the day as in the night. chap. 8.  
How to know the true Longitude of any place. chap. 9  
A ready way to find out the Longitude of any place invented by Gemma Phrizius. chap. 10.  
Another way taught by Appian to found out the Longitude of any place with the cross-staff, by knowing the distance betwixt the Moon and some known star that is situated nigh unto the Ecliptic line. chap. 11.  
How to know the distance of places, that is to say, how many miles one place is distant from another, and how many ways one place is said to differ in distance one from another. Chap. 12.  
A Table showing how many miles are answerable to one degree of every several Latitude.  
How to know by the help of the foresaid Table, the distance of two places differing only in Longitude. chap. 13.  
How to find out the true distance of two places differing both in Longitude and Latitude, by the Arithmetical way. chap. 14.  
Also how to found out the same by help of a Demicircle divided into 180. parts, without any Arithmetic.  
How to find out the distance of places by the Geometrical way. chap. 15.  
Of the five Zones. chap. 16.  
A figure of the foresaid five Zones, that is to say, two cold, two temperate, and one extremely hot.  
Of Parallels. chap: 17.  
A figure showing the 21. Parallels.  
The Table of Parallels, showing how many degrees and minutes every one is distant from the Equinoctial, made according to the rule of Ptolemy.  
Of Climes both old and new. chap. 18:  
A figure showing the order of the seven ancient common climes whereunto Appian addeth two more, making in all nine climes.  
A Table showing the degrees of Latitude in the beginning, midst, and end, of every one of the foresaid nine climes.  
A Table showing the longest day in every degree of Latitude, proceeding orderly from the Equinoctial to the North pole, by whole degrees without minutes from one degree to 90.  
A Table showing how many miles every one of the seven climes do contain, aswell in breadth, as in length.  
Of the diverse seasons and shadows incident to diverse climes and parallels, and first what seasons and shadows they have that devil right under the Equinoctial. chap. 19  
Of the seasons & shadows which they have that devil betwixt the Equinoctial and the Tropic of Cancer. chap. 20.  
Of the seasons and shadows which they have that devil right under the Tropic of Cancer. chap. 21.  
Of the seasons and shadows which they have that devil betwixt the Tropic of Cancer, and the circle Arctic. chap. 22.  
Of the seasons and shadows which they have that devil right under the circle Arctic, and how long their day is. chap. 23:  
What seasons, shadows, and length of day they have that devil betwixt the circle Arctic, and the pole Arctic. chap. 24.  
Of those that devil right under the Pole. chap. 25.  
By what names certain inhabitants of the earth are called, according to the diversity or likeness, as well of their shadow, as of their situation. chap. 26.  
By what names certain parts of the earth are called by reason of their diverse shapes. chap. 27:  
Of the wind, what it is, what motion it hath, and of the diverse names and divisions thereof. chap. 28.  
Of the nature and quality of the ancient twelve winds. chap. 29.  
Of the modern division of the winds. chap. 30.  
A figure of the 32. winds representing the Mariners compass.    

The first part of the Sphere.  

Of Cosmography, what it is, and what kinds of Sciences it comprehendeth, and who were first inventors thereof. Chap. 1.  
WHat is Cosmography?  
Cosmography is the description of the whole world, that is to say, of heaven and earth, and all that is contained therein.  
What special kinds of knowledge are comprehended under this Science.  
These four, Astronomy, Astrology, Geographie, and chorography.  
What is Astronomy?  
Astronomy is a Science, which considereth and describeth the magnitudes and motions of the celestial or superlour bodies.  
Which call you superior bodies?  
The Spheres or Heavens, and the Stars as well fixed as movable, which we shall define hereafter in their proper places.  
What is Astrology?  
It is a Science which by considering the motions, aspects, and influences of the stars, doth foresee and prognosticate things to come.  
What is Geographie?  
It is a knowledge teaching to describe the whole earth, and all the places contained therein, whereby universal Maps and Cards of the earth and sea are made.  
What is chorography?  
It is the description of some particular place, as Region, I'll, City, or such like portion of the earth severed by itself from the rest.  
Who were first inventors of these Sciences.  
Some say that Atlas was the first inventor, whom the Poets sane to bear up the heavens with his shoulders, having his head placed in the North Pole, and his feet in the South Pole, and his right hand bearing the East part, and his left hand the West part of the world. Albeit that some apply this fiction of the Poets to an high mountain in Africa called Atlas, which for his great height surmounting the clouds, is said to bear up the skies. But some say that Adam our first Parent, was the first inventor thereof: And some others affirm that Abraham was the first inventor thereof. But whosoever was the first inventor, it well appeareth by Ptolemy his book, called Almagesti, that he hath been no small furtherer thereof, and since his time in these latter days Georgius Purbacchius, johannes de Monte Regio, Copernicus, and diverse others have learnedly treated thereof. But leaving to speak of the first inventors, or of the furtherers of these sciences, I will speak of the shape, capacity, and unity of the world, and because the shape thereof is likened to a round body called a Sphere, I will first define what a Sphere is.   

The definition of a Sphere, and of the unity, roundness, and capacity of the world, also of the poles and axeltre thereof. Cap. 2.  
WHat is a Sphere?  
A Sphere is a round body, contained or covered with one only superficies or upperface, in the midst of which body is a point or prick, called the Centre, from which all right lines being drawn to any part of the superficies or upper face thereof, are of equal length, as you may perceive by this figure here following, the Centre whereof is marked with the letter A.  
And this Sphere before defined representeth the frame of the whole world, containing all things, and is contained of none.  
How prove you that there is but one world?  
By the authority of Aristotle, who saith, that if there were any other world out of this, than the earth of that world would move towards the Centre of this world, which cannot be, unless it should first ascend from his own Centre to the superficies or upper face of the same, which is against nature, for every heavy thing naturally tendeth downward, and every light thing upward, which common experience showeth: for a stone being cast upwards, will fall downwards, and the flame of a candle being turned downward, will fly upward.  
How prove you the frame of the world to be round?  
By three reasons, First by comparison, for the likeness which it hath with the chief Idea or shape of God's mind, which hath neither beginning nor ending, and therefore is compared to a Circle. Secondly by aptness aswell of moving as of containing, for if it were not round of shape, it should not be so apt to turn about as it continually doth, nor to contain so much as it doth, for the round figure is of greatest capacity and containeth most. Thirdly, necessity proveth it to be round, for if it were with angles or corners, it should not be so apt to turn about, and in turning about, it should leave void and empty places, which nature abhorreth, for no place by nature can be without a body, nor a body without a place.  
Whereupon is this Sphere or great round frame turned?  
Upon two most firm and immovable hooks called in Latin Cardines mundi, and in Greek Poli, derived of the verb Polo, which is as much to say, as to turn, for as a door turneth upon his hooks, so the world turneth upon his two Poles, whereof the one is fixed in the North, and the other in the South, & the North pole is called in Latin Polus arcticus, and the South pole is called Antarcticus, through which Poles from the one to the other passeth a right immaginative line called of the Astronomers the Axeltrée of the world, about the which the world continually turneth like a Cart wheel, which you may see lively expressed in a material Sphere made for the purpose, for in Plano it cannot be so well described.   

Of the division of the world, and of the two essential parts thereof, and what things each part containeth. Cap. 3.  
HOw is the world divided?  
Most writers do make a twofold division thereof, that is according to substance, and according to accidents, which accidents because they are none other thing, but points, lines, and circles, and so incident and necessarily belonging (by position though not by nature) to the substantial parts, as the one can not be described without the other, neither the subject without the accidents, nor the accidents without their subject, I will therefore make but one division of the world, dividing the same into two essential parts, that is, the celestial part, and the Elemental part, in the description of which parts, it shall also plainly appear what those lines and Circles be, and to what end they serve.  
What doth the celestial part contain?  
The eleven Heavens and Spheres.  
Make you no difference betwixt these Spheres, and the great Sphere which you last defined to be covered with one superficies or upperface.  
Yes verily, for these have two superficies or upper faces, that is, the Conuexe and Concave.  
Which call you Conuexe and which Concave?  
The Conuexe is the outermost upper face of any thing that is round, and therewith hollow, as of an oven or vault, and the Concave is the innermost or hollow superficies of the said oven or vault, or of any such like round body having concavity or hollowness.  
Which be those eleven Spheres or Heavens whereof you did last speak?  
In ascending orderly upwards from the Elements they be these, The first is the Sphere of the Moon, The second the Sphere of Mercury, The third the Sphere of Venus, The fourth the Sphere of the Sun, The fift the Sphere of Mars, The sixth the Sphere of jupiter, The seventh the Sphere of Saturn, The eight the Sphere of the fixed Stars, commonly called the firmament. The ninth is called the second movable or Crystal heaven, The tenth is called the first movable, and the eleveuth is called the Imperial heaven, where God and his Angels are said to devil.  
What doth the Elemental part contain?  
It containeth the four Elements.  
Which be those?  
The Element of the fire which is next to the Sphere of the Moon, and next to that more downward is the Element of the air, and next to that is the Element of the water, and next to that is the earth, which is the lowest of all. All which things you may see here plainly set forth in the figure following on the other side.  
A figure of the whole world, wherein are set forth the two essential parts before mentioned, that is to say, the eleven heavens and the four Elements.  


EARTH  
AER  
FIRE  
1 ☽ TE MOON COLD AN MOIST BENEVOLENT SILVER  
2 ☿ MERCURI SUCHE AS HE IS JOINED with QUCKSYLVER.  
3 ♀ VENUS COLD AN MOIST BENEVOLENT COPPER  
4 ☉ THE SON HOT AN DRY BENEVOLENT GOLD  
5 ♂ MARS HOT AN dry. MALEVOLENT IREN  
6 ♃ JUPITER HOT AND MOIST BENEVOLENT TYN  
7 ♄ SATURN · COULD AN DRY MALEVOLENT lead  
8 LEO * 4 / ♋ * 3 ♊ * 2 ♉ * 1 ♈ * 12 ♓ * 11 ♒ * 10 ♑ * 9 ♐ * 8 ♏ * 7 ♎ * 6 ♍ *  
9 THE CRISTALINE HEAVEN ●  
10 · THE FIRST · MOVABLE.  
11 · THE EMPYREAL HEAVEN. THE ABITATION · OF · THE BLESED·     

A demonstration to prove the plurality of the heavens. Chap. 4.  
HOw is it to be proved that there is such multiplicity of heavens, sith to the eye it seemeth one whole body?  
If there were not diverse heavens and diverse motions, there should neither be generation nor corruption of any thing, for all things than should be as one, and ever in one estate, but besides that reason men have found out by experience in observing the diverse rising and setting of diverse wandering stars called the Planets, that every such Planet hath a several Sphere or heaven by itself, for otherwise they should continually keep their places as the fixed stars do, which do move altogether according to the moving of the viii. heaven, wherein they are placed.  
I grant that there may be diverse heavens or Spheres. But how prove you that there be ten movable heavens, besides the imperial heaven before mentioned, and specially sith of the old writers, some affirm that there be but nine, and some have said that there be no more than eight.  
The latter writers have found by good observation (as they say) that the firmament or eight heaven hath three several motions wherewith it is moved and turned about, which could not be, unless there were two movable heavens higher than that, as shall be declared when we come to show the threefold motion of the said eight heaven, for here I mind briefly to describe unto you all the heavens beginning at the highest, and so proceed to the lowest.   

Of the highest Sphere or Heaven, called the imperial heaven. Chap. 5.  
THe imperial heaven (as our ancient Divines affirm) is immovable and being created by God the first day that he began the first creation of the world, was by him immediately replenished with his ministers the holy Angels, and this heaven being the foundation of the world, is most fine and pure in substance, most round of shape, most great in quantity, most clear in quality, and most high in place, where God and his Angels are said to devil.   

Of the tenth Sphere or heaven, called in Latin primum mobile, that is, the first movable, and what motion it hath. Chap. 6.  
THis heaven is also of a most pure and clear substance and without stars, and it continually moveth with an equal gate from East to West, making his revolution in 24. hours, which kind of moving is otherwise called the diurnal or daily moving, & by reason of the swiftness thereof, it violently carrieth & turneth about all the other heavens that are beneath it from East to West, in the self same space of 24. hours, whether they will or not, so as they are forced to make their own proper revolutions, which is contrary from West to East, every one in longer or shorter time, according as they be far or near placed to the same: yet some writers do not admit any violence in the heavens, but that ever the heaven hath his proper and natural moving without any violence.  
I do not well understand by what reason the first movable should have such force over the rest.  
For your better understanding how this may be, suppose that you turn with your hand from East to West a Grindstone, or some other turning wheel, whereon is placed a Fly or some other creeping worm which cometh towards you from West to East, & you shall easily conceive that by your swift turning of the wheel, you shall turn about the said Fly or Worm many times contrary to her own course, whether she will or not, before she can get about the said stone or wheel.  
That is true, proceed therefore to the rest, and tell the nature and moving of the ninth heaven.   

Of the ninth heaven, what motions and names it hath, and whether there be any waters above the firmament or not. Chap. 7.  
THe ninth heaven is also clear of substance, and without stars, having two moovinges, the one from East to West upon the poles of the world, according to the daily moving of the first movable, and the other from West to East upon his own poles, according to the succession of the signs of the zodiac, which is in the first movable, turning so slowly about as it maketh but one degree in 100 years, and accomplisheth his full revolution in 36000. years, or as Alfonsus saith in 49000. years.  
Is there then in the first movable any zodiac, that is to say, a broad Circle carrying the 12. signs and yet no stars?  
Yea though there be no stars, yet many superstitious fools do imagine that there be diverse Characters and lineaments not to be seen, but with a most sharp and subtle sight (by virtue of which Characters & feigned constellations) they imagine that they can work wonders and strange effects, enough to deceive themselves and others too.  
What is the cause that the ninth heaven is so long in making his proper revolution.  
Because as I said even now, this heaven is placed next to the first movable, which carrieth him about contrary to his own course with such violence as he cannot make his own proper revolution so soon as the other heavens which are placed further off.  
If the ninth heaven be so long in making his course, it will never be complete whilst the world lasteth, for the whole age of the world according to some, is but 6000. years?  
Ye Plato was of an other opinion, and therefore this revolution was called, magnus annus Platonis, that is to say, the great year of Plato, because he affirmed that when this revolution was once complete, all things should be in the same estate wherein they were at the first, and that he should then stand reading to his Scholars in the self same chair wherein he stood at that present, which fond opinion S. Augustine confuteth in his 12. Book de Civitate Dei, speaking in this manner, God forbidden (saith he) that we should credit these things, for Christ died once for our sins, and being risen again from the dead, he is no more to die, neithet can death have any more power over him, by virtue of whose resurrection, we also that believe shall rise again, and devil with God for ever.  
With what names is this ninth heaven called?  
Some do call it the Crystalline heaven, because of the cleareves thereof, and some the watery heaven, because the Scripture affirmeth that there be waters above the firmament as we read in Gen. cap. 1. let the firmament be made, and divide the waters from waters. Again we read in the Psalms, all ye waters that be above the firmament, bless ye the Lord.  
The natural Philosophers allow no waters to devil above the firmament.  
That is true, yet notwithstanding if the holy Scriptures do manifestly affirm that there be waters above the firmament, it behoveth a Christian to believe it, but question perhaps may be moved, what manner of waters they are that are above the firmament, whether such as breed rain, or whether they are only to be referred to the Crystalline heaven, because it is of a watery substance, and therefore of some is called the watery heaven, affirming it to be placed next to the primum mobile, or first movable, to the intent that with the coldness thereof, it might assuage and repress the extreme heat of the same primum mobile, which otherwise (as some affirm) with his swift & violent moving, would set all the heavens on fire, and yet no rain is bred therein, for the great rain that drowned the world in Noahes time, did not fall from above the firmament, but from the Air, which in the holy Scripture is many times signified by this Latin world Coelum, that is to say, Heaven, as when the Scripture saith, the flood gates of heaven were opened, is as much to say as the floodgates of the Air were opened, etc. but leaving this question, let us proceed to the eight heaven.   

Of the eight heaven what motions it hath, and what circles are imagined by the Astronomers to be in that heaven, and to what use and purpose they serve: Also in what time every one of the seven Planets maketh his revolution, & of what thickness their Spheres be. Chap. 8.  
THe eight heaven otherwise called the firmament, is a most glorious heaven adorned with all the fixed stars.  
Why are they called fixed?  
Because they are fastened in this heaven, like knots in a knotty board, having no moving of themselves, but are moved according to the moving of this eight Sphere or heaven wherein they are fixed.  
How many fold is the moving of this heaven?  
The moving of this heaven (as hath been said before) is threefold, for first it turneth about every day from East to West in 24. hours, according to the moving of the first movable, otherwise called the diurnal moving. Secondly it moveth from West to East, according to the moving of the ninth heaven which maketh but one degree in 100 years, and this moving is called motus angium Stellarum fixarum.  
What are Anges?  
They be certain imagined points in the heaven, notifying the furthest distance of any Orb or Sphere from the Centre of the world: Thirdly it moveth sometime towards the South, and sometime towards the North, by virtue of his own proper moving, called in Latin motus trepidationis, that is to say, the trembling moving, whereby it is moved upon two little Circles, the poles whereof are in the beginning of that Aries and that Libra which are imagined to be in the ninth heaven, the Semidiameter of which little Circles is 4. degrees 18′· and 43″· & maketh his whole revolution in 7000. years, and this moving is otherwise called motus accessus & recessus, that is to say, the moving of approaching and retiring, and is only proper to the 8. Sphere: for every natural body by order of nature, can have but one proper moving, and by reason of this moving the fixed stars be not always equally distant from the immovable poles of the tenth Sphere, nor always under the Ecliptic of the said 10. Sphere, but oftentimes clean without it. Nor the fixed stars are in equal times equally distant from the beginning of that Aries & Libra, which are imagined to be in the tenth heaven, but they seem to move sometimes towards the East, and sometimes towards the West, now more swiftly, and now more slowly. For Ptolemy in his time found them to be moved in 100 years one degree. And the latter observers have found them when they be in their swift motion to be moved one degree in 63. years, all which three movings are easy (as some affirm) to be demonstrated by an instrument made of purpose, showing every several moving of this heaven by itself, but such demonstrations as some others think be not altogether necessary conclusions: for it may be that the daily moving is common to the whole frame, and that the moving of the Angles of the fixed stars, is proper to the eight Sphere, having some other mover, and the like objection may be made touching the trembling moving, called motus trepidationis. But now you have to understand that for the better describing, dividing, and measuring of this heaven, and all the appearances thereof, the Astronomers with the Pencil of imagination, or rather of most necessary invention, do (as it were) trace the same with certain Circles both greater and lesser, whereof we shall speak so soon as we have briefly showed the moving of the rest of the heavens, contained in the celestial part of the world, in such order as they follow one another. For next to the firmament is the heaven of Saturn, which maketh his revolution from West to East in 30. years, next to him is the heaven of jupiter, who maketh his revolution from West to East in 12. years, next to him is the heaven of Mars, which maketh his revolution also from West to East in 2. years, next to him is the heaven of the Sun, which maketh his revolution from West to East in 365. days, and 6. hours lacking certain minutes, next to him is the heaven of Venus, which maketh her revolution from West to East in like time as doth the Sun, next to Venus is the heaven of Mercury, which maketh his revolution from West to East in the like space that Venus doth: And next to Mercury is the heaven of the Moon, which maketh her revolution from West to East in 28. days and certain minutes. Thus having briefly described the eleven heavens, I will now treat of the Circles that are imagined (as I said before) by the Astronomers to be in the eight heaven or firmament, to the intent that the measures and distances of the stars, images, signs, and other appearances therein contained, might be the better demonstrated.  
Before you deal with these Circles, I would gladly understand some reason why all these several heavens seem to the eye but as one entire body.  
That is because they are all clear and transparent like fine birall glass, or Crystal, through the which the sight doth easily pierce, though there were never so many coats of such clear substance, covering one another like the scales of an Onion, for so the heavens do cover and enclose one another, and every one is of an exceeding great thickness.  
Why how thick is every such heaven?  
The heaven of the Moon containeth in thickness.— 
105/222 2/33 miles.  The heaven of Mercury containeth— 
253372 ⅔.  The heaven of Venus— 
3274494 6/11.  The heaven of Sol,— 
343996 4/11  The heaven of Mars,— 
26/308/800.  The heaven of jupiter,— 
1899654 6/11.  The heaven of Saturn,— 
19604454 6/11.   
What need have the heavens to be so thick?  
Because otherwise they could not contain each one his star or stars, for there is no fixed star so little, but that it is far greater in compass then the earth, neither is there any wandering star, but that it is bigger than the earth, the Moon, Venus, and Mercury excepted.  
For the Sun containeth the earth— 
166. times.  Saturn containeth it— 
95. times.  jupiter containeth it— 
91. times.  Mars containeth it— 
2. times.  Venus is less than the earth— 
39 times.  Mercury is lest of all and is contained of the earth,— 
3143. times.   
You have well satisfied me touching these matters, and therefore now proceed with the Circles if you think so good.  
Before that I describe unto you the Circles whereof a Sphere is made, or declare the uses thereof: I think it not amiss here to set down the shape of a Sphere, together with the names of every Circle written upon the same, to the intent that you may be acquainted with all the parts of a Sphere before you come to use the same: which shape or figure was first drawn by Master Blagrave, and is set down in his book called the Mathematical jewel.   depiction of astrolabe  

Of the Circles whereof a material Sphere consisteth, and of their diverse divisions. Chap. 9  
OF Circles which are imagined to be in the firmament, and whereof a Sphere representing the shape of the world is commonly made, be in all 10. that is to say, the Equinoctial, the zodiac, the two Colours, the Horizon, the Meridian, the two Tropiques and the two polar Circles, of which Circles some be greater and some lesser.  
Which call you greater and which lesser?  
Those are called the greater Circles, which passing through the Centre or midst of the firmament, do divide the whole circuit thereof into two equal parts, and of such Circles there be six, that is to say, the Equinoctial, the zodiac, the Colour of the Solstices, the Colour of the Equinoxes, the Horizon and the Meridian. The lesser Circles are the four last mentioned, that is the two Tropiques, and the two polar Circles, which are called lesser, because they do not divide the world into two equal parts, but into unequal portions.  
What other divisions do the Astronomers make of the said Circles?  
diverse, for some are said to be parallels, some right, some obliqne, some movable, and some immovable.  
Which are said to be parallels?  
The two polar Circles, and the two Tropiques, and the Equinoctial which is in the very midst of them all.  
Which are said to be right and which obliqne?  
The right be the two Colours, the right Horizon, & the Meridian, because they cut the Sphere or Globe with right Angles: And the obliqne are these two, that is, the zodiac, and the obliqne Horizon, which are said to be obliqne because they cut the Sphere or globe with obliqne Angles.  
Which are said to be movable, and which immovable?  
The movable are these, the Equinoctial, the zodiac, the two Tropiques, & the 2. polar Circles which are said to be movable, because they continually move with the firmament & are like in all places. And these also having respect to a material Sphere, are said to be intrinsical or inward. The immovable are these 2. that is, the Horizon, & the Meridian, which are said to be immovable because in the turning of the Sphere they remain unmoved, for though we change both Meridian and Horizon by going from one habitation to another upon the earth, yet every place hath still his own Meridian and Horizon, which do remain immovable, and these two Circles having respect to a material Sphere, are said to be extrinsical or outward, because they do enclose on the outside all the other Circles of a material Sphere. Thus having set down the divisions of the said Circles, I will now describe them all in order, and show to what uses they serve, beginning first with the great Circles, and so proceed to the lesser, and first I will speak of the Equmoctiall.   

Of the Equinoctial line, why it is so called, and of the diverse uses thereof. Chap. 10.  
WHat is the Equinoctial?  
The Equinoctial is a great Circle, which being in every part equally distant from the two poles of the world, divideth the Sphere in the very midst thereof, into two equal parts, and therefore it is called of some, the girdle of the world.  
Wherhfore is it called by the name of the Equinoctial?  
Because that when the Sun toucheth this Circle, which is twice in the year, the day and night is of equal length throughout the whole world.  
At what time of the year is it Equinox?  
In the spring of the year, about the 11. day of March when as the Sun entereth into the first degree of Aries, and again in Autumn about the 13. of September when as the Sun entereth into the first degree of Libra, And by reason that this Circle divideth the world in the very midst, those that devil right under it, are said to have no Latitude either Northward or Southward, to whom the days and nights are always equal. But if they devil any thing distant from the Equinoctial, towards any of the poles, than they are said to have Latitude more or less, either Southward or Northward, as shall be declared hereafter more at large, when we come to treat of the Longitude and Latitude of the earth, both which are to be known by help of the Equinoctial.  
To what other uses serveth this Circle?  
This Circle hath many most necessary uses, for first it showeth the daily moving of the first movable, which maketh his revolution in 24. hours, which hours are equal, and are to be measured by the degrees of the Equinoctial, by allowing 15. degrees thereof to an hour. And therefore the degrees of the Equinoctial are commonly called in Latin of the Astronomers tempora, of which degrees it containeth 360. which being divided by 15. do make 24. hours, which is a natural day, containing both day and night: Moreover it showeth the declination of the fixed stars and their right ascensions, whereof we shall speak when we come to treat of the stars, and by his equal motion, all the inequalities of the zodiac, & of all the signs contained therein, are measured: And to be short, this Circle hath many other good uses not to be declared until you have some knowledge of the rest of the other Circles hereafter described.  
A figure showing the Equinoctial line, the two Poles and axle-tree of the world.   

Of the zodiac why it is so called, and of the 12. signs therein contained: Also of the Latitude, Longitude, and declination thereof. Chap. 11.  
WHat is the zodiac?  
It is a broad, obliqne, or stoup Circle, having a circular line in the midst thereof, called the Ecliptic line, and divideth the Sphere into two equal parts, by crossing the Equinoctial with obliqne Angles in two points, that is, in the beginning of Aries, and in the beginning of Libra, so as the one half of this broad Circle declineth towards the North, and the other half towards the South: in which Circle many things are to be considered, first the name, than what breadth it hath, and why it hath such breadth, and into what parts it is divided aswelt touching the breadth or Latitude, as circuit or Longitude thereof, also into how many parts the firmament is divided by the spaces described in the zodiac, and appointed to the 12. signs. Also how much it declineth from the Equinoctial towards the South or North, and upon what poles it turneth about, and why the line in the midst is called the Ecliptic line, and many other necessary points, which I mind here briefly to touch.  
Why is this Circle named the zodiac?  
It is named the zodiac either of this Greek word zoe, which is as much to say as life, because the sun being moved under this Circle, giveth life to the inferior bodies, or else of this Greek word Zodion, which is as much to say as a beast, because that 12. Images of stars, otherwise called the 12. signs, named by the name of certain beasts, are form in this Circle: and therefore the Latins do call this Circle Signifer because it beareth the 12. signs.  
How are these signs called, and with what Characters are they marked.  
Their names and Characters this table here following doth show, and also which be opposite one to another, as Aries to Libra, Taurus to Scorpio, and so forth.  
The six Northern signs: The six Southern signs. Aries ♈ Libra ♎ Taurus ♉ Scorpio ♏ Gemini ♊ Sagittarius ♐ Cancer ♋ Capricornus ♑ Leo ♌ Aquarius ♒ Virgo ♍ Pisces ♓  
Of which signs the first six on the left hand are called the Northern signs, because they are contained in that half of the zodiac, which declineth towards the North. And the other six on the right hand being right opposite to the first 6. are called the Southern signs, because they are contained in the other half of the zodiac, declining towards the South: And against every sign is set his proper Character.  
What divisions do Astronomers make of the 12. signs?  
diverse, as followeth: for some are said to be ascendent, and some descendent.  
Ascendent are those that rise from the South towards our Zenith, tending Northward, which be these, Capricornus, Aquarius, Pisces, Aries, Taurus, and Gemini.  
The descendent are these, Cancer, Leo, Virgo, Libra, Scorpius, and Sagittarius.  
Again some are said to be vernal, as Aries, Taurus, and Gemini.  
Some Estivall as Cancer, Leo, Virgo.  
Some Autumnal as, Libra, Scorpio and Sagittarius.  
And some Hiemall or Brumal, as Capricornus, Aquarius, and Pisces, signifying the four seasons of the year, that is to say, the Spring, summer, Autumn, and Winter.  
And some do make diverse other divisions, which because they appertame to Astrology rather than to the Treatise of a Sphere, I willingly omit them.  
Now tell what breadth the zodiac hath, and why it is imagined to have such breadth?  
It hath (according to the ancient writers) 12. degrees in breadth, that is to say, 6. degrees on the one side of the Ecliptic line, & 6. degrees on the other side of the Ecliptic line, but according to the modern writers, it hath 16. degrees in breadth, that is, eight degrees on each side of the Ecliptic line, which breadth was necessarily imagined, first to the intent that by the measures and degrees thereof it might be known, how much the Planets (otherwise called the wandering stars, whose course is to pass continually under this broad Circle) do wander at any time on either side of the Ecliptic line, for they all wander, but some more, some less, the Sun only excepted, which never swerveth from that line, but always goeth right under the same, and therefore the said line is called of some writers the way of the Sun: And secondly it hath such breadth to the intent it may contain within the same, the 12. signs aforesaid, by means of which signs the whole circuit or longitude of the said Circle is divided into 12. equal parts, and every such part is divided into 30. degrees, and every degree into 60. minutes, and every minute into 60. seconds, etc. so as the whole Longitude thereof containeth 360. degrees, according unto which division, all the rest of the Circles both greater and lesser described in the Sphere, are made to contain the like number of degrees, and every half Circle to contain 180. degrees, and every quarter of a Circle to contain 90. degrees, and by this division aswell of the breadth as of the length of the zodiac, it appeareth that every one of the 12. Signs hath 30. degrees in length, and 12. degrees in breadth, and thereof the Planets, Stars, and all other Celestial bodies are said to have both Longitude and Latitude, the sun only excepted.  
How is such Longitude and Latitude to be understood?  
The Longitude of any Planet or Star is to be counted in the Ecliptic line containing in circuit 360. degrees, reckoning from the first point of Aries, and so to Taurus, Gemini, and Carcer, & so forth according to the succession of the signs, until you come again unto the first point of Aries, at which point such Longitude both endeth & begimeth. The Latitude is counted from the said Ecliptic line towards any of the Poles of the zodiac. And hereof look how many degrees any fixed star or Planet is distant from the Ecliptiqueline towards any of the said poles, so much Latitude it is said to have either Northern or Southern: moreover, by this division of the signs the whole firmament is divided into 12. parts by reason of 6. Circles imagined to pass through the poles of the zodiac, and also through the beginning of every sign, whereby we know under what sign every star is situated though it be clean out of the zodiac as this figure here plainly showeth, marked with these letters A. B. C. D. A signifieth the North pole of the world, B. the North pole of the zodiac, C. the South pole of the world, D. the South pole of the zodiac.   

How much the zodiac declineth from the Equinoctial towards either of the Poles, and of the greatest declination of the Sun, what it is at this present, and what it hath been in times past. Chap. 12.  
YOu have to understand that the zodiac or rather the Ecliptic line, declineth from the Equinoctial towards either of the poles, 23. degrees 28′· and that is said in these days to be the greatest declination of the sun, which declination is twofold, that is Northern and Southern, for like as the Sun entering into the first point of Aries, beginneth then to decline from the Equinoctial Northward, to the quantity of 23. degrees and 28′·S so entering into the first point of Libra, he declineth again from the Equinoctial as much Southward. And note by the way that by reason of his slow motion, when he is in the Northern signs, he spendeth 7. days, and ⅗. of a day more in making his North declination then in making his South declination, because he is then in his swift motion, and the time hath been that he hath spent above ten days more in making his North declination, then in making his South declination: neither is the greatest declination of the Sun in all ages of like quantity. For in Ptolemy's time it was 23. degrees, 51′·S and 20″· ever since whose time it hath always continually decreased until this present, so as now the greatest declination is no more but 23. degrees and 28′·S And Copernicus maketh the declination of the sun in respect of quantity to be twofold, that is, greatest and lest, affirming the greatest to be 23. degrees and 52′·S and the lest to be 23. degrees and 28′·S as it is now accounted, the difference whereof is 24′· and whilst the Ecliptic departeth from the Equinoctial, and turneth again towards the Equinoctial, there do run (as he saith) 34●4. years.   

How to know the quantity of the suns declination be it Northward or Southward, every day throughout the year, as well by a Table as by help of the Sphere. Chap. 13.  
THis is chiefly to be known by Tables calculated of purpose, which Tables most commonly are either made to answer every day of the month, or else to the degree of the sign wherein the sun is every day, which kind of Table is contained in lesser room than the other, but to work by such a Table, you must first know in what sign and degree the sun is every day.  
How is that to be done?  
It is to be known most truly by the Ephemerideses or such like Table caluclated of purpose, showing not only the degree of the sign, but also the very minute wherein the sun is every day, and for want of such a Table, you may without consideration of the minutes, know it by such an instrument or figure as this following, which consisteth of diverse Circles, whereof the outermost containeth the degrees of the 12. signs, together with the names of the said signs, & the next the days of each month, together with the names 〈◊〉 ●he said months, much like the backside of an Astrolabe, in the centre or midst of which instrument or figure is a thread, which if you lay upon the day of the month which you seek, 〈◊〉 strait direct you to the degree of the sign wherein the Sun is that day, as for ex●●ple, if you would know in what sign and degree the Sun is the 4. of May, then by draw●●● the thread right upon the said day over and beyond the outermost circle, you shall find 〈◊〉 it will fall right upon the 23. degree of Taurus.  
An instrument to know thereby in what sign and degree the Sun is every day throughout the year.  
Than having found the degree of the Sun you must resort therewith to this Table following, made for the declination of the sun.  

A Table showing the declination of the sun every day throughout the year  Degrees of the Signs. ♈ ♎  ♉ ♏  ♊ ♐ Degrees of the signs. D M S  D M S  D M S 1  0 23 53  11 50 6  20 22 57  29 2  0 47 46  12 10 56  20 35 7  28 3  1 11 39  12 31 34  20 46 55  27 4  1 35 30  12 51 59  20 58 20  26 5  1 59 20  13 12 12  21 9 21  25 6  2 23 8  13 32 12  21 19 59  24 7  2 46 54  13 51 58  21 30 13  23 8  3 10 37  14 11 30  21 40 3  22 9  3 34 18  14 30 48  21 49 29  21 10  3 57 54  14 49 51  21 58 29  20 11  4 21 28  15 8 40  22 7 6  19 12  4 44 57  15 27 13  22 15 17  18 13  5 8 22  15 45 30  22 23 3  17 14  5 31 42  16 3 32  22 30 24  16 15  5 54 57  16 21 17  22 37 19  15 16  6 18 6  16 38 44  22 43 48  14 17  6 41 9  16 55 55  22 49 50  13 18  7 4 6  1● 12 48  22 55 27  12 19  7 26 57  ●● 29 23  22 0 38  11 20  7 49 40  17 45 40  23 5 22  10 21  8 12 16  18 1 39  23 9 39  9 22  8 34 45  18 17 18  23 13 29  8 23  8 57 5  18 32 37  23 16 53  7 24  9 16 16  18 47 38  23 19 50  6 25  9 41 19  19 2 18  23 22 19  5 26  10 3 12  19 16 37  23 24 22  4 27  10 24 56  19 30 36  23 25 57  3 28  10 46 30  19 44 14  23 27 5  2 29  11 7 53  19 57 30  23 27 46  1 30  11 29 5  20 10 25  23 28 9  0 ●  D M S  D M S  D M S  D   ♍ ♓  ♌ ♒  ♋ ♑    

The description and use of the Table.  
This Table consisteth of 5. collums, whereof the first on the left hand, and the last on the right hand do contain the degrees of the 12. signs of the zodiac, counting from one to 30. And the three middle collums do contain the degrees minutes, and seconds of declination, over the head of which three middle collums are set down the characters of these 6. opposite signs, Aries and Libra, Taurus and Scorpio, Gemini and Sagittarius, and at the foot of the said midlde columns are set down the characters of the other 6. opposite signs, that is, Virgo and Pisces, Leo & Aquarius, Cancer and Capricornus, for every 2. opposite signs, aswell above as beneath, have like declination, the use of which Table is thus: first having found out the degree of the Sun by the instrument before described, or rather by some true Ephemerideses, you must seek out the number of the said degree, either in the first or last collum, according as the character of the sign is placed. For if the sign or character be above, than you must seek for the said number in the first collum on the left hand, which numbers do descend from 1. to 30. but if the sign be beneath, than you must found it out in the the outermost colum on the right hand, the numbers whereof do ascend from 1. to 30. and the common angle or square, standing right against the said number will show you the degree, minutes and seconds of the declination, as for example, having found by the former instrument that the 4. day of May the Sun is in the 23. degree of Taurus, I seeing the character of Taurus to stand above, do seek my foresaid number of 23. degrees in the first collum on the left hand, and in the common angle or square right against that number, and under the sign Taurus, I found the declination of the Sun to be 18. degrees 32′· and 37″· But this Table cannot serve always: yea rather such tables are to be renewed as our Astronomers say every 30. years. Also you may know the daily declination of the Sun, by help of a material Sphere or globe: thus having set the Sphere at your latitude, bring the degree of the sign wherein the Sanne is that present day unto the movable Meridian, and staying it there, mark whether it falleth on the South side or on the North side of the Equinoctial: for if it be in any of the Northern signs, it will fall on the North side of the Equinoctial, and if it be in any of the Southern signs, it will fall on the south side of the Equinoctial, and by counting the degrees upon the Meridian, contained betwixt the degree of the Sun and the Equinoctial, you shall know what declination the Sun hath that day, as for example in the latitude 52. in the year 1590. the fift day of May, I found the Sun by the Ephemerideses to be in the 23. degree and 48′· of Taurus, which point I bring to the movable Meridian, and there staying it, I found that point to be distant from the Equinoctial northward 18. degrees & certain minutes, and so much of North declination I conclude the Sun to have that day.    

Upon what Poles the zodiac turneth about, also of the Ecliptic line and of the divers uses thereof. Chap. 14.  
THe zodiac turneth about upon his own proper Poles from West to East, whereof the one being placed in the Colour of the Solstices towards the North, is distant from the pole Arctique 23. degrees and 28′·S and the other is placed in the said Colour towards the South, being of like distance from the pole Antarctique, whereof the Astronomers have a general rule, affirming that the distance of the two Poles of the world from the Poles of the zodiac, is always equal to the greatest declination of the sun, which as hath been said before, is 23. degrees & 28′·S as you may plainly see by the Sphere. And note that these 2. Poles are otherwise called the Poles of the Ecliptic, for in considering the declination of the Sun or of the zodiac from the Equinoctial, you must have respect only to the Ecliptic line, which is in the midst of the zodiac, and not to any other part of the zodiac. And as the Equinoctial line showeth the moving of the first movable, which is from East to West, so the Ecliptic line showeth the moving of the second movable, which is from West to East, clean contrary to the first movable, the causes whereof have been before declared.  
What other uses hath this line more than you have already declared?  
It hath diverse, for in this line or circle are noted the degrees, wherewith any star riseth or goeth down, either rightly or obliquely, for all the appearances of the heavens are chief referred to this circle. Again, by this circle the chiefest distinctions and parts of times, as years and months are known, and also the four seasons of the year, as Spring, Summer, Autumn, and Winter: Moreover, the obliquity of this circle under which the Sun continually walketh, is cause that the days both natural and artificial are unquaell. Finally, this circle doth show the places and times of the Eclipses both Solar and Lunar, from whence this line taketh his name, of which Eclipses we mind here briefly to treat.   

Of the Eclipses both Solar and Lunar, and of the head and tail of the Dragon, with certain figures showing the same. Chap. 15.  
WHat signifieth this word Eclipse?  
It is as much to say, as to want light, & to be darkened or hidden from our sight, as when the Sun & Moon are both at one self time right under the Ecliptic line, the one of these 2. lights most commonly is eclipsed and darkened: for there be two Eclipses, the one of the Sun, and the other of the Moon, but sith that neither the eclipse of the Sun or of the Moon doth chance, but when they meet either in the head or tail of the Dragon, I think it good to show first what is meant by the head and tail of the Dragon. The Dragon than signifieth none other thing but the intersection of 2. circles, that is to say, of the Ecliptic, & of the circle that carrieth the Moon, called her Defferent▪ cutting one another in 2. points, whereof that intersection which is westward when as the Moon goeth towards the North, is called the head, and that which is Eastwardes when the Moon goeth towards the South is called the tail, marked with such characters as you see in the figure following, and that part towards the South is called of some the belly of the Dragon. And note that the Defferent of the Moon is at no time distant from the Ecliptic above 5. degrees at the most.  
The figure of the Dragon.  
This being presupposed, I will speak first of the Eclipse of the Moon, and then of the Sun, both which may be eclipsed either totally or in part.  
When is the Moon said to be totally eclipsed?  
When the Sun and Moon are opposite one to the other diame trallie, and the earth in the very midst between them both, for the body of the earth being thick and not transparent, casting his shadow to that point which is opposite to the place of the Sun, will not suffer the Moon to receive any light from the Sun, from whom she always borroweth her light.  
At what time is the Moon said to be diamtrallie opposite to the Sun?  
When a right line drawn from the centre of the Sun to the centre of the Moon, passeth through the centre of the earth: & note that every time that she is at the full, she is opposite to the Sun, and yet the earth is not at every such full diametrally betwixt her and the Sun, for than she should be eclipsed at every full, which indeed cannot be but when she is either in the head or tail of the Dragon.  
When is the Moon said to be eclipsed in part?  
When is the Sun said to be eclipsed?  
When the Moon is betwixt the Sun and the Earth, which chanceth in a Conjunction, and yet not in every conjunction, but when it falleth either in the head or tail of the Dragon, which may chance, as I said before, either totally or in part: totally I say, in respect of those parts of the earth whereon the shadow directly falleth. For sith the Moon is far lesser than the Earth, she cannot shadow all the Earth, and therefore the Eclipse of the Sun cannot be universal, but yet to some parts of the Earth totally, and to some partly, and to other some nothing at all, as you may plainly see by this figure following.  
Yet all the histories do affirm that the Eclipse of the Sun was universal at the death of Christ.  
Yea, that was miraculous, and also it was then at the full of the Moon, which was also as miraculous: and therefore Dionysius a Senator of Athens, beholding that Eclipse cried out, saying these words, Either God this day suffereth, or else the world must needs perish for ever: which Dionysius was the first that converted the Frenchmen to the faith of Christ, doing there great miracles: in honour of whom was erected the rich Abbey of S. Device, not far from Paris, whereas the Kings and Princes of France were wont to be buried.  
How is it to be proved that the Eclipse at Christ his death was at the full of the Moon?  
Aswell by ancient history, as by S. Augustine, who saith that the jews were wont to keep their feast of Passover (at which time God suffered) always at the full of the Moon.  
If the Sun and Moon be eclipsed but in part, how are such parts to be accounted?  
By the parts of the Diameter of the bodies of those two Planets, for the Astronomers do divide the Diameter aswell of the Sun, as of the Moon into 12. and some into 24. parts, which they call points, and therefore are wont to say, that the Sun or Moon are darkened or eclipsed 7. points, 8. points, 9 points, etc.   

Of the two Colours, and why they are so named, and whereto they serve: also of the four Cardinal points, that is, of the two Equinoctial, and the two Solstitial points, and of the entrance of the Sun into any of those points or into any other sign. Chap. 16.  
WHat be Colours?  
They be great movable circles passing through both the Poles of the world, which the Astronomers do otherwise call Circles of declination, whereof they make 180. which are half so many as there be degrees in the Equinoctial applying them to divers uses not needful here to be rehearsed, for sith that there are but two Colours accustomably set down in the Sphere, without the which a material Sphere cannot be made, I mind here therefore only to treat of them.  
Show first what ●his name Colour signifieth.  
This word Colour being compounded of Colos and Oura, is as much to say as unperfect or maimed, the tail being cut off, because none of those Circles are ever seen whole above our Horizon, but part thereof, for some part is always seen, and some part is always hidden, as that part which is above the Horizon, and nigh unto the Pole, is always seen, because it never goeth down under the Horizon: likewise that which is nigh unto the South pole is always hidden from us, because it never riseth above our Horizon, as by turning the Sphere about, you may easily perceive the same.  
Which be those Colours that are commonly set down in the Sphere, and how are they named?  
They are two great movable circles, passing through the Poles of the world, crossing one another in the said Poles with right spherical angles, by means whereof they divide the whole Sphere into four equal parts, of which two Colours the one is called the Colour of the Equinoxes, and the other the Colour of Solstices.  
Describe these two Circles, & show why they are so named.  
The Colour of the Equinoxes is so called because it cutteth the zodiac in the beginning of Aries, which is called the vernal Equinoxe: and also in the beginning of Libra, which is called the Autumnal Equinoxe, at which two times the days and nights be equal, as hath been said before when we did speak of the Equinoctial circle, and this circle divideth the Ecliptic into two halves, the one Septentrional, and the other Meridional and thereby showeth the signs▪ wherein the Sun maketh the days longer and shorter than the nights, for whilst he is an the 6. Northern signs, he maketh the days with us longer than the nights, and when he is in the 6. Southern signs he maketh the nights longer than the days: now you have to understand that the Colour of the Solstice, is so called because it cutteth the zodiac in the two Solistitiall points, that is to say, in the beginning of Cancer, and the beginning of Capricorn, as you may see in beholding and turning the Sphere about with your hand.  
Why are these two points called Solstitial?  
They take their name of these two Latin words Sol and stano, that is to say, the Sun and standing, for when the Sun is in any of the two points, he seemeth to stand still, or at the lest moveth so little, as his proper moving from West to East cannot be easily perceived, during the space of twelve days. And you have to note, that when the Sun entereth into the first degree of Cancer, which is about the 12. of june, than he is at the highest, and the days be at the longest, and therefore, it is called the Summer Solstice. Again, when he entereth into the first degree of Capricorn, which is about the 12. of December, than the Sun is at the lowest, and the nights are at the longest, and therefore it is called the Winter Solstice. And in this Colour there are set down the two Poles of the Ecliptic line being distant from the Poles of the world 23. degrees and 28′·S Moreover on this colour is measured the greatest declination of the Sun, which is always equal to the distance of the Pole of the Ecliptic, from the Pole of the world, as hath been said before. And you have to note that the 4. former points, that is to say, the 2. Equinoxes, and the 2. Solistices, are commonly called the four cardinal or principal points, and of some they are called, the four points or Change, signifying the 4. beginnings of the four divers seasons of the year: for betwixt the beginning of Aries and beginning of Cancer, is contained the Spring time, and betwixt the beginning of Cancer and the beginning of Libra, is the Summer time: and from the beginning of Libra to the beginning of Capricorn is the time called Autumn, or fall of the leaf: and from the beginning of Capricorn to the beginning of Aries is contained the winter season, albeit the Sun entereth not into any of these signs always at one self day or time of the year, for at Christ his incarnation, the Sun entered into Aries the 25. of March, and into Cancer the 24. of june, and into Libra the 27. of September, and into Capricorn the 25. of December, which was then the shortest day in the year, and the beginning of Winter, and therefore is called of the Latins, dies brumalis, on which day Christ our Saviour was born, so as from the time of his birth unto this present year, there are run almost 13. days, wherefore, unless the calendars be reform as well here in England as else where (for the Roman reformation is not so exactly true as it might be) we shall have in process of time, the Spring in Winter, and the Winter in Autumn.  
How shall I know this present year, or any year to come hereafter, at what day and hour the Sun entereth into any of the 12. Signs?  
First you must learn by some good Ephemerideses, or other Table, the true entrance of the Sun into every sign in any year passed before, then from the time of the entrance of the Sun into the sign which you desire to know, consider how many years are betwixt, & how many leap years are in the same contained, and subtract for so many times as there be leap years, 44′· of an hour, and ad to the hours remaining, so many times five hours, and 49′· as there be years remaining over and beside the leap years, and that sum shall show you the day, hour, and minutes of the true entrance of the Sun into that sign in the same year that you desire to know.   

Of the Horizon both right and obliqne, making thereby three kinds of Spheres, that is, the right, the Parallel, and the obliqne Sphere. Chap. 17.  
WHat is the Horizon?  
It is a great immoovable circle which divideth the upper Hemisphere, which is as much to say, as the upper half of the world which we see, from the neither Hemisphere which we see not, for standing in a plain field, or rather upon some high mountain void of bushes & trees, and looking round about, you shall see yourself environed as it were with a circle, and to be in the very midst or centre thereof, beneath or beyond which circle, your sight cannot pass, and therefore this circle in Greek is called Horizon, and in Latin Finitor, that is to say, that which determineth, limitteth or boundeth the sight, the Poles of which circle are imagined to be two points in the firmament, whereof the one standeth right over your head, called in Arabic Zenith: and the other directly under your feet, called in the same tongue Nadir, that is to say the point opposite, and from point to point you must imagine that there goeth a right line passing through the centre of the world, and also through your body both head and feet, which is called the Axletree of the Horizon, and you have to understand that of Orisons there be 2. kinds, that is, right, & obliqne, making 3. kinds of Spheres, that is to say, the right Sphere, the parallel Sphere, and the obliqne Sphere,  
When is the Horizon said to be right, and thereby to make a right Sphere?  
It may be said to be right two manner of ways, first, when the Horizon passeth through both the Poles of the world, cutting the Equinoctial with right angles, in which Sphere they that dwell have their Zenith in the Equinoctial, which passeth right over their heads, to whom the days and nights are always equal. Secondly, they are said to have a right Horizon, & to devil in a right Sphere, to whom one of the Poles of the world is their Zenith, and their Horizon is all one with the Equinoctial, cutting the Axletree of the world in the very midst with right angles, and because the Horizon & the Equinoctial are Parallels, this kind of Sphere is called a parallel Sphere, in which Sphere they that dwell have 6. months day, and 6. months night, as you may easily perceive by placing the Sphere, so as one of the Poles may stand right up in the midst of the Horizon, by means whereof you shall see 6. signs of the zodiac to be always above the Horizon, and 6. signs to be always under the Horizon: Again by placing the Sphere so as both the Poles may lie upon the Horizon, you shall see the shape of the first right Sphere, wherein the Horizon passeth through both the Poles of the world, and the Equinoctial passeth through the Poles of the Horizon, which are the two points called before the Zenith and Nadir.  
When is it said to be an obliqne Horizon, and thereby to make an obliqne Sphere?  
When the Pole of the world is elevated above the Horizon, be it never so little, so as the Horizon do cut the Equinoctial with obliqne angles, and look how much the Pole of the world is elevated above your Horizon, so much is your Zenith distant from the Equinoctial, and the nigher that your Horizon approacheth to the Pole, the nigher your Zenith approacheth to the Equinoctial. Again, look how much the Equinoctial is elevated above your Horizon, so much is your Zenith distant from the Pole, all which things this figure here following doth plainly show, whereby you may easily perceive that the latitude, which is the distance of your Zenith from the Equinoctial, is always equal to the altitude of the Pole, which is the distance betwixt your Horizon & the Pole, as for example, knowing the latitude of Norwich to be 52. degr. lay the zenith of this figure upon the 52. degrees, reckoning from the Equinoctial towards the pole Arctique on your left hand, and look what distance is betwixt the said zenith and the Equinoctial, the self same distance you shall found to be betwixt the Horizon and the foresaid Pole on your right hand, and you may do the like upon the Sphere itself by raising the movable Meridian above the Horizon at that altitude, so as the 52. degr. may be even with the Horizon.  
A Figure showing the latitude of any place to be equal to the elevation of the Pole.  
What other uses hath this circle?  
In this circle are set down the four quarters of the world, as East, West, North and South, and the rest of the winds: Again, this circle divideth the artificial day from the artificial night for all the while that the Sun is above the Horizon it is day, and whilst it is under the same it is night. And by this circle we know what stars do continually appear, and which are continually hidden, also what stars do rise and go down. Again, in taking the elevation of the Pole, this circle is chiefly to be considedered, for when we know how many degrees the Pole is raised above the Horizon, than we have the elevation thereof for that place. For to every several place, yea to every little moment of the earth in an obliqne Sphere, belongeth his proper Horizon and several altitude of the Pole, whereby it appeareth that the Orisons are infinite and without number.  
How shall I know in any place, having an obliqne Horizon, how much the Pole is elevated above the Horizon?  
That is declared in the second book of this Treatise, whereas I speak of the latitude and longitude of the earth, in the eight chapter.   

Of the Meridian, and of the uses thereof. Chap. 18.  
WHat is the Meridian?  
It is a great immoovable circle passing through the Poles of the world, and through the Poles of the Horizon.  
Why is it called the Meridian?  
Because that when the sun rising above the Horizon in the East, cometh to touch this line with the centre of his body, them it is midday or noontide to those, through whose Zenith that circle passeth And when the sun after his going down in the west cometh to touch the self line again in the point opposite, it is to them midnight, & note that divers cities, having divers latitudes, that is to say, being distant one from another North and South be it never so far, may have one self Meridian: but if they be distant one from another East and West, be it never so little, than they must needs have diverse Meridian's, and such distance betwixt the two several Meridian's, is called the difference of longitude, whereof we shall speak hereafter more at large when we come to treat of the longitude and latitude of the earth, which something differeth from the longitude and latitude of the Stars or Planets, whereof we have already spoken in the 11. chapter.  
How many Meridian's be there?  
The Astronomers do appoint for every two degrees of the Equinoctial a Meridian, so as they make in all 180. Albeit most commonly in the Sphere they set down but one, which serveth for all by turning the body of the Sphere to it, which for that cause is called the movable Meridian. And in such Spheres as have not a foot and a standing Horizon, there is no Meridian at all, but the two Colours are feign to supply their want, but all terrestrial Globes are commonly described with twelve Meridian's, cutting the Equinoctial in 24. points, and dividing the lame into 24. spaces, every space containing 15. degrees, which is an hour, by means whereof we know how much sooner or latter it is noontide in any place, for it is noontide sooner to those whose Meridian is more Eastward then to them whose Meridian is more Westward. And contrariwise the Eclpise of the Sun or Moon appeareth sooner to those whose Meridian is more Westward.  
What other uses hath this circle?  
This circle divideth the East part of the world from the West and also it showeth both the North and South, for by turning your face towards the East, you shall found the Sun being in that line at noon tide to be on your right hand right South, the opposite part of which circle showeth on your left hand the North. Also this circle by reason that it passeth through both the Poles of the world, divideth both the Equinoctial and all his Parallels into two equal parts aswell above the Horizon as under the Horizon, and by that means it divideth the artificial day and artificial night each of them into two parts, that is to say, into two semidiurnal & into two seminocturnal parts. For betwixt that part of the Horizon where the sun riseth, mounting still until he come to this circle, which is at noontide, is contained the first half of the day, & the other half is from the same circle to the going down of the Sun under the Horizon. And the first part of the night is the space betwixt the Sun's going down and his coming again to the Meridian, which is at midnight, and from thence to the time of his rising is the other half of the night, and also the Astronomers take the beginning of their natural day from this circle, counting either from noontide to noontide, or else from midnight to midnight. Again, this circle showeth the right ascensions and declinations of the stars, and the highest altitude, otherwise called the Meridian altitude of the Sun or of any star, or degree of the Ecliptic, or of any other point in the firmament, all which uses and many others more you shall better understand hereafter, when we come to show the uses of the globe aswell terrestrial as celestial.   

Of the vertical circle, and uses thereof. Chap. 19  
But here you have to note that though the most part of Geographers do set down in their Spheres but 6. great circles, yet there is another great circle called the circle Vertical, which passeth right over our heads through our zenith, wheresoever we be upon the land or sea, crossing our Horizon in 2. points opposite, and dividing the same into two equal parts, and such kind of circles are called in Arabic Azimuthes, whereof you may imagine that there be so many as there be rombes or winds in the Mariners Compass, which are in number 32. yea, and if you will, you may make half so many as there be degrees in the Horizon, which are in number 360. the half whereof is 180. If you be right under the Equinoctial, and do go or sail right East or West, than the Equinoctial is your Vertical circle, and if you go or sail right North or South, than the Meridian is your vertical circle, which two circles notwithstanding do always keep their names. But in sailing by any other rombe, that circle which is imagined to pass from the true East point right over your head unto the true West point, or which crosseth your Meridian in the zenith point with right Spherical angles, is most properly called the vertical circle, and the learned seamen have great respect to two special kinds of Vertical circles, that is, the Magnetical Meridian, and the Azimuth of the Sun.  
What manner of vertical Circles be those, and whereto serve they?  
M. Borough in his discourse of the variation of the Compass, defineth the Magnetical Meridian to be a great Circle, which passeth through the Zenith and the Pole of the load stone called in Latin Magnes, and divideth the Horizon into two equal parts, by crossing the same in two points opposite. Again the Azimuth of the sun is a great Circle, passing through the Zenith & the Centre of the sun in what part of the heaven so ever he be, so as he be above the Horizon, which Circle divideth the Horizon into two equal parts, by crossing the same in two points opposite, And by help of these two Circles and a certain instrument made of purpose to give a true shadow, he teacheth to find out the true Meridian of any place: And also to know how much any Mariner's Compass doth vary from the true North and South, in Northeasting or Northwesting, whereof I shall speak more at large hereafter in my treatise of Navigation.  
What use is there of the vertical Circle or Azimuthes?  
The vertical Circle showeth what time the Sun or any other star rising beyond the true East point, is passed before the Sun or said star, cometh to the true East or any other rombe. Also in what Coast or part of heaven, the Sun, Moon, or any other star is at any time being mounted above the Horizon, as whether it be Southeast or North-east, or in any other rombe: Also by help of the vertical Circle most properly so called, are the twelve houses of heaven set, according to Campanus and Gazula. And by help of these Circles you may also know how any place upon the earth beareth one from another either Eastward or Westward, and so forth, for every place hath his several Azimuth answerable to the Horizon and Zenith of the said place.  
Of certain Circles called Almicanterathes.  
Since I have spoken here somewhat of the vertical Circles called Azimuthes, it shall not be amiss to show you also that there be other Circles to be considered of in the Sphere aswell as in the Astrolabe called Almicanterathes, that is to say, Circles of Altitude, which though they be not all great Circles, for every one is lesser than other proceeding from the obliqne Horizon of any place to the Zenith of the said place, yet the first Almicanterath which is the very obliqne Horizon itself, is a great Circle dividing the Sphere into two equal parts, and all the rest are lesser and lesser, until you come to the very Zenith, and are parallels to the Horizon, even as the Tropiques and the other lesser Circles are parallels unto the Equinoctial and the Zenith in Spherical bodies is the Centre of them all, though it be not so in Astrolabes, for there every Almicanterath is fame to have his several Centre, of which Circles there be in all 90. according to the number of 90. degrees contained betwixt the obliqne Horizon and the Zenith, and these Circles do serve to show the Altitude of the Sun or Moon, or of any other star fixed or wandering, being mounted at any time above the obliqne Horizon, which is easy to be found by any Quadrant, cross-staff, or Astrolabe. But leaving to speak any further of these Circles, because they are not used to be described in Spheres but only in Astrolabes, I will now treat of the four lesser Circles before mentioned, which are commonly set down in every Sphere or Globe.   

Of the four lesser circles, that is to say, the circle Arctique, the circle Antarctique, the Tropic of Cancer, and the Tropic of Capricorn, and also of the five Zones, that is to say, two cold, two temperate, and one extremely hot. Chap. 20.  
WHich call you the lesser Circles?  
They are those that do not divide the Sphere into two equal parts, as the great Circles do, and of such there be four, that is the two Polar circles, and the two Tropiques, that is to say, the Tropic of Cancer, and the Tropic of Capricorn, of which Polar circles the one is called Arctique, and the other Antarctique, and are made by the turning about of the two Poles of the zodiac, which Poles being situated in the Colour of the Solstices are so far distant from the Poles of the world, as is the greatest declination of the Sun from the Equinoctial, which is 23. degrees, 28′· as hath been said before.  
Which is the Arctique Circle, and why is it so called?  
The Arctique Circle is that which is next to the North pole, and hath his name of this word Arctos which is the great bear or Charles wain, which are seven stars placed next to this Circle on the outside thereof, and it is otherwise called the Septentrional Circle of this word Septentrio, which is as much to say as seven Oxen, signified by the seven stars of the little Bear, which do move slowly like Oxen, and are placed all within the said Circle, and the bright star that is in the tip of the tail of the said little Bear, is called of the Mariners the load star or North star, whereby they sail on the Sea, and the Centre of this Circle is the North Pole of the world which is not to be seen with man's eye.  
What is the Antarctique Circle?  
It is that which is next unto the South Pole, and it is so called, because it is opposite or contrary to the Circle Arctique.  
Now describe the two Tropiques.  
The Tropic of Cancer is a Circle imagined to be betwixt the Equinoctial and the Circle Arctique, which Circle the Sun maketh when he entereth into the first degree of Cancer, which is about the twelfth or thirteenth day of june being then in his greatest declination from the Equinoctial Northward, and nighest to our Zenith, being ascended to the highest point that he can go, at which time the days with us be at the longest, and the nights at the shortest. And so from thence he declineth to the other Tropic called the Tropic of Capricorn, which is a Circle imagined to be betwixt the Equinoctial and the Circle Antarctique, which the Sun maketh when he entereth into the first degree of Capricorn, which is about the twelfth or thirteenth day of December, at which time he is again in his greatest declination from the Equinoctial southward, and furthest from our Zenith: whereby the days with us be then at the shortest, and the nights at the longest: And note that these two Circles are called Tropiques of this Greek word Tropos, which is as much to say as a conversion or turning, for when the Sun arriveth to any of these two Circles, he turneth back again either ascending or descending, by reason of which four Circles as well the firmament as the earth is divided into five Zones, that is to say, two cold, two temperate, and one extremely hot, otherwise called the burnt Zone, of which five Zones, the foresaid four Circles are the true bounds. For of the two cold Zones, the one lieth betwixt the North pole and the Circle Arctique, and the other lieth betwixt the South Pole and the Circle Antarctique, and of the two temperate Zones, the one lieth betwixt the Circle Arctique, and the Tropic of Cancer, and the other lieth betwixt the Circle Antarctique, and the Tropic of Capricorn, and the extreme hot Zone lieth betwixt the two Tropiques, in the midst of which two Tropiques, is the Equinoctial line, as you may see in this figure, and also in the Sphere or Globe itself.  
A figure showing the five foresaid Zones.  
Of which Zones the ancient men were wont to say that three were unhabitable, that is, the two cold, and the extreme hot, which experience showeth in these latter days to be untrue, as we shall declare more at large when we come to treat of the division of the earth: Again you have to understand that every one of these lesser Circles doth contain in length, 360. degrees as well as every one of the greater Circles, but the degrees are not of like bigness, no more than the Circles themselves are like in compass or circuit, for the lesser the Circles are in circuit, the lesser their degrees must needs be.  
Sigh every of the lesser Circles differ one from another in circuit, and thereby the degrees of every Circle be lesser than other, how shall I know the true quantity of every degree in each Circle, and how many minutes are required in every lesser degree proportionally to answer one degree of the Equinoctial.  
For the better knowledge hereof, you must first imagine that there may be as many Circles made from the Equinoctial towards any of the Poles, as there be degrees of Latitude, which are in number 90. as hath been said before: And the nigher that any Circle is to the Equinoctial, the greater it is in circuit, and the further from the Equinoctial towards any of the Poles, the lesser in circuit, and therefore more or less minutes are requisite to answer to one degree of the Equinoctial, as you may easily perceive by this Table following, consisting of 6. collums, every front or head whereof is noted with three great letters, D. M. S. signifying degrees, minutes and seconds, six times repeated, and in the beginning of the first colum on the left hand is set down one degree, which is the first degree of 90. & nighest unto the Equinoctial, right against which one degree is placed towards the right hand, 59 minutes, and 59 seconds: and so proceeding from degree to degree successively, until you come to 90. you shall find how many minutes and seconds do answer to one degree of the Equinoctial, and this Table will also serve to show the difference of miles in every sundry clime or parallel, whereof we shall speak hereafter when we come to treat of the earth.  
 

A Table showing how many minutes are requisite to make one degree, in every lesser circle answerable to one degree of the Equinoctial.  D M S 1 59 59 2 59 58 3 59 53 4 59 51 5 59 46 6 59 40 7 59 33 8 59 25 9 59 16 10 59 5 11 58 54 12 58 41 13 58 28 14 58 13 15 57 57 16 57 41 17 57 23 18 57 4 19 56 44 20 56 23 21 56 1 22 55 38 23 55 14 24 54 49 25 54 23 26 53 56 27 53 28 28 52 59 29 52 29 30 51 58 31 51 26 33 50 19 34 49 45 35 49 9 36 48 32 37 47 55 38 47 17 39 46 38 40 45 58 41 45 17 42 44 35 43 43 53 44 43 10 45 42 26 46 41 41 47 40 55 48 40 9 49 39 22 50 38 34 51 37 46 52 36 56 53 36 7 54 35 16 55 34 25 56 33 33 57 32 41 58 31 48 19 30 54 50 30 0 61 29 5 62 28 10 63 27 14 64 26 18 65 25 21 66 24 24 67 23 27 68 22 29 69 21 30 70 20 31 71 19 32 72 18 32 73 17 33 74 16 32 75 15 32 76 14 31 77 13 30 78 12 28 79 11 27 80 10 25 81 9 23 82 8 21 83 7 19 84 6 16 85 5 14 86 4 11 87 3 8 88 2 5 89 1 3 90 0 0   

Of the stars and celestial bodies contained in the firmament, and first of their substance. Chap. 21.  
Having briefly described all the Circles as well greater as lesser that are imagined to be in the 8. heaven, I think it good now to speak somewhat of the stars and celestial bodies placed in the said heaven, And first of their substance, & then of their moving, figure, shape, number, magnitude or greatness, also of their Longitude, Latitude, declination, ascension, descension both right and obliqne, and of the ascentionall difference, and finally of the threefold Poetical rising and going down of the stars, but first of their substance.  
Of what substance are the stars?  
The stars be of the same substance that the heavens are wherein they are placed, differing only from the same in thickness, and therefore some defining a star do say, that it is a bright and shining body, and the thickest part of his heaven, apt both to receive and to retain the light of the Sun, and thereby is visible and object to the sight: for the heaven itself being most pure, thin, transparent, and without colour is not visible, and for this cause the milk-white impression in heaven like unto a white way called of the Astronomers Galaxia, and of the common people our Lady's way is visible to the eye, by reason that it is thicker than any other part of the heaven.  
Why are not the stars seen as well in the day, as in the night.  
Because they are darkened by the excellent brightness of the Sun from whom they borrow their chiefest light.   

Of the moving and shape of the stars. Chap. 22.  
WHat moving have the stars?  
The self same moving that the heaven hath wherein they are placed.  
Whereby are the heavens moved?  
Some say that the first movable is turned by God himself, and all the rest of the heavens every one by his proper intelligence, which though it turneth his heaven about, yet it giveth neither life, sense, nor understanding thereunto, as some have untruly holden, affirming the heavens to be living and intelligible bodies.  
If the stars have no moving of themselves, whereof cometh it then, that some seem to our sight sometime nigher and sometime further off.  
All the fixed stars of the firmament are always of like distance, notwithstanding by reason of the manifold moving of the firmament, wherein they are placed, they seem to change their places, and sometime to be more towards the East or West, North or South. And whereas the seven. Planets called the wandering stars, do change their places now here now there, that chanceth not by their own moving, but by the moving of the heavens wherein they are placed: for a star being round of shape hath no members meet to walk from one place to another, but only changeth his place through the motion of his Sphere or heaven wherein such Planet is fixed.   

Of the number of the stars, and of their magnitude or greatness, and into how many Images they are divided, and how many stars every image containeth. Chap. 23.  
MAy the stars be numbered by man?  
Not, for as David saith, that belongeth only to God, who as he created them, so he can number them and call them all by their names, notwithstanding the Astronomers by their industry and diligent observation, have attained to the knowledge of many: as first they know the sevenplanetes, otherwise called the wandringe stars, and have made manifest demonstrations of their motions, and by continual observation have found out the manifold virtues, powers and influences of the same, but of the fixed stars they could never find more than 1022. and because the stars are not equal in greatness or bigness, they make fix differences of greatness, appointing to the first difference 15. stars, which are bigger than all the rest, whereof every one containeth the earth 207. times, to the second difference 45. stars, whereof every one containeth the earth 90. times. To the third they appoint 208. stars, whereof every one containeth the earth 72. times. To the fourth difference they appoint 474. stars, whereof every one containeth the earth 54. times. To the fift they assign 217. stars, whereof every one containeth the earth 57 times. To the sixth or last greatness they appoint 49. small stars, whereof every one containeth the earth 18. times, and some say 20. times. Besides these there be 14. others, whereof 5. be called cloudy and the other dark, because they are not to be seen but of a very quick and sharp sight. And you have to note that the ancient Astronomers do divide all the fixed stars to them known into 48. images, whereof they liken some to living things as to men, women, beasts, monsters, fowls, fishes, and creeping worms, and some to things without life, having some artificial shape, of which 48. images, they appoint 12. to the zodiac, commonly called the 12. signs, as Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricornus, Aquarius and Pisces. Again they place in the North part of the firmament 21. images, and in the South part thereof 15. images, which make in all 48. The description of all which images, together with their names hereafter followeth: and first I will describe unto you the twelve Images contained in the zodiac.   

Of the xii. Images or signs of the zodiac. Chap. 24.  
THe twelve signs (as some affirm) do contain of the foresaid number of fixed stars 273. For the first sign called Aries, that is to say, the Ram, containeth 13. stars, which Image or sign being placed in the conjunction of the zodiac with the Equinoctial, hath his back turned towards the North, and his head towards the East, and riseth with his head, and goeth down with his feet. The second sign called Taurus, that is to say, the Bull, containeth 33. stars, whereof there is one bright star of the first bigness called Oculus Tauri, that is to say, the Bulls eye, who hath his head inclining towards the West as though he looked towards the earth, and riseth and goeth down with his heels upward. The third Sign called Gemini, that is to say the twins, do contain 18. stars, their heads looking towards the North, and their backs being joined together do embrace one another, and do rise lying, and do go down with their feet. The fourth Sign called Cancer, that is to say, the Crab containeth nine stars, extending his feet towards both the Poles, and looking towards Leo, hath his belly turned towards the earth, and he riseth and falleth with his hinder part or back part of his body. The fifth Sign called Leo, that is to say the Lion, containeth ten stars, whereof there be two bright stars of the first bigness, the one in his breast called Cor Leonis, and Regulus, that is to say the lions heart, and the other in his tail called Cauda Leonis, that is to say, the lions tail, who looketh towards Cancer, and having his back turned towards the North, he riseth and goeth down with his head. The sixth Sign called Virgo, that is to say, the virgin, whose head is behind the Lion and toucheth the Equinoctial line with her left hand, holding in the same hand an ear of corn, she both riseth and goeth down with her head: this image containeth six and twenty stars, whereof there is one bright star of the first bigness called Spica Virgins, that is to say, an ear of corn. The seventh Sign called Libra, that is to say the Balance, containing eight stars hath two scales, whereof the one hangeth towards the North, and the other towards the South. The eight sign called Scorpius, that is to say the Scorpion, containeth one and twenty stars, who looketh towards, Virgo, and extendeth his feet towards both the Poles, he boweth his tail towards the North, having his belly turned towards the earth, and he riseth and goeth down bowing. The ninth Sign called Sagittarius, that is to say the Archer, containing one and thirty stars, hath his head towards the North, and looketh towards the Scorpion, having a bow and a shaft, whereof the bow toucheth his left hand and left foot, he riseth right up, and goeth down headlong. The tenth Sign called Capricornus, that is to say, the Goat containing eight and twenty stars, hath his back turned towards the North, and his head towards the Archer, and turning himself towards Aquarius, he riseth right up, and goeth down headlong. The eleventh Sign called Aquarius, that is to say, the water-bearer containing two and forty stars, hath his head towards the North, extending his left hand upon the back of Capricorn, and with his right hand poureth out water out of his pot, which bendeth towards the East, runneth even to Pisces, he riseth and goeth down with his head before any other of his members. The twelfth Sign called Pisces, that is to say, two Fishes, do contain four and thirty stars, whereof the back of the first is towards the North, and the back of the second towards the West arm of Andromeda, and one of the Fishes looketh towards Aquarius, and the other towards the North, and betwixt these two Fishes is a certain little line wherewith their tails are bound together as it were with a bond, the lower part of which Fishes, doth always both rise and go down, and not the upper part: And though the 12. Signs of the zodiac are said to be equal both in length and breadth, that is to say, having thirty degrees in length, and twelve degrees in breadth, as hath been said before, yet these Images are not equal, for some do extend further than the zodiac in breadth, and some are more than thirty degrees in length; As the Tables of Alphonsus do manifestly show, who sayeth there that the twelve Signs do contain three hundred and fifty stars, for he appointeth to Aries eighteen, to Taurus forty four, to Gemini twenty five, to Cancer thirteen, to Leo thirty five, to Virgo thirty two, to Libra seventeen, to Scorpio twenty four, to Sagittarius thirty and one, to Capricornus twenty eight, to Aquarius forty five, to Pisces thirty eight, which make in all three hundred and fifty, in which Tables are also set down the Longitude, Latitude, and Magnitude of the said stars, but the Longitude of the said stars, is far altered from that Longitude which they had in his tune, whereof we shall speak hereafter more at large.   

Of the xxj. Northern Images. Chap. 25.  
WHich be they?  
These here following, first Vrsa minor, that is to say, the lesser Bear containing 7. stars, the tail star whereof being a bright star of the third bigness is called the lodestar: The second is called Vrsa maior, that is to say, the great Bear containing 27. stars, whereof there be 7. principal, making a shape like unto a Cart with four wheels, and therefore it is commonly called Charles wain. The third is called Draco, that is to say the Dragon that kept junos' Orchard from robbing, containing 31. stars. The fourth is called Cepheus, the proper name of a King of Ethiope, containing 11. stars. The fifth is called Boots, that is to say the roaring keeper of the Bear, containing 22. stars, whereof there is one bright star betwixt his legs, of the first bigness called Arcturus. The sixth is called Corona Ariadnae, that is to say the Crown of Ariadna the daughter of king Minos, containing 8. stars. The seventh is Hercules, who lieth groveling with his heels towards the North pole, holding a club in his right hand, and a Lion's skin hanging on his left arm containing 28. stars. The eight is called Lira, that is to say an instrument of music like a Harp placed in heaven in the memory of Orpheus, containing 10. stars. The ninth is called Cignus, that is to say the Swan, into which jupiter transformed himself to deceive the Nymph Leda, containing 17. stars. The tenth is called Cassiopea sitting in her chair, wife to Cepheus, and mother to Andromeda, containing 13. stars. The eleventh is called Perseus, holding his sword in the one hand, and the head of Gorgon or Medusa in the other hand, whose hairs were all Serpents, containing 26. stars. The twelfth is Auriga & Ericthonius, who was the inventor of the first Chariot that ever was made, containing 13. stars. The 13. is called Serpentarius and Anguitenens, that is to say, he that holdeth the Serpent, who as some think was Aesculapius the famous Physician, containing 24. stars. The 14. is called Serpens or Anguis, that is to say, the Serpent of Esculapius containing 18 stars. The 15. is called Sagitta or Telum, that is to say the shaft or dart wherewith Hercules slew the Eagle tormenting Prometheus in the mount Caucasus, containing 5. stars. The 16. is called Aquila, that is to say, the Eagle which carried Ganymedes into heaven containing 9 stars. The 17. is called Delphinus that is to say the dolphin, which saved Arion that excellent physician, being cast by Pirates into the sea, containing 10. stars. The 18. is called minor Equus, that is to say, the lesser Horse, containing only 4. little dark stars in his head. The 19 is the great Horse called Paegasus, that is to say the winged Horse, wherewith Bellerophon conquered the monstrous beast called Chimaera, which was half a Lion and half a Dragon, and is adorned with 20. stars. The 20. is called Andromeda the daughter of Cepheus by Cassiopeia, and wife to Perseus, who for her constant love towards her husband, was placed in heaven nigh unto him, and was adorned with 23. stars. The 21. is called Triangulum, that is to say a Triangle or three cornered figure, which being like in shape to the isle Cicilia the Goddess Ceres obtained to be placed in heaven, and was adorned with 4. stars, that is to say, every corn one, and the fourth in the midst of the shortest side: To these (in mine opinion) aught to be added Bernices hair, called crinis Bernices, which in all celestial globes is placed not far from the right hinder foot of the great Bear, and this Image containeth 4. little stars.   

Of the 15. Southern Images. Chap. 26.  
WHich be they?  
These hereafter following, whereof the first is called Caetus, that is to say the Whale, that monstrous fish which by the appointment of Neptune, would have destroyed Andromeda, whom Perseus delivered by killing the Fish, and afterwards took Andromeda to wife, which Fish Neptune placed in heaven, adorning the same with two and twenty stars. The second is called Orion with his sword by his side, who afterward was slain by Diana by mishap against her will, for the which she placed him in heaven, adorning him with 38. stars, whereof there be two bright stars of the first bigness, the one in his right shoulder called Bed Alguze, and the other in his left foot called Rigell Alguze. The third is the flood called Eridanus, into the which Phaeton the son of Apollo was strooken with a thunder bolt by jupiter, for burning the earth by rashly driving his father's Chariot, which he was not able to guide, in memory whereof the flood was placed in heaven, and adorned with 34. stars, whereof one is a bright star of the first bigness called Acarnar. The fourth is called Lepus, that is to say the Hare, placed nigh unto Orion, because he was a hunter, adorned with 12. stars. The fift is called Canis maior, that is to say the great Dog, passing all others in swiftness, which was given by Aurora to Cepheus the son of Aeolus, and is placed next to the Hare, being adorned with 18. stars, whereof there is in his mouth a very fair star of the first bigness called Syrius. The sixth is called Canis minor, that is to say, the lesser Dog, without the which Orion his master would not be placed in heaven, which hath but two stars, whereof the one is in his flank, and is a bright star of the first bigness called Canicula and Protion. The seventh is the ship called Argos, in the which jason and his companions sailed to Cholcos' to win the Golden Fleece, which is adorned with 45. stars, whereof there is one bright star of the first bigness in the left oar called Canopus. The eight is called Hydra, that is to say, the Water-serpent which Hercules slew, or as some say which kept the water-bowle, and would not suffer the thirsty Crow to drink, which Crow Apollo sent for water to do sacrifice, and is adorned with 25. stars. The ninth is called Crater, that is to say the Cup or Bowl which the Crow brought too late unto Apollo, and therefore his feathers were made all black whereas before they were white, containing 7. stars. The tenth is called Coruus that is to say the Crow before mentioned, adorned with 7. stars. The 11. is called Chiron siue Centaurus, the son of Saturn passing all others in justice and religion, and therefore is figured in heaven as though he were offering sacrifice upon an altar adorned with 37. stars whereof there is one bright star of the first bigness in his right foot called Chiron and Centaurus. The twelfth is called Lupus, that is to say the Wolf, into which Lycaon the cruel Tyrant was turned by jupiter, or as some say that Wolf which Centaurus killed to do sacrifice upon the altar containing 19 stars. The thirteenth is called Ara, that is to say the Altar made by the Smiths of Vulcan, whereupon all the Gods swore to revenge the insolency and pride of the Giants, which altar is placed next to Centaurus being adorned with 7. stars. The fourteenth is called Corona Australis that is to say, the Southern Crown which Bacchus did wear when he fetched his mother Semele from hell, which is placed in heaven, & is adorned with thirteen stars. The fifteenth and last of the Southern signs is called Piscis Australis, that is to say, the Southern Fish which was placed in heaven in memorial of the fishes which the people of Syria did worship as their Gods, and is adorned with 12. stars, whereof there is one bright star of the first bigness in his mouth called Fomahant: All which Images and Signs before mentioned, aswell Northern as Southern, you may see plainly described in every celestial Globe, and also set forth in Plano in the neither end of Vopellius his universal Map, that is to say the Northern signs on the left side, and the Southern signs on the right side of the Map: Or in the front of Planctius his great universal Map, who in the rondle representing the Southern half of the celestial Globe, setteth down also certain Southern stars lately found out by the travelers into the Indieses, as the Cross, the Southern Triangle, No his Dove or Pigeon, and another in shape of a man called Polophilax, so as there be now in all 19 Southern Images.   

Of the longitude of the fixed stars, and of the procession of the vernal Equinoctial point, and what it is. Chap. 27.  
WHat is the Longitude of a star?  
The Longitude of any star, is that Ark or portion of the Ecliptic line which is contained betwixt the first point of Aries, and that Circle which passeth through the Poles of the zodiac, and also through the body of the star, as for example the star called Cor Leonis is distant in these days from the vernal Equinoctial point of the Ecliptic line 143. degrees and 32′·S and thereby is found to be in the 23. degree and 32′· of Leo, again the star called Spica Virgins in these days is distant from the first point of Aries 198. degrees, and so is found to be in the eighteenth of Libra.  
Why do you say in these days?  
Because the fixed stars in process of time, do change their Longitude by reason of their proper moving upon the Poles of the zodiac, which is from West to East, for whereas Spica Virgins in Ptolemy's time was in the 26. degree of Virgo, it is found now to be in the 18. of Libra, the cause whereof is the precession of the Equinoctial point or section.  
Define what that Precession is?  
It is an Ark or portion of the Ecliptic line, contained betwixt two great Circles, both passing through the Poles of the zodiac, in such sort as the one passeth through the first minute of the vernal Equinoctial point of the said Ecliptic, and the other Circle passeth through the first or former star of the Ram's horn, from which star the Astronomers do make all the celestial motions and revolutions to take their first beginning, and this star in old time past was known to be before the vernal Equinoctial point, which is the first moment of Aries, but now it is found to have passed that point so far towards the Solsticiall point, as in these days it is known to be in the 27. degree and 42′· of Aries, and in process of time it will be clean out of Aries, and enter into Taurus.   

Of the Latitude of the fixed stars. Chap. 28.  
WHat is the Latitude of a star?  
The Latitude is none other thing, but the distance of any star from the Ecliptic line either towards the North or South pole of the zodiac, and such Latitude never changeth or altereth, for as the star Spica Virgins is at this present two degrees distant from the Ecliptic line towards the South, so it ever hath been and ever shall be, and the like is to be said of all the rest of the fixed stars which do always keep their Latitude, be it Northward or Southward, near to the Ecliptic or far from the same.   

Of the Declination of the fixed stars. Chap. 29.  
WHat is the declination of a star?  
The declination is none other thing, but the distance of any fixed star from the Equinoctial, either Northward or Southward, which is mutable as well as the Longitude: for as the fixed stars do change their Longitudes, so also by little and little they decline either more or less from the Equinoctial: As for example, the declination of the star called Canicula, that is to say the lesser Dog in the year of our Lord, 138. when Ptolemy lived, was 15. degrees 44′· and 38″· towards the South. But in these days the declination of the said star is but six degrees and 7′·S towards the South, and by reason that the fixed stars in process of time do change their Longitude and declination, they are not always under one self sign, but do flit out of one sign into another.  
How is the Longitude, Latitude, and declination of any star to be known, and how are the stars themselves to be known in the firmament.  
The Longitude, Latitude, and declination of any star is to be known most truly by the Astronomical Tables calculated of purpose, and you may know the same also without having regard to every small minute, by help of the celestial Globe, all the necessary uses whereof I have set down in a little Treatise to be added hereafter to this Book, and there also I show you how to found out any star in the firmament that is described in the globe, in which Globe are set down as many stars as ever were known, a few excepted towards the South pole, which were found out but of late days, of which stars I shall have occasion to speak hereafter in my Treatise of Navigation. In the mean time I will proceed to the ascension and descension of the stars both right, mean, and obliqne.   

Of the ascension and descension, that is the rising and setting of the stars, aswell according to the Astronomers, as according to the Poets. Chap. 30.  
Do the Astronomers and Poets differ touching this matter?  
Yea they differ greatly, aswell in name as in matter: for whereas of the Poets it is called ortus & occasus Signorum, that is to say, the rising and falling of the Signs, so of the Astronomers it is called Ascentio & descentio Signorum, that is to say, the ascension and descension of the Signs, again they differ in matter, or rather in manner, for that the Astronomers do consider the rising and falling of the stars more exactly than the Poets, for the Astronomers do consider the degrees and minutes of the same, and also do ground their ascension and descension upon more certain demonstrations then the Poets. Moreover whereas the Poets by their manner of rising and falling, do simply set down the time of things done or to be done, the Astronomers do the same a great deal more exactly, and by their manner of ascension and descension do consider the increase and decrease of the days, of which Astronomical ascension and descension, I mind here to treat first in general and then in particular.   

Of the Astronomical ascension and descension in general both right, mean, and obliqne, and what a given Ark is. Chap. 31.  
DEfine what the Astronomical ascension and descension is.  
Astronomical ascension is that portion or Ark of the Equinoctial line which riseth together with some given Ark of the Ecliptic line above the Horizon, and the descension is that portion or Ark of the Equinoctial that goeth down or setteth together with some given Ark of the Ecliptic line under the Horizon, according to the moving of the world which is from East to West.  
What mean you by a given Ark?  
A given Ark is as much to say as some supposed portion of the Ecliptic or of any other Circle, as if you would know the ascension of some supposed portion of the Ecliptic, containing for example 25. or 30. degrees, here this portion of the Ecliptic containing that number of degrees, is called the given Ark, of which arks some are called continual, and some discrete or divided, which I mind to English here whole and broken, for so I do English quantitas, continua, & discreta in my Logic, That Ark is said to be continual or whole which taketh his beginning from the first point of Aries, and so proceeding orderly, endeth at some other degree of the said Ecliptic. And that Ark is called discrete or broken, which doth not take his beginning from the first point of Aries, but beginneth at some other degree of the Ecliptic, as for example, suppose that it beginneth at the fourteenth of Taurus, and endeth at the fifteenth of Gemini, this Ark is called a divided or broken Ark, because it doth not begin at the first point of Aries, and so proceed successively: moreover you have to understand that the ancient Astronomers do commonly make but two kinds of ascension and descension, that is, right and obliqne, but there be in deed three kinds of ascension, that is to say, right, obliqne, and mean ascension.  
When is any ascension said to be right, obliqne or mean?  
It is said to be right, when that portion of the Equinoctial which riseth or goeth down together with the Ecliptic, is greater or more in circuit then that of the Ecliptic. And it is said to be obliqne, when that portion of the Equinoctial which riseth or falleth together with the Ecliptic, is lesser than that of the Ecliptic, Again that is said to be mean ascension, when that portion of the Ecliptic which ascendeth, is neither greater nor lesser then that of the Equinoctial, for as in the right Sphere every quarter of the Ecliptic hath a mean ascension, and equal to every quarter of the Equinoctial, beginning the quarters at any of the four principal points, so if you take three signs in any other part of the zodiac, their ascensions will not agree with a quarter of the Equinoctial, sith there is no one sign that doth equally agree with the like portion of the Equinoctial, and all this matter dependeth upon the knowledge of the use of certain Circles before defined.  
Which be they?  
These three, the zodiac, the Equinoctial, and the Horizon: for first the zodiac doth show the place of the Sun, that is to say in what degree it is of any sign together with the minutes of the same, and turning about every day by the diurnal motion, doth both appear above the Horizon, and also is hidden under the Horizon. Secondly the Equinoctial with his equal rising and going down, doth measure the time of the sun whilst he maketh his abode unequally and diversely above the Horizon. Thirdly the Horizon divideth the one Hemisphere from the other, on which Horizon is to be considered what Angle any sign or star maketh therewith, in his ascension or descension, and according as any portion of the Ecliptic riseth or setteth rightly or obliquely, so in respect of the Angle which it maketh with the Horizon, it is called a right or obliqne ascension or descension.  
Why should the ascensions and descensions be measured rather by the equinoctial line then by the Ecliptic, sith the course of the sun measureth all times?  
The cause thereof is the obliquity of the zodiac, having diverse and variable situations, whereby the sun abideth sometimes a great while above the Horizon, and sometimes but a little while, all which inequality is only to be measured by the Equinoctial, which is always equally moved upon his Poles. Hitherto of the Astronomical ascension and descension in general, now of all three ascensions and descensions in particular.   

Of the right, obliqne, and mean ascension in particular, and of the chiefest causes of such diversity of ascensions. Chap. 32.  
ANd for the better understanding of the Astronomical ascension and descension, we will make this division, for either it is of some point or star, or else of some portion of a Circle chiefly of the Ecliptic line. In the ascension of any point or star, we consider two things. First what Angle it maketh with the Horizon either right or obliqne. Secondly the time from the rising of the first minute of Aries, which is the first beginning of the Longitude of any star or Circle in heaven, and in respect of the Angle every ascension is said to be right in a right Sphere, and obliqne in an obliqne Sphere: Again the time of the ascension is to be measured by the degrees of the Equinoctial from the first minute of Aries, unto that degree and minute of the Equinoctial which ascendeth together with the stars, And note by the way that 15. degrees of the Equinoctial do make an hour, and four do make one degree of the same Equinoctial, for four times 15. do make 60. minutes, which is an hour, Again every ascension considered, according to the time of his gate, is either right, obliqne, or mean: if it be right, it is slow: if it be obliqne, it is quick: if it be mean, it is equal. Now the ascension of any Ark or portion of a Circle is also either right, obliqne, or mean: if it be right, it ascendeth slowly: if obliqne, it ascendeth quickly: if mean, it ascendeth equally. And the better to understand all these three kinds of ascensions, I will set down these twelve rules here following, whereof five do belong to the right Sphere, and seven to the obliqne.  
In the right Sphere all the four quarters rising from the four principal points, have a mean ascension, and so hath all the four points themselves.  
In the right Sphere all those signs that be equally distant from the four principal points have equal ascensions.  
3 In the right Sphere all stars or points that be in the Solsticiall Colour have mean ascension.  
4 In the right Sphere those signs that do ascend rightly, do descend rightly, and those that do ascend obliquely, do descend obliquely.  
5 In the right Sphere, Gemini, Cancer, Sagittarius, and Capricornus, do ascend rightly, and all the rest obliquely.  
6 In the obliqne Sphere, the two Equinoctial points have mean ascension.  
7 In the obliqne Sphere each half of the Sphere, beginning at either of the Equinoxes have mean ascension: but this rule holdeth not, if that you begin any other where.  
8 In the obliqne Sphere, those signs that do ascend rightly, do descend obliquely, & those which ascend obliquely do descend rightly.  
9 In the obliqne Sphere, the ascension of any supposed sign is equal to the descension of his opposite sign, and the descension of any supposed sign, is equal to the ascension of his opposite sign.  
10 In the obliqne Sphere, the ascension of any sign being added to his descension, is equal to the ascension and descension of the same sign being in the right Sphere.  
11 In the obliqne Sphere every two signs equally distant from the two points of the Equinoctial, have equal ascensions and descensions.  
12 In the obliqne Sphere under the pole Arctique, all signs from Cancer to Capricorn, do ascend rightly, and all the rest obliquely, but contrariwise under the Pole Antarctique.  
What is the chief cause of the diversity of ascensions and descensions, aswell in the right as in the obliqne Sphere?  
The chief cause is the diversity of the Angles which the zodiac maketh with the Horizon, for the sharper that the Angles be the lesser portion of the Equinoctial riseth together with the Ecliptic, and the righter that the Angles be, the greater portion of the Equinoctial riseth, but the Equinoctial by reason of his uniformity, maketh his Angles always equal one to another, that is to say, in the right Sphere, it maketh right Angles, and in the obliqne Sphere, though not right, yet in every sign it maketh like Angles.   

How to know the diversities of the ascensions and descensions, as well in the right as obliqne Sphere. Chap. 33.  
THat is to be known most exactly by the Tables of ascensions calculated of purpose by johannes de Monte Regio, and by Reinholdus called in Latin Tabulae directionum, and you may know it also without having respect to every minute by marking and observing the same in a material Sphere or Globe, that hath a standing foot with a firm Horizon, for if you will know the diversities of ascensions in a right Sphere, than you must lay the Sphere or Globe so as the Horizon may pass through both the Poles, and in turning about with your hand the Equinoctial, together with the Ecliptic from East to West, mark with what degree of the Equinoctial any sign beginneth to ascend, & mark that degree of the Equinoctial with a little piece of wax, then turn the Globe or Sphere towards the West, until the last degree of the said sign do appear just with the upper edge of the Horizon, and then mark what degree of the Equinoctial is answerable to the said last degree of the foresaid sign, and there set another piece of wax, then count the degrees of the Equinoctial contained betwixt those two marks, and if it be more than 30. that sign is said to ascend rightly, if it be less than 30. then that sign ascendeth obliquely, if it be just 30. then it hath a mean ascension, & by allowing 15. degrees of the Equinoctial to an hour, and 4′· to a degree, you shall know in what time that sign riseth. As for example if you would know what ascension the whole sign Taurus hath in a right Sphere, and also in what time it riseth, do thus, First lay both the Poles of the Sphere just upon the Horizon, so as the same Horizon may pass through both the poles, then bring the first point of Taurus to the East part of the Horizon, so as it may touch the upper brim or edge of the Horizon, and staying it there with your hand, look what degree of the Equinoctial doth also touch the Horizon at that instant, which you shall find to be 27. degrees 54′· and mark that degree of the Equinoctial with a little piece of wax, or some other thing that may be easily put out or taken away, that done, put forward the foresaid sign Taurus still towards the West, until the last degree of the said sign be ascended up even to the upper edge of the Horizon, and there staying it with your hand, look again what degree of the Equinoctial doth rise withal, which you shall find to be 57 degrees 48′· and there set another mark upon the Equinoctial, then by telling the degrees contained in the Equinoctial betwixt the two marks, you shall find the number of degrees to be 29. degrees 54′· and by allowing 15. degrees to one hour, and 4′· to a degree, you shall find that the whole sign Taurus spendeth in his rising one hour, 59′· 36″· But now sith the Meridian in any place (as hath been said before) doth always show the right ascension of any star, sign, ark, or point, because that cutteth both the Equinoctial and the Horizon with right Angles: you may therefore found the right ascension of the said sign, or of any other sign or star without removing the Sphere from your own elevation or Latitude in this manner following, bring the first degree of Taurus close to the movable Meridian, and there staying it mark what degree of the Equinoctial the Meridian cutteth at that present, which you shall found to be 27. degrees 54′· which is the right ascension of the first point of Taurus, then having brought the last point of Taurus to the foresaid Meridian, mark what degree of the Equinoctial the said Meridian cutteth at that present, and you shall find it to be the 57 degrees 48′· now by counting upon the Equinoctial, the degrees contained betwixt those two marks, you shall find the number to be 29. degrees 54′· and you may find the self same number by subtracting the right ascension of the first point of Taurus, out of the right ascension of the last point of Taurus, & thereby you shall know the time of his rising to be the same that you found in the right Sphere. Now if you would know the ascension of any sign in an obliqne Sphere, then having placed your Sphere according to your Latitude, which for example sake suppose to be 52. degrees, and that in such Latitude you would know what ascension the whole sign Taurus hath, and in what time he riseth, you must first bring the first degree of Taurus to the East part of the Horizon, so as it may meet even with the upper edge of the Horizon, and there staying it, mark what degree of the Equinoctial riseth therewith, which you shall find to be 12. degrees 48′· and having marked that degree, put forward the foresaid sign Taurus towards the West until the last degree thereof be ascended up to the upper edge of the Horizon, and then make another mark upon the point of the Equinoctial, which riseth at that instant with the last degree of Taurus, which you shall find to be 29. degrees 42′· and by counting the degrees contained in the Equinoctial betwixt the two marks, or by taking the lesser ascension out of the greater, you shall find the number of degrees to be 16. and 54′·S whereby you may conclude that the ascension of Taurus in that Latitude is obliqne, and that he spendeth in his rising one whole hour 7′· 36″· And look what order is here taught to find out the ascension of any sign, the same order is to be observed for the finding out of the descension of any sign, saving that you must seek for the descension of any sign in the West part of the Horizon of the Sphere or Globe, and not in the East part. As for example, if you would know what descension Taurus hath, and in what time he descendeth in the foresaid Latitude: here having brought the first degree of Taurus to the West part of the Horizon, so as it may touch the upper edge thereof, and having also marked what point or degree of the Equinoctial toucheth the same Horizon at that instant, which you shall found to be 42. degrees 30′· cease not to turn the Sphere or Globe, until all the whole sign of Taurus be descended under the Horizon, and that the last degree thereof do meet just with the upper edge of the Horizon, and there stay it until you have again marked that point of the Equinoctial which toucheth the Horizon at that instant, which you shall find to be 84. degrees 54′· and by counting the degrees contained betwixt the two marks on the Equinoctial, you shall found the number of degrees to be 42. degrees 24′· so as you may conclude that the descension of Taurus in that Latitude is right, and that he spendeth in his going down two hours 48′·S  
How shall I know the right or obliqne ascension of any of the fixed stars, and also at what hour of the day or night they rise & set, and how long they abide above the Horizon: finally when they are at the highest, and when they are at the lowest, called the depression or lowest Meridian Altitude of the stars?  
All these things are most truly known by Tables calculated of purpose, and also they are to be known by help of the celestial Globe in such manner as shall be declared hereafter when we come to treat of the said Globe.   

Of the ascentionall difference and uses thereof. Chap. 34.  
WHat is the ascentionall difference?  
It is a portion of the Equinoctial, whereby is known how much the right ascension and obliqne ascension of any star, or portion of the Ecliptic line, or any other point in the firmament, doth differ one from another, As for example in that place where the Pole is elevated 52. degrees, the right ascension of the first point of Taurus is 27. degrees and 54′·S and the obliqne ascension of the same point is 12. degrees 48′· here by taking the lesser out of the greater, that is 12. degrees 48′· out of 27. degrees and 54′·S there will remain 15. degrees and 6′·S which is the ascentionall difference.  
What uses hath the ascentionall difference?  
The ascentionall difference being known, all the obliqne ascensions and descensions of the stars are easily known by the Tables of directions, again by this difference is known the increase and decrease of the artificial day in every Latitude, and therefore it is called of some incrementum diêi. Moreover it showeth the semidiurnall Ark of the artificial day, for in every obliqne Sphere the artificial day is always either longer or shorter than the Equinoctial day throughout the year, unless the sun be in either of the Equinoctial points.  
How is the increase or decrease of the day to be known by the ascentionall defference?  
That shall be declared hereafter in the 50. Chap. of this first book, whereas we treat of the artificial day and night, in the mean time we will speak somewhat of the poetical rising and setting of the stars.   

Of the Poetical rising and setting of the stars. Chap. 35.  
DEfine what the Poetical rising and setting is.  
The Poetical rising is the appearing of some star above the Horizon, determined by the sun, and the Poetical setting, is either the going down of some star under the Horizon, or else the hiding thereof under the beams of the sun.  
How manifold is the Poetical rising and setting?  
threefold, that is, Cosmicus, Acronicus, and Heliacus, the signification of which words shall appear unto you by the definitions of the foresaid three kinds here following: For ortus Cosmicus, called in Latin mundanus, which is as much to say here as the worldly or morning rising, is when any star riseth in the morning above the Horizon, together with the sun, or rather with that point of the Ecliptic line wherein the sun is at that time. And the Cosmical setting, called in Latin occasus Cosmicus, is when a star goeth down under the Horizon at such time as the sun riseth, so as this kind of rising and setting is wholly to be referred to the rising of the sun.  
What is Ortus and occasus Acronicus?  
Ortus Acronicus which is as much to say as the Evening or temporal rising, is when any star riseth above the Horizon in the Evening at the going down of the sun: And occasus Acronicus, that is to say, the Evening setting is when any star goeth down under the Horizon, together with the sun, and therefore this kind is always to be referred to the going down of the sun, and not to his rising: And whatsoever sign or star doth rise Acronicè, the same goeth down Cosmicè, and whatsoever star doth rise Cosmicè, the same goeth down Acromicè. And generally all stars that rise in the day time, are said to rise Cosmicè, and all those that rise in the Evening after the sun set, are said to rise Acronicè.  
What is Ortus & occasus Heliacus.  
Ortus Heliacus, that is to say, the Solar rising, is when any star by departing from the beams of the Sun appeareth, & may be seen, which before being darkened by the sun could not be seen, And occasus Heliacus, is when any star by the nigh approaching of the sun ceaseth to be seen, for by reason that the sun by his yearly course & obliqne motion of the Ecliptic, doth sometime approach to diverse stars, and sometime by little and little retireth back again from the same, it falleth out that those stars to whom he approacheth, are by nighness of his great light, darkened and not seen, and by his departing from them, and especially when the sun is in the East or West part of the firmament, they begin again to be seen. And therefore as in the other 2. kinds, the Horizon together with the rising and setting of the sun, are to be considered as chief causes thereof, so in this last kind the chief cause is to be referred to the highness or farrenesse of the sun from the star.  
Whereto serveth the knowledge of this threefold Poetical rising and setting of the stars?  
It serveth chiefly to understand thereby those Poets and Histroriographers, which in showing the time of any act done or to be done, do not set down the day of the month, but are wont to describe the time by the rising or setting of some notable star, which they think most meet for their purpose, and thereby do greatly adorn their style, and specially being poetical: And because that the times wherein such stars did rise or set, do greatly differ in these days from the ancient times. Many therefore of our modern writers, as Garceus and others, have made diveses Tables of purpose to find out the difference, and thereby to come to the true knowledge of the times by the ancient men described, of which matter I leave to speak, thinking it not meet to trouble young Sailors therewith, for whom I chiefly wrote this Treatise of the Sphere.  
Yet some affirm that the ancient men did use the foresaid poetical rising and setting of certain stars, and specially of the Pleyades, Hiades, Orion, Arcturus, Capella, and Lira, (which stars were to them best known) as a Calendar not only to know thereby the difference of times, and seasons of the year: but also by their manner of rising, setting, hiding, and appearing to prognosticate and to foresee tempests and storms, yea and that in these days we also (as some writ) might do the like, though there were neither Calendar nor Ephemerideses, and in that respect the knowledge hereof seemeth most necessary for Mariners.  
All such things are to be known more exactly by the Astronomical ascension and descension, then by the Poetical rising or setting of the stars: And you have to understand that the stars since those days have changed their places, their longitudes, and declinations, and thereby in diverse respects have altered their Natures and qualities, yea and the very signs themselves: As for example, neither Taurus, Gemini, nor Cancer, is so hot and dry now, as in times past, neither doth Scorpio cause so much thunder now, as in times past, some again are more or less cold and moist than they have been heretofore, the causes whereof I leave to the discussing of the Astrologers, and so once again end with this matter.   

Of time, what it is, and into what parts it is divided. Chap. 36.  
MOst men that writ of the Sphere, after they have spoken of the ascensions, do immediately treat of the diversity and inequality of days and nights, but sith days, nights, and hours, are but parts of time, like as be weeks, months, and years, I mind here therefore first briefly to treat of time, and then of all his chiefest parts in order, for if you will be instructed at large of these matters, then read the book of johannes de sacro Busto de anni ratione, and also johannes Garceus his book de tempore.  
How define you time?  
Leaving to speak of time, without time, that is to say everlasting and infinite, called of the Latins Eternitas, ascribed chiefly to God, & therefore not contained within the movable Spheres or heavens: I mind to speak here only of that time which is a number measuring the moving of the first movable, and of all other mutable things, which time had his beginning with the world, and shall end with the same, and this time consisteth of two parts, that is first, and last, or rather before or after, successively following one another, and these two parts are knit together with a common bond called of the Latins Nunc, that is to say now, or at this present, which is the end of that which went before, and the beginning of that which followeth after, and therefore some do divide time into three parts, that is, time past, time present, and time to come, but the time present is a moment indivisible, and is the beginning of time, even as a point or prick is the beginning of all Magnitudes, & yet least part thereof itself: Again time is divided of some into greater and lesser parts, the greater are such as these: kalends, Nones, Ideses, a week, a month, a year, the space of five years, called of the Romans' Lustrum, and of the Greeks Olympias, the Romans' did call it Lustrum a lustrando, that is to say, of going about, because that they used in the end of every five years, with lights and torches of wax to go in precession round about the City, and did purge the same by sacrificing a Dog, a Sow, and an Ox, and at that time also they did choose their Dictator in a place called the field of Mars, but the space of five years called Olimpias, took his name of the high mount Olympus in Greece, whereas in the end of every five years were celebrated all kind of martial plays, as Fencing, Wrestling, Running, and such like in the honour of jupiter Olympicus, also the space of 15. years called indictio, in which space those foreign Nations that dwelled far off, and were tributary to the Roman Empire, paid their tributes, that is to say, in the first five years they paid only gold, in token of their obedience to the Empire: In the second five years they paid silver for soldiers wages, and in the last five years they paid brass towards the reparation of armour and munition. Item the space of an hundred years, called in Latin seculum, and in English an age, whereof the plays that were celebrated in Rome every hundred year, were called Ludi seculares, and last of all the space of a thousand years, called aewm, containing ten ages, again the lesser parts (as johannes de sacro Busto saith) are these five, the first is called in Latin quadrants, which is the fourth part of a day, that is six hours: The second punctus, which is the fourth part of an hour in the suns account, but in the Moons accounted the fift part of an hour: The third is called momentum, which is the tenth part of punctus: The fourth is called uncia, which is the twelfth part of momentum: The fift is called Atomus, which is the 48. part of uncia. But because in all the greater parts of time, there is no greater variation or difference, then in that which in Latin is called annus, and in English a year. I mind here therefore first to treat of a year, and then of months, weeks, days, nights and hours.   

Of the year, and of his diverse kinds, and of the diverse computations had thereof in diverse ages, and amongst divers Nations. Chap. 37.  
BE there diverse kinds of differences of years?  
Yea indeed, but I will speak here only of three kinds or differences, that is of the great year, the Solar year, and the Lunar year, whereof the two last are most necessary for our purpose.  
What is the great year?  
The great year is a space of time in the which not only all the Planets, but also all the fixed stars that are in the firmament, having ended all their revolutions do return again to the self same places in the heavens, which they had at the first beginning of the world: And therefore it is called of some the year of the world, and of some the great year of Plato, which containeth according to Alphonsus, 49/000. years, whereof we have spoken before, yet some affirm that the perfect year of the world containeth but 36000. years, whose revolution is after one degree in 100 years, but leaving this matter as not greatly profitable, we will speak now of the year Solar.   

Of the suns year called in Latin annus solaris, and of the diverse kinds thereof, and first of the Tropical year, both equal and unequal. Chap. 38.  
WHat is the year Solar?  
It is that space of time in which the sun departing from any point of the Ecliptic line, or from some fixed star of the zodiac, goeth round about the zodiac by his own proper moving, which is from West to East, and so returneth again to the self same point or star from which he first departed, and the Astronomers do make diverse divisions of the Solar year, for first they say that it is either Astronomical or Political: Secondly that the Astronomical year is either Tropical or Syderall. Thirdly that the Tropical is either equal or unequal, which unequal year they otherwise call the apparent year, and true year, all which kinds have in a manner one self definition, saving that the Tropical year taketh his beginning from the vernal Equinoctial point, and the Syderall year from the former star of the Ram's horn, and do differ chiefly in quantity.  
Show then what quantity, that is to say, how many days, hours, and minutes every such year containeth.  
The equal Tropical year being counted always from the middle point of the vernal Equinoxe, containeth 365. days, five hours, 49′· 15″· and 46‴· But the unequal or apparent Tropical year containeth sometime more and sometime less than the equal year, for sometime besides the 365. days and five hours, it amounteth to 56′· 53″· and 1‴· so as it is more than the equal Tropical year by 7′· 37″· and 15‴· and sometime over and beside the foresaid 365. days and five hours, it only containeth 42′· 38″· and 27‴· which is less than the equal Tropical year by 6′· and 3‴· which inequality chiefly chanceth by reason of the unequal precession of the two Equinoctial points before defined in the 26. Chapter.   

Of the Syderall year, and how much it containeth. Chap. 39  
WHat is the Syderall year?  
The Syderal or starry year is that space of time wherein the sun walking under the firmament, departeth from the first or foremost star of the Ram's horn, and returning to the same star again, which space of time always and equally containeth 365. days, six hours, 9′·S and 39″· so as this year is always greater than the Tropical year, and by his equality doth always rule and rectify the inequality of the Tropical year.   

Of the Political year, and diverse kinds thereof. Chap. 40.  
WHat is the Political year?  
It is a yearly space of time which any people or nation attributeth to the course of the Sun or of the Moon or of either of them which is diverse and manifold, according to the diverse customs of the Nations, of all which I mean not to speak at this present particularly, but of certain special and necessary to be known, as of the julian year, of the Egyptian year, of the jews year, and of the Athenian year.   

Of the julian year, and why it is so called. Chap. 41.  
WHat is the julian year?  
The julian year is that which we use at this present day which of all other years draweth nighest unto the Tropical year, for this year consisteth of 365. days and six hours, which six hours, if it should be reckoned every year, it would make a great confusion, and therefore it is reckoned at the end of every fourth year, which year consisteth of 366. days, for four times six doth make 24. hours, which is one whole natural day, whereof that year is called the leap year, And thereby the julian year is said to be two fold, that is common, containing 365. days, and the other bissextile or leap year, containing 366. days This word bissextile is derived of bis and sextus, because the sixth day next before the Kalends of March is twice repeated or reckoned, which indeed is the 25. of February, upon which day the feast of Saint Mathias commonly falleth.  
Why was it called the julian year?  
Because julius Caesar the first Monarch of the Roman Empire caused the year (according to the course of the Sun) to be reduced to the number of days and hours before expressed, who brought an excellent Astronomer with him at his coming from Egypt, aswell for that purpose as to teach the Mathematical disciplines unto the Romans', yet you have to consider that the julian year being greater than the Tropical year, doth cause great diversity in that it maketh aswell the Equinoctial & Solsticiall points, as also the entrance of the Sun into the other signs by little & little to anticipate or to run before, for whereas in julius Caesar's time the vernal Equinox was the 23. day of March, the same Equinox is now about the 11. day of March, which is sooner by 12. days.   

Of the Egyptian year, and how many days it containeth. Chap. 42.  
THe Egyptian year containeth the just number of 365. days, by reason of which equality this year is very fit to serve the Astronomers turn in making their Astronomical computations, but the Egyptian year hath no certain place of beginning: For by omitting the six hours which is the julian year, it doth anticipate in the space of 4. years one whole day in such sort, as 1460. julian years do make of the Egyptian years. 1461.   

How many Moons the jews year, and the Athenians year doth contain. Chap. 43.  
THe jews year containeth for the most part twelve Moons, and sometimes thirteen Moons, which kind of years did agree with the years of the Greeks', and of the Athenians, and also of the ancient Romans' before julius Caesar's time: and the ancient Romans' did begin their year from March, but the latter Romans' from the winter Solstice. Again the jews did begin their year at the first new Moon that followed next after the vernal Equinox: But the Athenians began their year at the new Moon that followed next after the Summer Solstice: The most people of Asia began their year at the Autumnal Equinox: But the most part of those that dwell in these parts of the world, following the custom of the Roman Church, do begin their year at the Kalends of january, which in old time was not much distant from the Winter Solstice, which Solstice at Christ's birth was the 25. of December, but now the same Solstice is about the twelfth day of December, so as the Winter Solstice falleth sooner by thirteen days than it did at that time. But we here in England do begin the year at the 25. of March.   

Of the year Lunar, and of the kinds thereof. Chap. 44.  
HOw many kinds of Lunar years be there, and which be they?  
Of Lunar years there be two kinds, whereof the one is ordinary called in Latin Annus communis, and the other extraordinary or excessive, called by a Greek name Embolismalis, The ordinary or common year, is the space of twelve Moons or changes, passing by course within the year Solar, and is called common because it hath twelve Moons Lunar, even as the Solar year hath twelve months Solar, and consisteth of 354. days and a little more, so as the Solar year exceedeth the Lunar year by 11. days, for the year Solar containeth (as hath been said before) 365. days, in which account the Fractions in both years are omitted: And therefore if these two years should begin together at one self time, the Lunar year would end his course sooner by 11. days then the year Solar.  
What is the extraordinary Lunar year called Embolismalis?  
It is the space of thirteen Moons or changes containing 384 days, so as this year exceedeth the common Lunar year by 30. days, and is more than the year Solar by 19 days.   

Of the divers kinds of months, and into what parts every Solar month is divided according to the Romans, that is, into Kalends, Nones, and Ideses. Chap. 45.  
HOw many kinds of months be there, and which be they?  
There be three kinds, that is, the month Solar, the month Lunar, and the month Usual.  
The month Solar is that space of time which the Sun spendeth in passing through any one of the 12. Signs.  
The Lunar month is that space of time which the Moon spendeth whilst she departing from the Sun returneth to him again.  
The Usual month is that number of days which are set down in our common Calendars, whereof some contain thirty days, some thirty and one, and the month of February hath but 28. days. But if you will readily know which contain more days and which less, keep always in memory these old English verses here following.  
Thirty days hath November,  
April, june, and September:  
February hath 28. alone,  
And all the rest have thirty and one.  
But when it is leap year, February hath 29. days, Again the Romans' divided the Solar month into Kalends, Nones, and Ideses.  
What be Kalends?  
The Kalends are the first day of every month from which the Romans' counted the days of the month proceeding backward, As for example the first day of April they named the Kalends of April, and the last day of March next before they called in Latin pridie Kalendas Aprilis, that is the day before the Kalends of April, and the next day before that, the third Kalends of April, and the next day before that the fourth Kalends, and so forth until they come to the Ideses.  
Whereof sprang this name Kalends?  
Of the Greek verb Calo, which is as much to say as call, for the first day of every month the crier standing in a high place made four calls or more to signify thereby to the people how many days in that month the fairs or markets called Nundinae should endure, and of Nundinae sprang this word Nonae, that is to say the days of the fairs: For look how many Nones there were in every month, so many fairs there were, during which time the Romans' never worshipped any God because there was no holy day during that time, and therefore Ovid saith that Nonarum tutela Deo caret, that is to say, no God had tuition of the Nones.  
What are Ideses?  
They are those days by which the Nones are divided from the rest, and these Ideses do divide in a manner the whole month into two equal parts, for the first Ideses most commonly falleth either on the 13. 14. or 15. day of the month.  
How many Ideses, Nones, and kalends do belong to every month?  
Of Ideses every month hath eight, but of Nones March, May, July, and October, have six, and all the rest of the months have but four Nones, but they differ most in the number of Kalends, as you may perceive by this Table following, which showeth how many kalends, Ideses, and Nones, do belong to every month. Thus far of the month Solar, now I will speak of the month Lunar.  

The Table.  Months. Kal. Ide. None. januarie. 19 8 4 February. 16 8 4 March. 17 8 6 April. 18 8 4 May. 17 8 6 june. 18 8 4 july. 17 8 6 August. 19 8 4 September 18 8 4 October. 17 8 6 November. 18 8 4 December. 19 8 4   

Of the diverse kinds of months Lunar. Chap. 46.  
HOw many kinds be there, and which be they?  
johannes de Sacro Busto saith that there be four kinds, that is the month of paragration, the month of apparation, the month medicinal, and the month of consecution.  
The month of Paragration, is that space of time in which the Moon departing from any one point of the zodiac, goeth by her proper moving about the zodiac, and returneth again to the said point from which she first departed which her revolution is accomplished in 27. days, & 8. hours. And this revolution of some is called a year, and by this account the Moon tarrieth in every sign two days six hours and 29′·S  
The month of Apparation consisteth of 28. days, divided commonly by four weeks, every week containing seven days, for four times seven maketh 28. of which four weeks the first is counted from her first appearance unto the end of the seventh day, and so forth from week to week, so as the fourth week endeth at the 28. day, in which account the odd hours during the moons abode under the beams of the sun, when as she is said to be combust, are not reckoned.  
The month Medicinal containeth but 26. days, and a half (as Galen saith) and is divided also into four weeks, the division being made by minutes.  
The month of Consecution is that space of time wherein the Moon being in conjunction with the sun, goeth about her Circle and returneth again to the same point, and not finding the sun there because he hath in that while passed through one whole sign, she hasteth after and in two days and four hours 44′·S and a little more she over taketh the sun, and is again with him in conjunction, of which her following and overtaking, the sun this month is called the month of Consecution, which month consisteth of 29. days and a half: during which time as the sun by his own proper course passeth through one sign or there abouts, so the Moon by her course in the self same time passeth through the whole zodiac and one sign more: And note by the way that the sun in making his own proper course doth not enter into any sign in the very beginning of any month, but rather about the midst of every month, or at the lest not much over or under that day.   

Of a week. Chap. 47.  
NOw because that every month aswell Solar as Lunar is divided into four weeks, I will speak somewhat of a week.  
What is a week?  
A week called most commonly in Latin Septimana, which is as much to say as seven mornings, is the space of seven days, whereof the first is called Sunday, the second Monday and so forth to Saturday, which names the Gentiles gave to these seven days in honour of the seven Planets whom they worshipped as Gods, for they called the first day the day of the sun, the second the day of the Moon, the third the day of Mars, the fourth the day of Mercury, the fifth the day of jupiter, the sixth the day of Venus, the seventh the day of Saturn.  
Are there any more names belonging to this word week?  
Yea, it is called also by a Greek name Hebdoma, that is to say, containing seven days, and in the scripture it is called sometime Sabbatum, as when the Pharisee said that he fasted Bis in Sabbato that is to say, twice a week, for the jews called Sunday the first of the Sabaoth, and Monday the second of the Sabaoth, and so forth in order until Saturday, which in deed was their true Sabaoth or day of rest.  
Why is it not still so counted amongst us Christians, but changed into Sunday?  
For two causes, first to avoid the superstition of the jews, and partly it was done in the honour of Christ, whose day of birth, resurrection, and sending of the holy Ghost, was on the Sunday.   

Of days and nights both natural and artificial. Chap. 48.  
THe Astronomers do divide the days into two kinds, whereof the one is called natural, and the other artificial.  
Which call you a natural day?  
A natural day is one entire revolution of the Equinoctial about the earth, whereunto must be added such portion of the zodiac, as the sun in the mean while maketh by his proper motion, which is from West to East.  
In what time doth the Equinoctial every day make his revolution?  
In 24. hours, which space containeth both day and night, according to which revolution and number of hours, the most part of Horologies or clocks in the East countries do go, and are set to show the hours of the day, but yet diversly, for some begin their natural day at the rising of the sun, as the Bohemians, and the Persians', and some at the going down of the sun, as the Italians, the Athenians, the jews, the Egyptians, and the Arabians, but the Astronomers reckon their natural day from noontide to noontide.  
If the Equinoctial doth make his revolution just in 24. hours, than all natural days are equal?  
That is true if ye only consider the motion of the Equinoctial, but if you add thereunto (as I said before) that portion which the sun in the mean while maketh by his own proper motion, you shall find them to be unequal, because that portion is sometime more, sometime less, according to the swift or slow ascension of the sign wherein the sun is.  
What is the artificial day?  
It is the distance or space that is betwixt the rising of the Sun, or going down of the same.   

The causes why the time betwixt the rising & going down of the sun is unequal. Chap. 49.  
IT must needs be unequal, because the abode of the sun above the Horizon is variable, aswell for the unequal ascension and descension of the signs, as also for the obliquity of the Horizon and zodiac: and therefore the spaces of the artificial days must needs be unequal: for the sun aswell in ascending from the beginning of Capricorn, to the beginning of Cancer, as also in descending from the beginning of Cancer to the beginning of Capricorn, describeth on each part 182. Circles or Parallels, the middlemost whereof is the Equinoctial. All which Parallels are divided into two parts by the Horizon, and the arks which are above the Horizon are called the artificial days, and the arks beneath the Horizon are called the artificial nights, which arks to those that devil under the Equinoctial in a right Sphere are always equal, that is to say, the diurnal Ark is equal to the nocturnal, because their Horizon passeth through the Poles of the world, but to those that devil in an obliqne Sphere, above whose Horizon the Pole is any thing elevated, be it never so little, the arks or Parallels are unequal one to another, that is to say, either making short days and long nights, or else long days and short nights, the Equinoctial only excepted, which aswell in the obliqne Sphere as in the right, is always divided by the Horizon into two equal parts, and so maketh the days and nights equal in all places of the world. All which things you shall easily comprehend by these two figures following, whereof that on the left hand representeth the right Sphere, and the other on the right hand representeth the obliqne Sphere.  
A figure of the right Sphere.  
A figure of the obliqne Sphere.  
And here I think it not amiss to show you how to find out the length of every artificial day and night throughout the year in every Latitude by the material Sphere, and likewise how to know at what part of the Horizon the sun riseth and setteth every day, and also how to find out his Meridian Altitude, whereby you shall know how nigh or how far from your Zenith he is every day.   

How to found out by the material Sphere or Globe, and by help of the ascentionall difference before defined, the increase and decrease of every day throughout the year in every several Latitude, and at what hour the sun riseth and setteth. Chap. 50.  
FIrst having set the Sphere at your Latitude, learn to know by some Table or instrument in what sign and degree thereof the sun is at such day of the month and year as you seek, and bring that degree close to the movable meridian, & there mark in what point or degree the said Meridian cutteth the Equinoctial, & what number it hath, for so shall you have the right ascension of the degree of the sun for that day. That done, bring the said degree of the sun to the East part of the Horizon, so as it may meet even with the upper edge thereof, and staying it there, mark what degree of the Equinoctial at that instant doth also touch the Horizon, and what number it hath, and that is the obliqne ascension of the foresaid degree of the sun, then subtract the lesser number out of the greater, and that which remaineth shall be the ascentionall difference, as for example in this present year 1590. the xi. day of April the sun is in the first degree of Taurus, whose right ascension in the Latitude 52. by doing as before is taught, you shall find to be 27. degrees and 54′·S and the obliqne ascension thereof to be 12. degrees 48′· and the ascentionall difference to be 15. degrees 6′· which difference you must first double, and then convert the same into hours and minutes by allowing 15. degrees to an hour, and 4′· to a degree, that done, add those hours & minutes to the Equinoctial day, which is always 12. hours, & the sum of that addition will show you the length of the day, when the sun is in the first point of Taurus, which is 14. hours & 12′·S and that is the true length of that day. But you have to note by the way that as you have to add the sum of the hours to the Equinoctial day, the sun being in any of the 6. Northern signs, so must you subtract the said hours from the Equinoctial day, the sun being in any of the six Northern signs, As for example, suppose the sun to be in the first point of Scorpio, the right ascension whereof is 207. degrees 54′· and the obliqne ascension is 223. degrees, here by taking the lesser out of the greater you shall find the ascentionall difference to be 15. degrees 6′· which being doubled, maketh 30. degrees and 12′·S that is two hours and 12′·S which if you subtract from 12. there will remain 9 hours 48′· which is the length of the artificial day, when the sun is in the first degree of Scorpio, the one half whereof is called the semidiurnall Ark of that artificial day, which is 4. hours 54. minutes, whereby you may gather that the sun at that time riseth four minutes after seven of the clock, and setteth again four minutes before five: so likewise in the former example where the ascentionall difference was 15. degrees six minutes, which being doubled made 30. degrees and 12′·S that was two hours and 12′·S which being added to 12. hours made 14. hours and 12′·S the half whereof is seven hours and six minutes, whereby you may gather that the sun did then rise 6′· before 5. of the clock in the morning, and did set 6′· after 7. of the clock in the Evening, and so you have both the forenoon & afternoon of the day, which two times are best to be reckoned always from 12. a clock at noon, that is to say, the forenoon hours and minutes, from twelve backward, and the after moon hours and minutes from twelve forward, & by subtracting the whole length of the artificial day from 24. hours, you shall have the length of the artificial night, as in the former example take 14. hours and 12′·S from 24. hours, and there will remain for the length of the artificial night 9 hours and 48′·S   

How to know by the material Sphere or Globe, in what part of the Horizon the sun riseth and setteth every day, and thereby the length of the day. Also how to know the Meridian altitude of the sun every day throughout the year, & being at his Meridian altitude, to know how far distant he is from the Zenith every day. Chap. 51.  
FIrst then having set the Sphere or Globe at your Latitude, which suppose to be 52. degr. bring the degree of the sun for that day, to the East part of the Horizon, so as it may meet just with the upper edge thereof, and there set a little piece of wax upon the Horizon, that dove turn the said degree of the sun to the West part of the Horizon until it meet again with the upper edge of the Horizon, and there set another piece of wax upon the Horizon, and those two marks will show you in what part of the Horizon the sun riseth and setteth: As for example, I would know this present year 1594. in what part of the Horizon the sun doth rise and set the twelfth of june, here finding by the Ephemerideses that the sun is entered 16′·S into Cancer at that day, I bring that point of the Ecliptic to the East part of the Horizon, so as it may meet just with the upper edge thereof, and there I set a little piece of wax upon the Horizon: that done I turn the said point of the Ecliptic to the West part of the Horizon, and whereas that point toucheth the upper edge of the said Horizon, I set there another piece of wax upon the Horizon, then counting the degrees upon the Horizon from the true East point thereof to the first piece of wax Northward, I find the number of the degrees to be 40. degrees or thereabouts, which containeth three points and a half and somewhat more, of the Mariner's Compass, whereby I gather that the sun riseth at that time very near to the North-east, and setteth near to the Northwest, because the number of the degrees in both parts of the Horizon are like.  
How shall I know how many points of the Mariner's compass are contained in any number of degrees, exceeding the number of degrees and minutes contained in one point?  
The Mariner's Compass containeth 32. points, and every point containeth 11. degrees and ¼. of a degree which is 15. minutes, wherefore whensoever you would know how many points of the Compass are contained in any number of degrees be it great or small, multiply that number by four, and divide the product thereof by 45. and the quotient will show the number of the points, and if there be any remainder, then because the Mariner doth make every point to have four quarters, multiply that remainder by four, and divide the product by 45. which is the common divisor, and the quotient will show the quarters, or if the remainder be but small, then multiply that remainder by eight, which are half quarters, and divide the product thereof by 45. as before, and the quotient will show the half quarters of a point: As in the former example in multiplying 40. degr. by 4. the product is 160. which if you divide by 45. you shall find in the quotient 3. whole points, & the remainder to be 25. which being multiplied by 4. the product will be 100 which if you divide by 45. you shall found in the quotient 2. quarters of a point, & the remainder to be 10/45. of a quarter, that is to say, if you can divide one quarter into 45. parts, than you must take 10. of those parts, and add them to the former sum, which being of small importance is not to be regarded. But now to return to my first matter, I say that by counting the degrees upon the Horizon, from the first piece of wax to the South point of the Horizon, I found the number of degrees to be 130. degr. & by allowing 15. degr. to an hour, I find the half day to contain 8. hours 10′· which being doubled maketh 16. hours 20′· Than to know the Meridian altitude of the sun, you must bring the degree of the sun right under the Meridian, and the number of the degrees contained in the Meridian betwixt the South point of the Horizon, and the said degree of the sun will show the Meridian Altitude of the sun, so shall you found the Meridian Altitude of the sun, being in the first point of Cancer to be 61. degrees, which once had, the distance from the Zenith is soon known: for if you subtract 61. degrees from 90. you shall found the distance of the sun from the Zenith to be 29. degrees. Again contrariwise the distance from the Zenith being subtracted from 90. the remainder will show the Meridian Altitude, for those two numbers being added together, do always make a just quarter of the great Circle, which is 90. degrees.   

Of hours as well equal as unequal, and into what parts they are divided. Chap. 52.  
AS of days and nights there be 2. kinds, that is natural and artificial before defined, so likewise are there 2. sorts of hours, that is equal, and unequal: An equal hour is the 24. part of a natural day, and every such hour containeth 15. degrees of the Equinoctial, for 15. times 24. maketh 360. degrees, which is the whole circuit or Longitude of the Equinoctial, which according to the diurnal moving of the first movable maketh his revolution in 24. hours as hath been said before, and therefore this equal hour is also called an Equinoctial or natural hour. The unequal hour is the twelfth part of an artificial day or artificial night, which days and nights as they be sometimes long and sometimes short, according to the time of the year, so are the hours of the same. For both day and night, be it never so short, is divided by the Astronomers into twelve hours, so as when here in some part of England towards the North, the artificial day is 17. hours long, and the night but seven hours, if you divide either of these into twelve parts, you shall found that every such twelfth part of the day shall contain more than the natural or Equinoctial hour by 25′· and the twelfth part of the said artificial night to contain but the one half of an equal hour and five minutes, which notwithstanding being both added together, will make in all 24, equal hours, which is a natural day, so as by this means you may easily perceive that a natural day comprehendeth both kinds of hours, aswell unequal as equal. Note also that the unequal hours are called sometime artificial and sometime temporal hours, artificial, because they are daily changed by the variety of the artificial days and nights, and are never equal but twice in the year, when the sun is in either of the Equinoxes: and they are called temporal, because the ancient observers of time were wont to make diverse clocks and Horologies to show these unequal and temporal hours, of which clocks there are yet some to be seen at this day. Moreover the hours both equal and unequal are divided not only into quarters of hours, but also intominutes, for every hour be it long or short, is divided into 60. minutes, and every minute into 60. seconds, and every second into 60. thirds, and so forth to fourth's, fifts, sixthes, and into as many as you will, so as you make your division always by 60. But you have to note, that as the Astronomers do divide the artificial day and artificial night into hours both equal and unequal, so the jews do divide each of them into four quarters, in manner and form following.   

How and in what manner the jews do divide the artificial day and night, each of them into four quarters. Chap. 53.  
THey divide the artificial day into four quarters by allowing to every quarter 3. hours, accounting the first hour of the first quarter at the rising of the sun, and the third hour of the said quarter they called the third hour, and the third hour of the second quarter they called the sixth hour, which was midday or noontide, again they called the third hour of the third quarter, the ninth hour, and they called the second hour of the fourth quarter the eleventh hour, and th●● called the last hour which was the twelfth hour of the day, eventide because then the sun went down.  
Whereto serveth the knowledge hereof?  
By knowing this account you may the better understand certain places of the Scripture making mention of things done at certain hours not like unto our common hours, For whereas it is said in the Gospel of Saint john, that Christ healed the Ruler's son that was sick of an Ague in Capernaum, and that the Ague left him the seventh hour, is as much to say, as the Ague left him at one of the clock in the afternoon: Again where as mention is made in the Gospel of S. Matthew, of the labourers that came to work in the Vineyard at the eleventh hour is to be understood at five of the clock in the afternoon, or rather one hour before the sun set, for you must think that the sun riseth not nor yet goeth down in jewrie always at six of the clock, for than they should have no artificial day nor night, but all days and nights should be a like. Also they divided the artificial night into four quarters, otherwise called by them the four watches of the night, for the first three hours was the first watch, during which time all the Soldiers both young and old of any fortified Town were wont to watch, the second three hours they called the second watch, which was about the dead of the night, at which time the young Soldiers only watched, and the third quarter of the night containing also three hours, and was called the third watch the Soldiers of middle age did watch, and the last three hours called the 4. watch which was about the break of day, the old Soldiers only watched. But now because the ancient Astronomers do appoint the government of the unequal hours to the seven Planets, it shall not be amiss to show you here what Planet reigneth every hour both day and night.   

How to know what Planet reigneth in every hour of the day or night artificial, as well by help of a table, as by a rule contained in one verse. Chap. 54.  
But first I will describe unto you the Table, and then briefly set down the use thereof. In the first collum of this Table on the left hand are set down the seven days of the week, whereof the first is Sunday, and the second Monday, and so forth downward to Saturday, against every which day the Planets are placed towards the right hand, every one in course one after another, and in the first row of this Table which is the head or front thereof, are placed the hours of the day written in Arithmetical figures, and in the next row of the said front, are set down the hours of the night written as you see in common numeral letters.  

The Table.  Hours of ●●e day. 1 2 3 4 5 6 7 8 9 10 11 12   Hours of ●●e night. iij iiij v uj seven viii ix x xj xii   i ij Sunday. ☉ ♀ ☿ ☽ ♄ ♃ ♂ ☉ ♀ ☿ ☽ ♃ ♂ ●●nday ☽ ♄ ♃ ♂ ☉ ♀ ☿ ☽ ♄ ♃ ♂ ☉ ♀ ☿ ●●esday ♂ ☉ ♀ ☿ ☽ ♄ ♃ ♂ ♀ ☿ ☽ ♄ ♃ W●●nesday. ☿ ☽ ♄ ♃ ♂ ☉ ♀ ☿ ☽ ♄ ♃ ♂ ☉ ♀ ●hursday. ♃ ♂ ☉ ♀ ☿ ☽ ♄ ♃ ♂ ☉ ♀ ☿ ☽ ♄ Friday ♀ ☿ ☽ ♄ ♃ ♂ ☉ ♀ ☿ ☽ ♄ ♃ ♂ ☉ Saturday ♄ ♃ ♂ ☉ ♀ ☿ ☽ ♄ ♃ ♂ ☉ ♀ ☿ ☽  

The use of the Table.  
Now the use of the said Table is thus: whensoever you would know in what hour of the day or night any Planet reigneth, you must first seek out the hour of the day or night, and if it be of the day, than you shall find it in the first row of the front, if of the night, in the second row of the front, as hath been said before: and from that hour descend with your finger to the common Angle standing right against the day which you seek, and that will show you what Planet then reigneth. As for example, if you would know on Wednesday at 8. of the clock of the day what Planet reigneth, then having found the number of 8. in the front, written in Arithmetical figure, come strait down from thence with your finger to the common Angle standing right against Wednesday, and you shall find that Mercury reigneth. And if you would know what Planet reigneth the same day at the eight hour of the night, then descend from the hour of the night down to the common Angle, and you shall find that the sun reigneth, and so forth of all the rest.  
The rule contained in one verse, and the use thereof.  
The rule in verse is thus:  
Sol, We, Mer, Luna, Saturnus, jupiter, & Mars.  
These seven. words (the conjunction & being left out) do signify the seven Planets: For Sol is the Sun, We standeth for Venus, Mer for Mercurius, Luna is the Moon, and the other three Planets following, as Saturnus, jupiter, and Mars, do make up the number of seven, which must always follow one another, in such order as they are here set down in the foresaid verse, and to have the true use of this rule, you must first apply every Planet to his own proper day, as Sol to Sunday, Luna to Monday, Mars to Tuesday, Mercurius to Wednesday, jupiter to Thursday, Venus to Friday, and Saturnus to Saturday: for every one of these Planets governeth the first hour of his own proper day, and the Planet placed next to him in the verse, governeth the second hour of the same day, and so forth orderly, as for example, if you would know what Planet shall reign on Sunday at the third hour of the day, you must first say that Sol doth reign the first hour because that is his day, and Venus reigneth the second hour, and Mercury the third hour according to your rule, & so by keeping the order of the verse, you shall easily appoint to every hour both of the day and of the night artificial his own governor: For though both day and night be divided each of them into 12. hours, making in all 24. hours, and that there be but seven Planets, yet by appointing every Planet to his own proper day as governor of the first hour of the same day, and by observing the order of the verse in repeating the said Planets, you shall not fail to give to every hour his proper Planet. Thus having sufficiently spoken of the celestial part, I will now proceed to the Elemental part of the world, contained in the second Book of this Treatise.     

The second part of the Sphere.  

Of the Elemental part of the world. Chap. 1.  
WHat doth the Elemental part contain?  
I told you before that as the celestial part doth contain the eleven heavens before described, so the Elemental part containeth the 4. Elements, that is to say, fire, Air, Water, & Earth, which are of themselves pure substances, and the first & next beginnings whereof all mixed bodies are compound, and therefore not to be seen with our outward eyes: for as we ourselves are bodies compound, so with our outward senses we can discern nothing but that which is compound: and therefore the fire, air, water or earth which we daily feel or see, are not the Elements themselves, but things compounded of them. The natures and properties of which Elements I mind here but briefly to touch, sith the exact handling thereof belongeth rather to natural Philosophers & to Physicians, then to Geographers, who have to deal only with the situations of the earth with Zones, Parallels, Climes, Longitudes, Latitudes, distances & such like things belonging to the measure and description of this earth here which we inhabit.  
Define these Elements.   

Of the Fire and of his nature and motion. Chap. 2.  
THe Fire is an Element most hot and dry, pure, subtle, and so clear as it doth not hinder our sight looking through the same towards the stars, and is placed next to the Sphere of the Moon, under the which it is turned about like a celestial Sphere.   

Of the Air and into how many Regions it is divided. Chap. 3.  
NExt to the Fire is the Air whichis an Element hot and moist, & also most fluxible, pure & clear, notwithstanding it is far thicker & grosser as some say, towards the Poles then elsewhere, by reason that those parts are farthest from the sun: And this Element is divided of the natural Philosophers into three Regions, that is to say, the highest Region, the Middle Region, and the lowest Region, which highest Region being turned about by the fire, is thereby made the hotter, wherein all fiery impressions are bred, as lightnings, fire drakes, blazing stars and such like.  
The middle Region is extreme cold by contra opposition by reason that it is placed in the midst betwixt two hot Regions, and therefore in this Region are bred all cold watery impressions, as frost, snow, ice, hail, and such like.  
The lowest Region is hot by the reflex of the sun, whose beams first striking the earth, do rebound back again to that Region, wherein are bred clouds, dews, rains, and such like moderate watery impressions, which three Regions of the air with the rest of the Elements this figure doth plainly show.  


The highest region of the air  
Middle region  
Loest region  
Earth.     

Of the Water, and whether it be round or not. Chap. 4.  
NExt to the Air, is the Water which is cold, moist, and fluxible, and being lighter than the Earth would of his own nature surmount and cover the whole earth, had not God in the creation of the world divided waters from waters (as the Book of Genes. saith) and gathered together those waters that are under the firmament into certain concavities of the earth, leaving other parts of the earth dry, and discovered, that man and beast might inhabit the same, & have food necessary for their behoof, so as now both water and earth doth make one entire and Spherical body, which is environed with the Air.  
Is not the water a round body of itself without the earth?  
Many late writers do deny the whole body to be round, affirming only the Conuere superficies or upper face of the water to be round, for (say they) the earth being not altogether round the Concave superficies of the water cannot be round, notwithstanding the most part of the ancient writers do affirm the whole body of the water to be round, saying that the water hath the like shape in his whole, that it hath in his parts: For the parts which are drops, are round, ergo the whole is round. Again they prove the water of the sea to be round by demonstration thus, suppose a ship to departed from the shore whereon some mark is set, which you may see with a right leveled line standing at the stern of the said ship, but saying further from the shore, you cannot see the mark any more standing upon the stern, but shall be feign to go up to the top of the mast to see it, by reason that the water being a round body riseth and swelleth in the midst, and so letteth your sight as this figure plainly showeth.   

Of the Earth and whether it be all round or not. Chap. 5.  
NExt to the Water is the Element of the Earth, which of his nature is thick, heavy, cold, dry, and not fluxible as is the water and air, but is firm and apt to keep his place, and though some deny the earth to be, round because of the high mountains, and deep dales and valleys therein, which are nothing in comparison of the whole earth to altar that roundness which it hath by nature, yet Aristotle affirmeth in his second book de coelo & mundo, the fourteenth Chapter, that the earth of her own nature is round, proving the same as well for that the Moon when she is eclipsed in part, could not have such horned shape as this figure representeth.  
Unless the earth were also round by the interposition whereof she is eclipsed, either totally or in part as hath been said before, Again he proveth the roundness of the earth by the altering of the Horizon, for in going from North to South, our Horizon altereth in such sort as we discover those stars which we could not see before, but were clean hidden from our sight, some also deny that the earth is in the midst of the world, and some affirm that it is movable, as also Copernicus by way of supposition, and not for that he thought so in deed: who affirmeth that the earth turneth about, and that the sun standeth still in the midst of the heavens, by help of which false supposition he hath made truer demonstrations of the motions & revolutions of the celestial Spheres, than ever were made before, as plainly appeareth by his book de Revolutionibus dedicated to Paulus Tertius the Pope, in the year of our Lord 1536. But Ptolemy, Aristotle, and all other old writers affirm the earth to be in the midst, and to remain unmovable and to be in the very Centre of the world, proving the same with many most strong reasons not need full here to be rehearsed, because I think few or none do doubt thereof, and specially the holy Scripture affirming the foundations of the earth to be laid so sure, 
Psal. 104.  that in never should move at any time: Again you shall find in the self same Psalm these words, He appointed the Moon for certain seasons, and the Sun knoweth his going down, whereby it appeareth that the Sun moveth and not the earth. But leaving this matter, we will now speak of the compass of the earth, and of the Longitude and Latitude thereof.   

Of the compass of the earth, and of the diversity of measures according to diverse countries. Chap. 6.  
CAn the whole earth be measured?  
Yea very well, for sith the earth and the water (as hath been said before) do make together, one whole Spherical or round body, and that every great Circle as well thereof as of the heavens, containeth 360. degrees, there is no more than to be done, but to allow for every such degree 60. Italian miles, which differ not much from our English miles, so as in multiplying 360. degrees by 60. you shall find the whole compass of the earth to be 2160. miles, of which compass if you would know the true Diameter, then having multiplied the said compass or circuit of 21600. miles by seven, divide the product thereof by 22. and the quotient together with the remainder, will show the true Diameter which is 6872. miles, five furlongs, and 9/11 of a furlong, and the half of that is the semi-Diameter of the earth, which is 3436. miles, and 4/11 of a mile: and as the Italians and we English men do measure great distances on the earth by miles, so the French, the Spanish, and the high Almains, do measure such distances by leagues both by land and sea, and every one differeth from other: for the French league containeth two of our miles, the Spanish league three, and the common league of Germany four, and the great league of Germany containeth five of our miles, yea in some places of Germany, as in Suevia, the leagues are so long as a man shall scant ride three of them in a whole day. Again the Grecians did measure the distances of the earth by furlongs, the Egyptians by signs, and the Persians' by parasanges, all which measures do greatly differ even in the smallest parts, from whence all measures do take their first original: for as well amongst the ancient men, as amongst them of latter days, four barley kernels couched close together side by side, and not end long, are said to make a finger breadth, and three finger breadthes an inch, and four finger breadthes a palm or hand breadth, and three palms or nine inches a span, and four palms or hand breadthes a foot, and two foot and a half to make a common pace, and five foot to make a Geometrical pace, of which kind of paces, 125. do make a furlong, and eight furlongs do make an Italian mile, and four such miles do make a common German league, as hath been said before, but by reason that the barley kernels be not in all Countries of like bigness, neither finger breadthes, inches, handbreadths, feet, nor any of the other measures are found any where to be equal: for the French foot of Paris is longer than ours by an inch, and the Italian foot is longer by two inches and more, and yet their miles are somewhat shorter than ours: and the German foot (according to Stophlerus.) is less than ours by two inches and a half. But to show the diversity of measures would require a long discourse more intricable than profitable, and therefore I lean to talk any further thereof, wishing you when we speak of miles, furlongs, paces, or feet, to consider the measure thereof according to the inch or foot of our English standard.   

Of the Longitude and Latitude of the earth. Chap. 7.  
WHat mean you by the Longitude and Latitude of the earth?  
Longitude is as much to say as length, and Latitude signifieth breadth, for sith the earth is a body, it must needs have both length, breadth, and depth.  
How define you such Longitude and Latitude, and how is it to be counted?  
The Longitude of the earth in general is that space or upper face of the earth which extendeth from West to East, and again from East to West: And the Latitude in general is that space which extendeth North & South even from the one pole to tother. Now to know how such Longitude and Latitude is to be accounted, you must first understand that the Equinoctial Circle girding the earth in the very midst, is divided into 360. degrees by reason of certain Meridian's which passing through the Poles of the world, do cut each half of the Equinoctial in eighteen points, which being doubled do make 36. spaces, every space containing ten degrees: and some do divide the Equinoctial with 36. Meridiaans', cutting each half thereof in 36. points, which being doubled do make 72. spaces, every space containing 5. degrees, which cometh all to one reckoning, for five times 72. do make 360. as well as ten times 36. of which Meridian's, be there never so few or many (for you may if you will make half as many Meridian's as there be degrees in the Equinoctial which amounteth to a 180.) yet according to Ptolemy, that Meridian is said to be first and furthest Westward which passeth through the islands called Insulae fortunatae, for the West Indies were not known nor discovered in his days, nor yet long time after, since the discovery whereof, the late Cosmographers of these days do make the first Meridian to pass through the Islands called Azores, which Islands, as appeareth by their Cards are situated more Westward from the foresaid Insulae fortunatae, by five degrees, the reason that moveth them so to do, is because the Mariner's Compass as they say, will never incline to the true North pole, but when they sail either by the isle S. Mary or S. Michael, affirming that in every other place the Compass doth vary from the true North, either by north asting or Northwesting. And by thus altering the ancient placing of the first Meridian, they must likewise altar all the Longitudes set down heretofore by Ptolemy or any other ancient writer, notwithstanding the matter is easily helped, for by adding to every Longitude Eastward five degrees, or by subtracting five degrees from every Longitude Westward, you shall not greatly vary from those ancient Longitudes that be truly set down. But to return again to my first purpose, I say tha● wheresoever this first Meridian cutteth the Equinoctial, there beginneth the first degree of Longitude, which proceedeth Eastward until you come to 180. degrees, which being the one half of the earth is as far as you can go Eastward, for then the earth being round, you must needs turn again Westward until you come to the 360. degree, which is the last degree of Longitude, and endeth where the first degree beginneth: and therefore the Cosmographers measuring always the Longitude by the degrees of the Equinoctial, do define Longitude to be that portion of the Equinoctial Circle, which is contained betwixt the first Meridian and the Meridian of any place supposed, but the distance betwixt any two supposed Meridian's (neither of them being the first Meridian) is not called of them Longitude, but the difference of Longitude: For suppose the distance of the one Meridian to be twenty degrees distant from the first Meridian, and the other but ten, these Longitudes you see are not like but do differ, and therefore the distance betwixt any such two places may be very well called the difference of Longitude, and not Longitude itself, which hath always regard to the first Meridian and to none other.  
Define once again what Latitude is?  
Latitude is none other thing but the distance of any place from the Equinoctial either towards the North pole or towards the South pole, so as there be two kinds of Latitudes, the one Northern, and the other Southern: And such Latitude is measured upon the Meridian which passeth through any place supposed. For every Meridian is also divided into 360. degrees, and by reason that the Equinoctial girdeth all such Meridian's in the very midst, it divideth them all into four equal quarters every quarter containing 90. degrees, which is the greatest Latitude that any place can have, as you may see in this figure following, whereof the first Meridian on the left hand is put to signify according to Ptolemy, that which passeth through the Fortunate Islands, or by the Azores according to the modern Cosmographers (if you will have it so) containing the degrees of the Latitude both Northward and Southward, and through the midst of all the Meridian's passeth the Equinoctial containing the degrees of the Longitude.  
The figure of the longitude and latitude of the world.  
What be the other manifold Circles in this figure dividing the Meridian's on each side of the Equinoctial, as well towards the North pole, as towards the South pole.  
They be called Parallels, whereof we shall speak in the next Chapter, in the mean time mark well this figure that thereby you may the better conceive what is the Longitude and Latitude of the earth.  
Understanding now the Longitude & Latitude of the whole earth, I am desirous to know how the Longitude and Latitude of every several place of the earth or sea is to be found out, & how far any place is distant one from another.   

How to know the Latitude of any place, aswell in the day as in the night. Chap. 8.  
BEcause the Latitude of any place is more easy to be found (as most men think) then the Longitude, I will first treat of Latitude. The Latitude them is to be known by the Astrolabe, Quadrant, Crossestaffe, and by such like Mathematical instruments, & that diverse ways whereof the most easy is thus: first with your Astrolabe or Quadrant, or any such like instrument, take the height of the sun right at noon, when the sun is in the first point of Aries or of Libra, which height if you subtract from 90. that which remaineth is the true Latitude of that place. But if you would know the Latitude at any other day or time of the year, then after that you have taken the height of the sun at noon, otherwise called the Meridian altitude, you must first learn to know the true degree of the suns declination by the Table of the declinations before set down, together with the use thereof in the 13. Chap. of the first book, or by some other Table more lately calculated, and if such declination be Northernly, than you must subtract that from the foresaid Altitude or height: but if the declination be Southernly, them you must add the same unto the foresaid height, and by such subtraction or addition, you shall have the height of the Equinoctial above your Horizon, which being subtracted from 90. that which remaineth is the true Latitude of that place: and to be sure in taking the Meridian altitude, it shall be needful to take it diverse times one after another, with some little pause betwixt, to see whether it increaseth or decreaseth, for if it doth increase, than it is not yet full noon, but if it decreaseth than it is past noon. This last way of finding out the Latitude, is, and hath been most commonly taught, as well by the ancient, as modern writers, as a most sure and ready way of finding the Latitude of any place.  
What if the sun do not shine at noon, nor perhaps all that day?  
Than you must tarry until night that some star appear, which you perfectly know, and such a one as both riseth and setteth. And having taken the Meridian altitude of that star with your Astrolabe or Quadrant, you must learn what declination he hath, and whether it be Northern or Southern. For if the star hath North declination, than you must subtract his declination from his Meridian altitude, and the remainder shall be the Altitude of the Equinoctial, which being taken out of ninety, shall be the Latitude of the place, or elevation of the pole: but if the declination of the star be Southernly, than you must add his declination to his Meridian altitude, and that sum shall be the Altitude of the Equinoctial, which being taken out of 90. the remainder shall be the elevation of the Pole. As for example, supposing that you know the star called Arcturus, or Bubulcus, and that you find his Meridian altitude by your Astrolabe or Quadrant to be 59 degrees 30. minutes and also that you have learned by some Table, that his declination to the Northward is 21. degrees 30. minutes: here by taking his said declination because it is Northernly, out of his Meridian altitude, you find the remainder to be 38. degrees which is the Altitude of the Equinoctial, which being taken from 90. the remainder willbe 52. degrees, which is the Latitude of the place whereas you made your observation, and this is a far more ready way then to wait all night to take the Meridian altitude, and also the depression of such a star as never setteth, which is séeledome done in one self night. And therefore I would wish all Mariners to acquaint themselves with many stars that do both rise and set, and so shall they be sure to find one such star or other, to be at his Meridian altitude at any hour of the night that they desire, if the stars do show.   

How to know the true Longitude of any place. Chap. 9  
THough the Longitude may be found out by divers ways not easy for every man's capacity, yet because Gemma Frisius thinketh none so sure as to know the same by the eclipse of the Moon, (which also as he saith) may sometime fail by reason of the diversity of aspects and Latitude of the Moon, and for that cause hath invented a more ready way to find out at all times the Longitude of any place, I mind here therefore briefly to show you first the order of finding out the Longitude by the Eclipse of the Moon, and then how to find out the same by that ready way which he hath invented: the order then to know it by the Eclipse of the Moon is thus: First you must learn by some Ephemerideses at what hour the Eclipse shall be in some place, where you know already by some Table the Longitude, that done, you yourself or some other for you, must the same day of the Eclipse observe by the Astrolabe at what hour the Eclipse beginneth in that place, whereof you know not the Longitude: For if the Eclipse do begin in both places at one self hour, then assure yourself that both places have one self Longitude, but if it begin sooner or later, than there is difference betwixt them, according to the variety of the time, which difference is thus to be known: Take the lesser sum of hours out of the greater, and there shall remain either hours, or minutes, or both, if hours, then multiply the same by fifteen, if minutes, divide the same by four, (for in this account fifteen degrees do make one hour, and four minutes do make one degree) and add the difference so found to the Longitude, if the Eclipse do appear there sooner: if later then subtract the said difference from the known Longitude, and that which remaineth will show the unknown longitude. But note by the way, that if there remain any minutes after the division, you must multiply those minutes by 15. and so shall you have the minutes of degrees.  
Show the use of this rule by some example.  
For example I find by the tables of Ptolomey that the longitude of Paris in France is 23. degrees, and by some Almanac or Ephemerideses I find that the Eclipse doth begin there at three hours after midnight, now by this I would know the longitude of Tubing a famous city in Suevia, which is a region of Germany, at which town in the very day of the Eclipse I cause to be observed by Astrolabe at what hour the Eclipse beginneth there, and I find that it beginneth at 3. of the clock and 24′· after midnight, then by subtracting the lesser number of time out of the greater, I found the remainder to be 24′· which being divided by 4′· which do make one degree, the quotient shall be 6. degrees, and that is the difference, which being added to the known longitude of Paris (because the Eclipse is sooner there then at Tubing) it maketh in all 9 degrees, whereby I gather that the longitude of Tubing is 29. degrees, by this means all the Tables of Cosmographers are most commonly made, and yet many times they greatly differ in their longitudes for lack perhaps of using diligence in taking the right hour and moment of the Eclipse, and for not duly considering the divers aspects, and what latitude the Moon hath at that instant which may 'cause great error.   

A ready way to find out the longitude of any place, invented by Gemma Frisius. Chap. 10.  
THat way is done by the help of some true Horologie or watch apt to be carried in journeying, which by an Astrolabe is to be rectified and set just at such hour as you departed from the place where you are, to go to any other place whereof you are desirous to know the longitude, in which your going you must be diligent to see that your watch never cease going, and being arrived at that place whereof you seek to know the longitude, you must tarry until the Index do justly touch the prick of some perfect hour, and also at that instant, to see what hour it is by your Astrolabe, for if your Astrolabe and watch do both agree in one, then assure yourself that there is no difference of longitude, but that you have traveled still under one self Meridian either towards the North or South. But if they differ one hour or certain minutes, then reduce them to degrees, or to the minutes of degrees in such order as is before taught, and thereby you shall find the longitude which you desire to know. But to take the longitude of any place upon the sea by this manner of way, most men think it were a great deal better to do it by the help of a great hour glass, made to run 24. hours, which must be watched when it is ready to run out, that it may be immediately turned: for watches made of Iron or Steel will soon rust upon the sea. This way of taking the longitude and many others I doubt not but that M. borough hath tried, & thereby is able to judge which is best, wherefore in mine opinion he may do his country great good by setting down in writing that way which he by experience knoweth to be best and readiest, for want whereof the Mariners cards do make the sea men often to err in their voyages.   

Another way taught by Appian to find out the longitude of any place with the cross staff by knowing the distance betwixt the Moon and some known star that is situated nigh unto the Ecliptic line. Chap. 11.  
FIrst seek to know by the Astronomical Tables the true moving of the Moon according to the longitude at the time of your observation at some certain place, for whose Meridian the roots of those Tables are calculated and verified. Also you must know the degree of longitude of some fixed Star nigh unto the Ecliptic, going either next before or else next after the moving of the Moon, than you must seek out the distance of the moving of the Moon and of the said Star, which distance once had, apply the cross staff to your eye, and move the cross up and down until you may see the centre of the body of the Moon with the one end of the cross and the foresaid fixed Star with the other end of the cross, so shall the cross show you by the degrees and minutes marked upon the staff, the distance of the Moon and of the foresaid Star answerable to the place of your observation, which being set down, set down also the distance betwixt the Moon and the foresaid Star that was first calculated, and then take the lesser out of the greater, so shall remain the last difference, which may be rightly called the diversity of aspects, which difference if you divide by the moving which the Moon maketh in one hour, you shall know thereby the time in which the Moon is or was joined with the first distance of the foresaid Star, then having converted that time into degrees and minutes, add or subtract the product thereof to or from that Meridian, unto which the Tables (whereby you first calculated the moving of the Moon) were verified, that is to say, if the distance betwixt the Moon and the fixed Star of your observation be lesser, then add the degrees and minutes to the known longitude, so shall you found the place of your observation to be more Eastward, but if it be greater, then subtract the degrees and minutes from the known longitude, and the place of your observation shall be more westward. All which rules Gemma Frisius affirmeth to be true, so as the Moon be more westward than the fixed Star: for if at the time of your observation the Moon be more Eastward, than you must work clean contrary, that is to say, if the distance betwixt the Moon and the fixed Star be lesser, you must subtract the degrees and minutes from the known longitude, so shall the place of your observation be more Westward: but if it be greater, than you must add the degrees and minutes unto the known longitude, and you shall find the place of your observation to be more Eastward.   

How to know the distance of places, that is to say, how many miles one place is distant from another, and how many ways places are said to differ in distance one from another. Chap. 12.  
THe distance may be known diverse ways, that is, either Arithmetically, Geometrically, or by the Tables of Sinus. But before I show you the order of any of these ways, you have to understand that any two places do differ in distance one from another one of these 3. manner of ways, that is, either in latitude only, or in longitude only, or else in both: if two places having one self longitude do differ only in latitude, then according to the Arithmetical way you must subtract the lesser latitude out of the greater, and the remainder shall be the difference, which being multiplied by 60. will show the number of miles as for example, London and Rouen having in a manner one self longitude, do differ only in latitude, for the latitude of London is 51. degrees 32′· & the latitude of Rouen is 49. degrees and 10′·S which being the lesser latitude and therefore to be taken out of 51. degrees 32′· there remaineth 2. degrees 22′· which 2. degrees being multiplied by 60. maketh 120. whereunto if for the 22′·S annexed to the degrees you add 22. miles (for every minute is a mile) it shall make in all 142. miles, which by a right line is the true distance betwixt London and Rouen. But you have to note that the difference of 2. sundry latitudes is not to be known by subtracting the lesser out of the greater, unless both the places be so situated, as both may have either North latitude or South latitude, for if the one place have North latitude and the other South latitude, the difference is to be known by addition, and not by subtraction: as for example, Naples in Italy hath 41. degrees of North latitude, and la Madalena in Aphricke not far from Manicongo, hath 8. degrees of south latitude, both places having one self Meridian, here the difference of these two latitudes is to be known by addition, and not by subtracting the lesser out of the greater, for 8. and 41. being added together do make 49. and that is the true difference, which being multiplied by 60. maketh 2940. miles. So likewise the difference of the two longitudes is not always known by subtracting the lesser out of the greater, unless the two places have both East longitude or else both West longitude: As for example Lisbona in Spain hath in East. longitude 13. degrees and Cap de losslavos in the West Indies hath in west longitude 334. degrees: here the difference of these two longitudes is not to be known by taking the lesser out of the greater, but thus, first take 334. out of 360. and there will remain 26. degrees, whereunto if you add the East longitude for Lisbona, which is 13. degrees, it will make in all 39 degrees, which is the true difference of the 2. longitudes, for if you should take 13. degrees out of 334. there would remain 321. which is not the true difference. But to know the distance of two places differing in longitude, this Table here following is needful.  

The Table of miles answerable to one degree of every several latitude.  1 2 3 4 5 6 D M S D M S D M S D M S D M S D M S 1 59 59 16 57 41 31 51 26 46 41 41 61 29 5 76 14 31 2 59 58 17 57 23 32 50 53 47 40 55 62 28 01 77 13 30 3 59 55 18 57 4 33 50 19 48 40 9 63 27 14 78 12 28 4 59 51 19 56 44 34 49 45 49 39 22 64 26 18 79 11 27 5 59 46 20 56 23 35 49 9 50 38 34 65 25 21 80 10 25 6 59 40 21 56 1 36 48 32 51 37 46 66 24 24 81 9 23 7 59 33 22 55 38 37 47 55 52 36 56 97 23 27 82 8 21 8 59 25 23 55 14 38 47 17 53 36 7 68 22 29 83 7 19 9 59 16 24 54 49 39 46 38 54 35 16 69 21 30 84 6 16 10 59 5 25 54 23 40 45 58 55 34 25 70 20 31 85 5 14 11 58 54 26 53 56 41 45 17 56 33 33 71 19 32 86 4 11 12 58 41 27 53 28 42 44 35 57 32 41 72 18 32 87 3 8 13 58 28 28 52 59 43 43 53 58 31 48 73 17 33 88 2 5 14 58 13 29 52 29 44 43 10 59 30 54 74 16 32 89 1 ● 15 57 57 30 51 58 45 42 26 60 30 0 75 15 32 90 0 0  
Describe this Table.  
This Table is divided into 6. collums, every collum containing, first the degrees of longitude, and then the miles and seconds of miles answerable to every degree, for every degree of the very Equinoctial itself is in value 60. miles, but the further you go from the Equinoctial either Northward or Southward, every degree of latitude is lesser in value than other, and containeth fewer miles, as you may easily see by the said Table, proceeding from one degree to 90. which is the greatest latitude that any place can have.   

How to know by the help of the foresaid Table the distance of two places differing only in longitude. Chap. 13.  
YOu must multiply the difference of the longitudes by the number of miles answerable to the latitudes of the said places, omitting always the seconds of miles set down in the said Table, because in this account they are of small importance: as for example, London and Antwerp, having both in a manner one self latitude, do differ only in longitude 6. degrees 42′· which difference being multiplied by 37. miles answerable to 51. degrees of latitude, as you see in the Table, do make in all 247. miles and 54″·S of a mile. But in making this multiplication, you must first multiply the 6. whole degrees by 37. and the product thereof will amount to 222. then by the rule of proportion you may found out the value of the minutes annexed to the degrees of the difference of longitude in saying thus, if 60′· which is one degree, do require 37. miles, what shall 42′· require. And by working according to the rule of proportion, you shall found the fourth number which you seek to be 25. miles and 54″·S which being added to the first product 222. maketh in all 247. miles and 54″·S of a mile, which wanteth but 6″· to make up another mile.   


How to find out the true distance of two places differing both in longitude and latitude by the Arithmetical way. Chap. 14.  
HOw is that done?  
First, take the difference of the longitudes and latitudes of both places by subtracting the lesser out of the greater, then convert the same into miles by multiplying the difference of the two longitudes into the miles that be answerable to the latitude of each place, which miles you shall found in the Table aforesaid, and if there be any minutes annexed to the degrees of the difference of longitude, then reduce the same also to miles by the rule of proportion, as before is taught, and having added the two products together, half the sum, and set it by itself. Than multiply the difference of the latitudes into 60. miles, and add thereunto the fraction of minutes annexed to the said difference if it hath any fraction, allowing for every minute one mile, and set that number also by itself: that done, square the sums reserved, that is to say, multiply each one part by itself into itself, and having added the two products together into one sum, seek out the square root thereof, & that shall be the true distance of the two places. As for example, if you would know the true distance betwixt London and Venice, first you must know by some Table the longitude and latitude of both towns, wherefore finding the longitude of London to be 19 degrees, 54′·S and the latitude thereof to be 51. degrees, 32′·S And the longitude of Venice to be 35. degrees, 30′·S and the latitude thereof to be 44. degrees, 45′· Now by subtracting the lesser longitude out of the greater, I find the difference of longitude to be 15. degrees, 36′·S and by subtracting the lesser latitude out of the greater, I found the difference of latitude to be 6. degrees, 47′· Than knowing the latitude of London to be 51. degrees, I resort to the table of miles appointed for every degree of latitude before set down, and theer I found that to 51. degrees of latitude do answer 37. miles and certain seconds, which being of small moment are not wont to be reckoned. Than in multiplying the difference of the longitudes which is 15. degrees 36′· by 37. miles I found the product of the 15. degrees so multiplied to be 555. and because there be 36′· annexed to the foresaid 15. degrees, I seek by the rule of proportion to know how many miles that fraction containeth, in saying thus, if 60. require 37. what shall 36. require? and I found 22. miles, which being added to 555. maketh 577. then by seeking in the foresaid table how many miles be answerable to the latitude of Venice, which is 44. degrees, I found the number of miles to be 43. by which number I multiply once again the difference of longitude, which is 15. degrees. 36′· the product whereof, together with the fraction annexed thereunto being converted into miles by the rule of proportion, as before, doth amount to 670. which sum being added to the former converted longitude, which is 577. maketh in all 1247. the half whereof is 623. which half number I reserve by itself, that done I multiply the difference of the latitude, which is 6. degrees 47′· by 60. miles, in saying 6. times 60. maketh 360. whereunto I add for the 47′·S annexed 47. miles, & it maketh in all 407. which sum I reserve also by itself. Than I multiply the first reserved number into itself, the product whereof is 388129. That done, I multiply the second reserved number also into itself, the product whereof is 165649. which two last productes being added together, do make in all 553778. whereof the square root being taken, is 744. miles, which is the true distance of Venice by a right line from London. And to the intent that the order and working herein may more plainly appear unto you, I have set down all the particular numbers of the same here by themselves, as it were in a Table.  
 longitude. latitude London. 19 degr. 54′· 51. degr. 32′· Venice. 35. degr. 30′· 44. degr. 45′· The difference of their longitudes and latitudes. 15. degr. 36′· 6. degr. 47′·  
The difference of the longitudes converted into miles: for London is 577. For Venice. 670. The sum of the two converted longitudes added together, is 1247. The half whereof, which is the first reserved number, is, 623 The second reserved number, which is the difference of the latitudes, converted into miles, is 407 The sum of the first reserved number multiplied into itself, is 388129 The sum of the second reserved number multiplied into itself, is 165649 The sum or both added together is 553778 The square root whereof, which is the sum of the miles, is 744   

How to find out the distance betwixt two places, differing both in longitude and latitude by help only of a demi-circle divided into 180. degrees without any Arithmetic. Chap. 14.  
But now because the way before taught to find out such distance by the Arithmetical way may seem perhaps to some folks very busy and tedious, I have thought good therefore to set down this other way which was sent me not long since from my loving friend M. Wright of Cayes college in Cambridge who is well learned in the Mathematicals, & is so apt thereunto by nature, as he is like inogh to attain to such perfect knowledge therein as he may be able thereby hereafter greatly to perfect his country, if for want of sufficient exhibition he be not forced to leave so noble a study, wherefore I wish with all my heart that all Gentlemen of ability were minded to show their liberality towards him in that behalf. But to return to my matter, I say that the way to found out the foresaid distance is this here following: first having drawn a demi-circle upon a right Diameter (the larger that the demi-circle is the better,) and divided the same into 180. degrees, like unto this hereafter described, and marked with the letters a b c d. whereof d. is the centre and a. c. the Diameter. Than learn first by some Table to know the longitude the latitude of both places, and the difference of their longitudes, as you did before in seeking to know by the Arithmetical way the distance betwixt London and Venice, the difference of whose two longitudes is 15. degrees, and 36′· as you may see in the former Table: for in working by this way, you have chief to seek out in the circumference of the demicircle but three things, that is, first, the difference of the two longitudes, secondly, the lesser latitude, and last of all the greatest latitude. Knowing therefore the difference of the said two places in longitude to be 15. degrees, 36′·S seek out the same in the demicircle, beginning to count at A. and so proceed towards B. And at the end of those degrees and minutes set down a prick marked with the letter e. unto which prick draw a right line by your ruler from D. the centre of the demicircle. That done, seek out the lesser latitude, which is 44. degrees, and 45′· in the foresaid demi-circle, beginning to accounted the same from the prick e. and so proceed towards the letter B. and at the end of the said lesser latitude, set down another prick marked with the letter g. from which prick or point draw a perpendicular line which by help of your squire or compasses may fall with right angles upon the former right line drawn from D. to e. and where it falleth, there set down a prick marked with the letter h. That done, seek out the greater latitude, which is 51. degrees, and 32′· in the foresaid demicircle, beginning to accounted the same from A. towards B. and at the end of that latitude set down another prick marked with the letter I. from whence draw another perpendicular line that may fall by help of your squire or compasses with right angles upon the Diameter A. C. and there make a prick marked with the letter K. That done, take with your compass the distance that is betwixt k. and h. which distance you must set down upon your said Diameter A. C. setting the one foot of your compass upon k. and the other towards the centre D. and there make a prick marked with the letter L. Than take with your Compass the length of the shorter perpendicular line g. h. and apply that wideness upon the longer perpendicular line I. K. setting the one foot of your Compass at I. which is the end of the greater latitude, and extend the other foot towards K. and there make a prick marked with the letter M. That done, take the distance betwixt L. and M. with your compass, and apply the same to the demicircle, setting the one foot of your Compass in A. and the other towards B. and there make a prick marked with the letter N. And the number of degrees contained betwixt A. & N. will show the true distance of the two places, which you shall found to be 12. degrees and almost 24′· Now by allowing for every degree 60. miles, and for every minute a mile, the sum of miles will agree with the former distance found out by the Arithmetical way which was 744. miles. And thus you have to deal to know the distance of any other two places whatsoever, differing both in longitude and latitude. But you have to note by the way, that if the difference of the longitudes doth exceed the number of 180. then you must subtract that exceeding difference out of 360. and the remainder shall be the difference of the longitudes, and then work in all points as is before taught. And this way is as Geometrical, as that which Appian setteth down in his book of Geography, to be done by the help of the terrestrial Globe, the order whereof here followeth.  
 Longitude. latitude. London. 19 deg. 54 '. 52. deg. 32 '. Venice. 15. deg. 30 '. 44. deg. 45 '. The difference of their longitudes and latitudes is 15. deg. 36 '. 6. deg. 47 '.    

How to found out the distance of the places by the Geometrical way. Chap. 15.  
HOw is that done?  
Most readily and easily by help of a terrestrial globe in this manner following. First take the distance of the 2. places by extending your compass upon the globe, from the one place to the other, which if you would know how many miles it comprehendeth, apply the same distance so taken unto the Equinoctial line, setting the first foot of your compass upon the first Meridian in that point, whereas it cutteth the Equinoctial, then see how many degrees of the Equinoctial are comprehended betwixt the two feet of your Compass, and multiply those degrees by 60. & the product thereof shall show you how many Italian miles such distance is in length. But if either of the places or both, be wanting and not expressed in the Globe, than you must learn by the tables of Ptolomey or of some others, as of Appian, Gemma Frisius, Orontius or such like, the longitude and latitude of the said places, that done, having sought out the longitude of the first place in the Equinoctial, turn the globe about with your hand until you have brought the longitude right under the brazen Meridian, which being stayed there, seek out in the said Meridian the latitude of the said place, and there set a mark upon the globe, for there the place should stand, and do in like manner to find out the second place: Than by extending your Compass from the one mark to the other, you shall have the true distance, which distance if you apply to the Equinoctial like as before is taught, the degrees thereof being multiplied by 60. will show you how many miles those two places are distant one from another.  
May not the distance of places be found out aswell by an universal Map as by the globe terrestrial?  
Yes indeed and more readily by reason that for the most part every Map hath his proper scale, so as you need to do no more but to take the distance of the 2. places with your Compass, and to apply the same to the scale showing the miles or leagues.  
What if the Map have no scale?  
Than you must seek out the distance by such means as I do show in my Treatise of the use of universal Maps, and also in my description of Planctius his Map.  
I pray you in the mean time proceed in showing me the third way of finding out of the distance of places which you said was per tabulas Sinuum.  
The order of finding out the distance of 2. places differing both in longitude and latitude per tabu●as Sinuum is plainly set down before in the end of my Arithm●ticke, where as I do make a plain description of the said Tables, and do show the use thereof aswell by this as by divers other examples, wherefore I wish you to resort to that treatise and you shall have your desire. For having for this time sufficiently spoken of the longitude, latitude and distance of places, and how the same is to be found out, I think it meet now to treat of the 5. Zones, of Climes and Parallels, whereinto the amient Cosmographers thought good to divide the earth, to the intent that every part thereof night be the better known how it is situated either Northward or Southward, whether it be hot or cold, or betwixt both, and of what length the day and night is in every place, and what manner of shadow the Sun yieldeth every where, and such like accidents, and first of the 5. Zones.   

Of the 5. Zones. Chap. 16.  
THe most ancient Cosmographers considering how the Sun by his obliqne, and bariable course did warm with his beams one part of the earth more than another, gathered thereby that the earth had three temperatures that is to say, extreme hot, extreme cold, and a mean temperature, that is, neither too hot, nor too cold: And therefore to show us unto which of these temperatures, any part of the earth was subject, they divided the earth into 5. Zones, answerable to the 5. Zones of the firmament, by help of the Equinoctial and the 4. lesser circles before described in the first part of this treatise, the 20. Chapter, and are there set forth in figure which figure I thought good to set down again in this place, to the intent you might the better remember what was said there touching the Zones.  
But other Cosmographers coming afterward, not satisfied with the 5. Zones, because they show nothing but the situation and the three temperatures of the earth, did divide the earth into certain Climes and Paraliels, to found out thereby ● the length of the day and night in every place, and the true latitude thereof as Ptolomey and many others after him have done, making such divisions as we shall speak of hereafter. But truly I think with Mercator, that the best and most exact way of dividing the earth to serve all purposes is to be made by degrees and minutes, wherein is less error than in Climes and Parallels, neither can Climes or Parallels be so well described when you draw night to any of the Poles, for that the spaces as well betwixt the Parallels as betwixt the Meridian's do grow continually straighter & straighter, as you may see in the figure of Parallels hereafter following.  
I remember that in the 20. chapter whereas you described the 5. Zones, you said that the ancient men did greatly err in affirming 3. of the Zones to be unhabitable, that is, the two cold Zones and the hot Zone, I pray you therefore show me here the cause why they erred?  
They erred for lack of experience, because they had never traveled into those regions, but in these latter days men of divers nations, specially the Spanish, English, French, & Flemish have traveled very far, some towards the North pole, and some towards the South pole, and also through the burnt Zone, for those that sail from the North parres towards the South pole, or from the South parts towards the North pole, must needs pass in their voyage through the burnt Zone: and these men do affirm that they have found all the 3. Zones, that is to say, the two cold and hot Zones to be well inhabited. And our late Cosmographers do not let to tender cause why they should be habitable, for (say they) though the cold be very extreme in those regions that lie next unto the Poles, yet the Sun appearing and giving shine unto them both day and night, doth greatly qualify and moderate the extreme cold of those regions. But truly, in mine opinion, they have small comfort of the Sun, sith it striketh almost round about their feet, without yielding any warm reflex from above, and especially to those that do inhabit nigh to either of the Poles. Again, they say that the burnt Zone is habitable, by reason that the night to them is continually as long as the day, the coolness whereof doth greatly refresh the extreme heat of the day. But now let us return to our purpose, and speak somewhat of the Climes and Parallels, & because every Clime consists of 2. Parallels, I think it best to speak first of Parallels.   

Of Parallels. Chap. 17.  
WHat be Parallels.  
Parallels be lines either right or circular always equally distant one from another so as they can never meet. And of Parallels that are to be considered in the Sphere, some make 3. kinds, according to their threefold signification, for some are called the Parallels of the sun, who in departing from the Equinoctial towards any of the Poles, maketh every day throughout the year one Parallel, so as in going from the Equinoctial to the Tropic of Cancer, he maketh 182 Parallels. And as many again in going from the Equinoctial to the Tropic of Capricorn. The second kind of Parallels are called the Parallels of Latitude, And the third the Parallels of the longest day, which two last are in effect both one, for the further that any Parallel is situated from the Equinoctial towards either of the Poles, the more Latitude it hath, & so by consequent maketh the day longer to those that devil under that Parallel, of which Parallels the ancient Cosmographers do make in all but 21. proceeding proportionally either towards the North pole or South pole, as you may see by this figure here following the middle line or Circle whereof is the Equinoctial.  
And every Parallel proceeding from the said Equinoctial, either Northward or Southward, doth lengthen the day by one quarter of an hour in such proportion as this Table here following showeth, appointing to the first Parallel, and next unto the Equinoctial 4. degr. & 15′· and to the second parallel 8. degr. 30′·S and so forth until you come to the 21. Parallel passing through the Island, which Parallel is distant from the Equinoctial 63. degr. 16′·S further than which Parallel Northward, the Tables of Ptolemy do not extend, and Southward they extend no further then to that Parallel which hath less 20. degrees of Latitude.  

The Table of Parallels showing how many degrees and minutes every one is distant from the Equinoctial, made according to the rule of Ptolemy.  Parallels D M the first 4 15 the second 8 30 the third 12 45 the fourth 16 35 the fifth 20 30 the sixth 24 15 seventh 27 30 the eight 30 45 the ninth 33 40 the tenth 36 24 eleventh 39 0 twelfth 41 20 thirteenth 43 15 fourteenth 45 24 fifteenth 48 40 sixteenth 51 50 seventeenth 54 30 eighteenth 56 30 ninetenth 58 20 twentieth 61 10 the xxj. 63 16   

Of Climes both old and new. Chap. 18.  
WHat is a Clime?  
A Clime is a space of the earth comprehended betwixt two Parallels, in which space the longest day doth vary by half an hour.  
How many Climes be there?  
The ancient Cosmographers divided aswell that part of the earth which lieth betwixt the Equinoctial and the North pole, so much I say as they thought to be habitable, as also the habitable part which lieth betwixt the Equinoctial and the South pole, each of them into 7. Climes, to every of which Northern Climes they gave a several proper name, according to the name of the place through which the midst of the said Clime did pass, for they called the first Clime Diameroes', Dia is a Greek preposition signifying in English By or Through, and Meroes' is a City of Egypt situated in a certain Isle enclosed with the flood Nilus, which Isle hath also the like name. The second Clime is called Dia Syenes, which is also a City in Egypt situated right under the Tropic of Cancer. The third Clime is called Dia Alexandrias, which is another City of Egypt situated upon the West mouth of Nilus, falling into the sea of Egypt. The fourth Clime is called Dia Rhodou, Rhodos is the chiefest City of an isle called Rhodos, standing in the sea called Mare Carpathium, washing the South-west end of Anatolia, sometime called Asia minor, which Island, together with the City Rhodos, Suliman the great Turk won not many years since from the Christians. The people of this isle in S. Paul's time were called Colossians, to whom he wrote one Epistle, and they were so called of a great brazen Image called in Latin Colossus, containing in Altitude 105. feet, which was dedicated to the Sun, or as some say to jupiter. The first Clime is called Dia Rhomes, that is to say by Rome that famous City of Italy, and sometime head of all the world. The sixth is called of the old men Dia Boristhenes, notwithstanding the modern writers think that it may be more rightly called Dia Pontou, Pontus is both a Sea and Country lying Eastward right against Constantinople. The seventh Clime is called of the old men Dia Ripheos, but of the modern writers Dia Boristhenes which is a great flood of Scythia in the South part of Sarmatia, which falleth into the sea called Mare Euxinum. To these seven Climes Appian addeth two others, so as in all he setteth down nine Climes, making the eight Clime to pass through Ripheos which are mountains environing Sarmatia on the North side, and the ninth through Denmark.  
What names did they give to the Southern Climes?  
The self same that the Northern Climes have, saving that they put before every such name this Greek word Anti, which is as much to say as Contrary or Right against, as Anti Meroes', Anti Syenes, etc. as you may easily perceive by this figure of Climes here following, notwithstanding none of those Southern Climes were known to Ptolemy more than Anti-meroes, and hardly all that, the furthest part whereof hath not twenty degrees of Latitude.  
And note that every such Clime is divided into 3. parts, that is the beginning, the midst, & the end. And if you would know what degrees and minutes of Latitude every such part hath, that is to say, how many degrees every such part is distant from the Equinoctial, then consider well this Table following, which both briefly and plainly showeth the same.  
 The beginning. The midst. The end. Degrees and Minutes of hours. D M D M D M The first Clime. 12 45 16 35 20 30 The second Clime. 20 30 24 15 27 30 The third Clime. 27 30 30 45 33 40 The fourth Clime. 33 40 36 24 39 C The fift Clime. 39 0 41 20 43 30 The sixth Clime. 43 30 45 24 47 15 The seventh Clime. 47 15 48 40 50 30 The eight Clime. 50 30 51 50 53 10 The ninth Clime. 53 10 55 30 56 30  
May not the North and South part of the world each of them be divided into more than 9 Climes?  
Yes indeed as our latter writers affirm, for betwixt the Equinoctial and the 66. degree and 30′· of Latitude in which the longest day containeth full 24. hours without having any night, they make 48. Parallels, and thereby 24. Climes, for every Clime containeth two Parallels, and every Parallel maketh the day to increase by one quarter of an hour as hath been said before. But from thence forth, though you may continued the Parallels almost to the very pole, yet you can make no more but 24. Climes by reason that the spaces of the Parallels towards either of the Poles, do grow more narrow every one than other, so as from 66. degrees and 30. minutes of Latitude under which the Circle Arctique passeth, the longest day is not to be counted any more by hours, but by whole days, weeks, and months, in so much as they which devil right under the North pole, whose Zenith is the pole itself, have six months day whilst the Sun abideth in the six Northern signs, and 6. months night, the Sun being in the six Southern signs. And contrariwise they that devil right under the South pole have 6. months' day, the Sun being in the 6. Southern signs, and 6. months' night whilst the Sun remaineth in the 6. Northern signs. But because I think it a more ready way to accounted the length of the day by the degrees of Latitude, then by Climes or Parallels, I thought good here to set down Orontius his Table made for that purpose, which from the 67. degree of Latitude to the North pole agreeth in all points with the Table of johannes de Sacro Busto, in which Table from the said 67. degree is not only set down the longest day, but also what portion of the zodiac always appeareth above the Horizon, which portion when it containeth one whole sign, than the day is one month long, and the night as much, if two signs, than the day is two months' long and the night as much, and so forth successively until you come to 6 signs, which is the one half of the zodiac, making both day and night each of them six months long, as I have said before, and as you may plainly see with your eye by placing the Sphere on his Horizon at every such Latitude.  

A Table showing the longest day in every degree of Laditude, proceeding orderly from the Equinoctial to the North pole, by whole degrees without minutes from one degree to 90.  
Degrees of latitude. The longest day. D M H M S 1 0 12 3 28 2 0 12 6 56 3 0 12 10 24 4 0 12 14 0 5  12 17 28 6  12 20 56 7  12 24 46 8  12 28 0 9  12 31 36 10  12 35 12 11  12 38 48 12  12 42 44 13  12 46 8 14  12 49 44 15  12 53 28 16  12 57 20 17  13 1 4 18  13 4 46 19  13 8 56 20  13 12 46 21  13 16 48 22  13 21 4 23  13 25 4 24  13 29 20 25  13 33 35 26  13 38 0 27  13 42 24 28  13 46 16 29  13 51 30 30  13 56 16 31  14 1 12 32  14 6 8 33  14 11 12 34 0 14 16 24 35 0 14 21 52 36 0 14 27 20 37 0 14 33 4 38  14 37 36 39  14 44 56 40  14 51 12 41  14 57 44 42  15 4 24 43  15 11 20 44  15 18 40 45  15 26 8 46  15 34 8 47  15 42 24 48  15 51 4 49  16 0 8 50  16 9 44 51  16 19 52 52  16 30 32 53  16 41 52 54  16 54 8 55  17 7 4 56  17 21 4 57  17 36 16 58  17 52 48 59  18 10 48 60  18 30 56 61  18 53 20 62  19 18 24 63  19 48 40 64  20 24 24 65  21 10 32 66  22 20 40  
Degrees of latitude The longest day The arch of the zodiac always appearing above the Horizon. D M Days H M De. M 67 0 24 1 40 22 50 68 0 42 1 16 40 0 69 0 54 16 25 52 0 70 0 64 13 46 61 26 71  74 0 0 70 26 72  82 6 39 78 22 73  89 4 58 84 56 74  96 17 0 92 12 75  104 1 4 96 20 76  110 7 27 105 16 77  116 14 22 111 20 78  122 17 6 117 6 79  127 9 55 122 46 80  134 4 58 128 22 81  139 31 36 133 50 82  145 6 43 139 6 83  151 2 6 144 22 84  156 3 3 149 36 85  11 5 24 154 42 86  166 11 23 159 50 87  171 21 47 164 52 88  176 5 29 169 58 89  181 21 58 174 58 90  187 6 39 180 0  
What things else do the Cosmographers teach to be considered in Climes and Parallels?  
diverse things, as first how many Italian miles every clime hath in breadth and length, also what seasons of the year, and what shadows the sun yieldeth to those that dwell under divers Climes and Parallels.  
What breadth and length do the ancient writers appoint to every one of the seven climes.  
They appoint such breadth and length as this Table following showeth, in which also is set down the degrees and minutes of Latitude through which the middle Parallel for every Clime passeth, and also the number of miles answerable to one degree of every such Latitude.  
Climes. miles in breadth. miles in length. The degrees and minutes of Latitude through which the middle parallel of every clime passeth. The number of miles answerable to one degree of every such Latitude.    De. M.  1 465 20555 16 37 57 2 420 19453 24 15 54 3 370 18398 30 45 51 4 350 17299 36 24 48 5 270 16215 41 20 45 6 225 15136 45 24 42 7 195 14426 48 40 40  
How the breadth & length of every Clime is to be known.  
The breadth is to be known by multiplying the degrees of difference contained betwixt the beginning & end of every Clime by 60. miles: And if the degrees of difference have any minutes annexed thereunto, than you must add so many miles as there be minutes, to the product of the former multiplication. Now to have the length of every Clime, you must seek to know by the Table of miles, how many miles be answerable to one degree of that Parallel of Latitude which passeth through the midst of the Clime, and by that number of miles multiply 360. and the product thereof shall be then length. But if the degrees of that Latitude have any minutes annexed thereunto, than you must find out the miles of those minutes by the rule of proportion in saying thus: if 60. do require so many miles as you found in the Table of miles, what shall so many minutes require, and add the quotient thereof to the product of the former multiplication, so shall you have the true length of the Clime, all which things are observed in the foresaid Table, which Table by observing like order, you may extend (if you will) to the number of 24. Climes, set down by Orontius and by diverse other modern writers.    

Of the diverse seasons and shadows incident to diverse Climes and Parallels, and first what seasons and shadows they have that devil right under the Equinoctial. Chap. 19  
THose that dwell right under the first Clime, and specially right under the Equinoctial, which people at one instant may see both the Poles, have two summers and two winters, for the sun having to pass right over their heads twice in the year, which is when he is in Aries, and again in Libra, than they must needs have two summers, because the sun at both those times is nighest unto them, but when he is in Cancer or in Capricorn, than he is furthest from them, and thereby maketh unto them two Winters. But yet neither of them so cold as our winter is, whereby it appeareth that our two times of Spring and Autumn are to them 2. summers, & our two times of summer and winter are to them two winters.  
What shadows have those inhabitants?  
They have five sundry shadows, for when the Sun is in either of the Equinoxes, they cast their shadow in the morning when the sun riseth towards the West. And at night when he goeth down towards the East, and at noon day they have no shadow at all, but a perpendicular shadow, which stréeketh right down from head to foot, because the sun being then in the Equinoctial, must needs at that time of the day be right over their heads, as you may plainly see in every material Sphere, having a foot with a firm Horizon, and dwely placed to show a right Sphere, but when the sun is in any of the Southern signs, than the foresaid inhabitants do cast their shadow towards the North, And when he is in the Northern signs, they cast their shadow towards the South, and because they devil in a right Sphere, their days and nights be always equal.   

Of the seasons and shadows which they have that devil betwixt the equinoctial and the Tropic of Cancer. Chap. 20.  
THese have also two summers and two winters, and 5. shadows like unto the others because the sun passeth twice in the year right over their heads, first in his declining from the Equinoctial towards Cancer, and again in his returning from Cancer towards the Equinoctial, under which Clime Arabia foelix is said to be situated. For Lucan writeth that the Arabians coming to Rome to aid Pompey, marveled much to see that the trees did never cast their shadows on the left hand, because in their country their shadow is sometime on the right hand, and sometime on the left, sometime perpendicular, sometime oriental, and sometime occidental, but in Rome and in all other places beyond the Tropic of Cancer the shadow always at noontide tendeth Northward, the words of Lucan be thus, Ignotum vobis Arabes venistis in orbem umbras mirati nemorum non ire sinistras.  
How is Lucan to be understood here in using this speech on the left hand?  
As all other Poets are, for they having regard to the West, do always make the North part their right hand, and the South part the left. But the Astronomers having regard to the South, do make the West their right hand, and the East their left hand: again the Geographers contrariwise having regard to the North, do make the East their right and the West their left hand, whereby you may see that the right hand and left hand may be taken three manner of ways, that is according to the manner of the Poets, of the Astronomers, and of the Geographers.  
Sigh these inhabitants have seasons and shadows like to those that dwell right under the Equinoctial, wherein then do they differ?  
They differ in that their days & nights be not always equal, by reason that their Horizon declineth from the poles of the world, and by cutting the Parallels of the sun with obliqne Angles it divideth those Parallels into unequal parts, neither can their two summers be so extremely hot as the others, because the sun for the most part is further from them.   

Of the seasons and shadows which they have that devil right under the Tropic of Cancer. Chap. 21.  
THey have but one summer and one winter, by reason that the sun never passeth right over their heads but once in the year, and that is when he entereth into the first degree of Cancer: In which time when the sun riseth, the shadow tendeth towards the West, and at noon the shadow is perpendicular inclining on neither side, but falling right down. And at night when the sun goeth down, the shadow tendeth towards the East: and at all other times of the year the shadow at noontide falleth always Northward, and as touching their days and nights, they shorten and lengthen according as the sun either approacheth towards Cancer, or retireth towards Capricorn.   

Of the seasons and shadows which they have that devil betwixt the Tropic of Cancer and the circle Arctique. Chap. 22.  
THey have also one summer and one winter as we have here in England, for the sun never passeth right over their heads, by reason that so soon as he hath made his last and highest Parallel in Cancer, he returneth again Southward, and therefore their shadow is never at noontide perpendicular but shooteth Northward, and the days and nights do lengthen and shorten according as the sun maketh his course either through the Northern or Southern signs. For whilst he passeth through the Northern signs the days are longer than the nights, and in passing through the Southern signs he maketh the nights longer than the days.   

Of the seasons and shadows which they have that devil right under the Circle Arctique, and how long their day is. Chap. 23.  
THey have but one Winter and one Summer, and their shadow always tendeth sidelong & Northward. And because their Zenith being in the circle Arctique, is at all times of the year all one with the Pole of the zodiac: the Ecliptic line therefore must needs be all one with their Horizon, whereby the one half of the zodiac in a very moment doth rise above the Horizon, and the other half in the same instant goeth down, and as the whole Tropic of Cancer appeareth always above the Horizon: So the whole Tropic of Capricorn is always hidden under the Horizon, so as when the sun entereth into the first degree of Cancer, their day is 24. hours long and their night but a moment, so contrariwise when the sun entereth into the first degree of Capricorn, their night is 24. hours long, and their day but a moment, as you may plainly see by placing the Sphere at the 66. degree and 30′ of Latitude.   

What seasons, shadows and length of day they have that devil betwixt the circle Arctique and the pole Arctique. Chap. 24.  
THese have like seasons and shadows as those that devil under the circle Arctique, saving that their Winter is colder and longer, because the nigher they approach to the Pole, the further are they from the sun, and also have both longer days and nights, for the more that the Pole is elevated above their Horizon, the greater portion of the zodiac doth always appear above the same, which portion if it contain one whole sign, than their day is a month long, and their night as much, if two whole signs, than their day is two months long, and their night as much, and so forth as hath been said before.   

Of those that devil right under the Pole. Chap. 25.  
WHat seasons, shadows, and length of day have they that devil right under the Pole, if there be any such people?  
Truly you do well to doubt thereof, for in mine opinion human nature is not able to suffer the extreme cold that by reason must needs be in those parts, neither do I think that ever any man either Christian or Heathen did ever sail so far as to discover any land there. Notwithstanding if there by any such people their season is always so extreme cold as no part thereof is worthy to be called a summer, but rather a continual winter, and as for their shadow sigh the sun when he is at the nighest doth never mount above their Horizon more than 23. degrees 30′ at the most, their shadow must needs go round about them nigher, for the most part to their feet then to their heads, for the pole is their Zenith, and the Equinoctial their Horizon, whereby 6. signs which is the one half of the zodiac, doth always appear above their Horizon, and the other half is always hidden under their Horizon, and thereby they have 6. months day & 6. months night, day I say whilst the sun is in the 6. Northern signs, & night, whilst the sun is in the 6. Southern signs, and yet the night can not be so dark there as elsewhere, by reason that the sun is never distant from their Horizon above 23. degrees 30′· which is only when he entereth into the first degree of Capricorn. So likewise the day with them can never be so clear as elsewhere, by reason that the sun mounting no higher above their Horizon then 23. degr. 30′·S which only is when he entereth into the first degree of Cancer, hath no power to dissolve their gross, thick, cloudy and misty air, yet have they some pre-eminence in that they may (if their cloudy air be not the let) always see all the fixed stars that are placed in the sky betwixt the Equinoctial and the Pole, because they never go down but are always remaining above their Horizon: whereas in all other parts of the world, the said stars can not be seen all at once, for that they both rise and set, more or less in number according as the Zenith of the inhabitants of every place is more or less distant from the Equinoctial.   

By what names certain inhabitants of the earth are called, as well according to the diversity or likeness of shadows as of situation. Chap. 26.  
YOu shall understand that according to the diversity or likeness of shadows, the ancient Cosmographers have given to the inhabitants certain Greek names, whereof some are called Aphiscij, some Heteroscij, & some Periscij.  
Show what these names do dignify.  
Amphiscij be those that cast their shadows both ways, that is sometime towards the North, and sometime towards the South, as those that inhabit the burnt Zone.  
Heteroscij be those that cast their shadow only one way, as those that devil in either of the temperate Zones, for if they devil in the North temperate Zone, they do cast their shadow always at noontide towards the North. And if they devil in the South temperate Zone they cast their shadow at noontide towards the South.  
Periscij are those that cast their shadow round about them as those that devil in either of the cold Zones, to whom the Pole is their Zenith: again they give them certain names according to the diversity or likeness of their situation and of the seasons incident to those places, whereof some are called Antoeci, some Perioeci, and some Antipodes siue Antichtones.  
What do these names signify?  
Antoeci be two sundry Nations, the one dwelling towards the North pole, and the other towards the South pole, having one self Meridian and one self Latitude, that is to say be of like distance from the Equinoctial, the one Southward and the other Northward, as the letters a. b. in the figure following do show.  
Perioeci be those that devil in one self Parallel, how distant soever they be East and West as the letters b. c. in the said figure do show.  
Antipodes be those that devil feet, so as a right line being drawn from the one to the other, passeth through the Centre of the world as you may see by the Letters a. c. in the figure following.  
And the first of these called Antoeci have contrary seasons of the year, for when it is summer to the one, it is winter to the other: Again Perioeci, though they have like seasons of the year, yet because they are so far distant in length one from an other, it is therefore midday to the one when it is midnight to the other. But Antipodes be contrary in seasons and in all other things, having nothing common, more than that they have one self Horizon.   

By what names certain parts of the earth are called by reason of their diverse shapes. Chap. 27.  
NOw besides the foresaid names attributed to the inhabitants of the earth for such respect as is above said, the Cosmographers do give also to diverse parts of the earth according to the divers shapes thereof divers names also: for if any part of the earth be environed round with water either salt or fresh, it is called Insula, that is an Island, as England, Ireland, & such like, but if the water go round about it saving in one part, then it is called peninsula, that is to say, almost an Island, as Denmark, Italy, Morea, and such like, and if it be a narrow strait enclosed with the sea on both sides, than it is called Isthmus as the narrow strait of Corinth lying betwixt Boetia and Achaia in Gréece which divers Emperors of Rome have in vain attempted to cut, to th'intent to make there through a navigable passage: finally, when it is neither insula, peninsula, nor Isthmus, than it is called Continens, that is to say firm land, as Saxony, Boemia, Suevia, and such like, but these be special continents, for the Cosmographers of these days do make but threé general continents, that is first so much as was known to Ptolemy, and to the rest of the ancient writers, secondly the West Indies, lately found out, and thirdly the South part of the world not yet wholly discovered: Again the ancient men divided that portion of the world, which was known in their days into 3. parts, that is, Europe, Asia, and afric, whereunto the modern writers have added a fourth part called America, containing the West Indies. Now if you would know what Kingdoms, Regions, Cities, Towns, Seas with their Havens, Ports, Bays, and Capes, Isles, Floods, Marshes, and Mountains, are contained in every one of these four parts, then peruse often the universal Maps, and Terrestrial Globes, as well of the modern as of the ancient Writers, and also the Tables of Ptolemy and of Ortelius, which I wish that they had been made in such form as the Tables of Ptolemy are: for having the North always set in the front, it should be the readier to compare the shape or situation of any place or Region to the universal Map, and by knowing the Longitude and Latitude of any place, it should be the easier to find the same, as well in the special Table as in the universal Map or Globe. The use of which universal Maps, I have already written in a several Treatise by itself printed not long since. But now for so much as the knowledge of the winds hath been always thought a thing meet to be treated of by him that writeth of Cosmography, and specially of the Sphere, I will here speak some what of them, neither do I mind to make any great discourse thereof, as natural Philosophers are wont to do, but only to define what the wind is, and to show into how many parts the same is divided, as well by the ancient as modern writers, and by what names they are called, and somewhat of the qualities thereof.   

Of the wind, what it is, what motion it hath, and of the divers names and divisions thereof. Chap. 29.  
FIrst than you have to understand that Aristotle and the rest of his sect, do define the wind to be an exhalation hot and dry, engendered in the bowels of the earth, and being gotten out, is carried sidelong upon the face of the earth.  
Why is not his motion right up & down, aswell as sidelong?  
Because that whilst by his heat he striveth to mount up, and to pass through all the three Regions of the air, the middle Region by his extreme cold, doth always beat him back, so as by such strife & by the meeting of other exhalations rising out of the earth, his motion is forced to be rather round then right.  
What is the cause why he bloweth more sharply at one time then another, and in one place more than another, and sometime not at all.  
As the fumes that rise of new exhalations, & out of floods and waters, may increase his force, so lack of heat and fumes doth diminish the same. Again the roundness of the earth is cause that he bloweth sometime more in one place then in another: also mountains, hills, and great woods may hinder his force in some place of the earth whereas upon the plain or upon the broad sea he bloweth most sharply, and as for his not blowing at all, it may chance diverse ways, as either for lack of sufficient heat to open the pores of the earth to let himself out, or for that some extreme frost and cold doth close the pores of the earth so strait as he cannot get forth, or for that the sun with his extreme heat consumeth the fumes and vapours that should maintain him. But leaving to answer any more to these questions, I will show you how many winds were observed by sailors in old time, and how many are observed at this present day, and how they were named. True it is that all Nations do agree in placing the four principal winds, according to the four quarters of the Horizon or Angles of the world, that is to say, East, West, North and South: but in the subdivisions of the said four quarters they differ, for some divide every quarter of the Horizon into two, making only eight winds, and some into three, and there by do make 12. winds which the ancient Greeks and the Romans' did chiefly observe, the names of the 8. winds are commonly expressed in the Italian tongue thus: Tramontano, North, Mezzodi, South, Lenante, East, Ponente, West, Griego, Northeast, Garbino, Southwest, Maistro, Northwest, Syroccho, Southeast, which names are often used by Christopher Columbus, Albertus Vesputius, and others that sailed first into the East and west Indieses, and if you will know the names of the 12. winds used by the ancient Greeks' and Romans', then behold this figure here following, wherein you shall found them set down in English, Greek, and Latin, that is to say, the English names without the Circle, and the Greek and Latin names within the Circle, the Greek upon the right side of every line pointing the wind, and the Latin name upon the left side of every such line.   

Of the nature and qualities of the foresaid 12. winds. Chap. 29.  
THe North wind called Septentrio or Aparctias is extremely cold and dry, prohibiting rain, it preserveth health by cleansing the Air of all pestiferous infections, but it causeth dry colds, and hurteth the fruits and flowers of the earth.  
2 The North-east and by North, called Aquilo or Boreas, is also cold and dry without rain, it hurteth the flowers and fruits of the earth, and specially the Vines when they bud.  
3 The North-east and by East, called Helispontus or Caecias, is hot drying up all things.  
4 The East wind called Subsolanus, is hot and dry, temperate, sweet, pure, subtle, and healthful, and specially in the morning when the sun riseth, by whom he is made more pure and subtle, causing no infection to man's body.  
5 The East Southeast, called Eurus or Vulturnus, is also hot and dry, he bloweth loud, and therefore is called of Lucretius, Altitonans vulturnus.  
6 The Southeast and by South called Euroauster or Euronotus, is hot and moist, and breedeth clouds and sickness.  
7 The South wind called Austere or Notus, is hot and moist breeding thick clouds, great rains, and pestiferous air.  
8 The South-west and by South, called Austro aphricus, is temperately hot, & yet breedeth sickens & rain as some writ.  
9 The South-west and by West named Aphricus or Libs, is cold and moist causing rain.  
10 The West wind called Favonius or Zephyrus is temperately hot and moist, and wholesome in the Evening, it dissolveth frost, ice, and snow, and maketh flowers and grass to spring, and some writ that it causeth Thunder.  
11 The West Northwest, called Corus or Syrus, is cold and moist without any great rigour.  
12 The Northwest and by North called Syrus or Trachias is cold and dry of earthly nature, breeding snow and winds.   

Of the modern division of the winds. Chap. 30.  
But the Mariners of these our latter days, to be the better assured of their routs and courses on the sea, do divide every quarter of the Horizon into 8. several winds, so as they make in all 32. winds, which of the Spaniards are called Rombes, which win des together with their names, you may see plainly set forth in this figure following representing the Mariner's Compass.  
A figure of the 32. winds representing the Mariner's Compass.  
And note that eight of these winds are called principal winds, that is North, South, East, West, Northeast, Southwest, Northwest, and Southeast, and all the rest are called Collateral winds, but the first four are chiefest, from whence the names of all the rest are derived, neither do the learned Pilots in their Tables, call the first four winds, Rombes, but will say the first, second or third Rombe from North, from East, South, or West, unto the number of seven, for so many Rombes they make in every quarter betwixt the four chief and principal winds, of which matter I shall speak hereafter in my treatise of Navigation. In the mean time I hearty pray all those that shall vouchsafe to read this my treatise of the Sphere, to take my labour therein bestowed in good part, and where any fault is, friendly to correct the same without any scorn or disdain.    

A plain description of Mercator his two Globes, that is to say, of the Terrestrial Globe and of the Celestial Globe and of either of them: Together with the most necessary uses thereof written by M. Blundevill. Whereunto is added a brief description of the two great Globes lately set forth by M. Molinaxe: and of Sir Frances Drake his first voyage into the Indieses.  He telleth the number of the stars and calleth them by their names. Psal. 147.  depiction of astrolabe 
Imprinted at London by john Windet.   

A plain description of the two globes of Mercator, that is to say, of the Terrestrial Globe, and of the Celestial Globe, and of either of them, together with the most necessary uses thereof, and first of the Terrestrial Globe, written by M. Blundevill.  
THe Terrestrial Globe is a round body covered with an universal Map containing both sea and land, which is divided by the later Cosmographers into four principal parts, that is, Europe, Africa, Asia, and America, the Longitude and Latitude of every which part is already set down in my description of universal Maps and Cards. And in this Globe are also set down certain stars, some towards either of the poles, & some nigh to the Ecliptic line, whereof we shall speak hereafter when we come to treat of the Celestial Globe. But you have to understand that the one end of this Globe is called the pole Arctique, that is to say, the North pole, and the other end the pole Antarctique, that is to say, the South pole: upon which two poles it is turned about. And this Globe is traced with certain circular lines, whereof some be greater and some lesser, those greater which pass through both the poles, are called Meridian's of this word Meridies, which is as much to say as noontide, for when the Sun toucheth any of those Circles, it is noontide to all those that devil right under that Circle, of which Meridian's though you may imagine to be half so many as there be degrees in the Equinoctial, which amounteth to the number of 180. yet there are set down in this Globe no more but twelve which do cut the Equinoctial in 24. points, making thereby 24. spaces, every space containing 15. degrees of the Equinoctial, which fifteen degrees do make one hour, for the Equinoctial is a great Circle which guirding the Globe in the very midst betwixt the two Poles, and representing the motion of the first movable, maketh his daily revolution from East to West in four and twenty hours. And this great Circle by dividing the Globe into two equal parts, causeth the same to have two Latitudes or breadthes, the one Northern and the other Southern, for that space of the half Globe which lieth betwixt the Equinoctial and the North pole, is called the North Latitude, and the other half of the Globe which lieth betwixt the Equinoctial and the South pole, is called the South Latitude, each Latitude containing every where from the Equinoctial to either of the Poles ninety degrees. Again this Circle is divided into 360. degrees containing the whole Longitude of the earth, and every degree is 60. Italian miles, which being multiplied into 360. maketh in all 21600. miles, and such Longitude is to be counted from West to East, beginning the same at the first Meridian which passeth through the Isles of Canaria, otherwise called Insulae fortunatae, crossing the Equinoctial and the Ecliptic line, in the first degree of Aries, and also in the first degree of Libra, which are the two Equinoctial points, so as the one half of the Equinoctial which goeth from West to East, containeth 180. degrees, and the other half returning again from East to West containeth also 180. degrees, which maketh in all 360. degrees, ending where the first degree of Longitude did begin: For though Mercator and others in their several Maps do in these days make the first Meridian to pass through the Isle's Azores which are five degrees more to the West. Yet Mercator in his great Terrestrial Globe dedicated first to the Lord Granuella, and afterwards to the Emperor Charles the fift, Anno Dom. 1541. placeth the first Meridian as before is said, both whose Globes my Worshipful friend Sir Thomas Knyvet of Ashwelthorpe Knight, did courteously lend me I thank him, who as he is very well learned in all the liberal Sciences himself, so is he a great favourer and furtherer of all such as delight in any learned or virtuous exercise: And by help of those Globes I wrote this Treatise. But to return to my matter, there is another great slooping and overthwart Circle, called the Ecliptic line, under which the Sun continually walketh, and this line is marked with the Characters of the twelve signs, every sign containing in length thirty such degrees as the Equinoctial hath, so as in all, the Ecliptic line containeth 360. degrees, which is the Longitude of heaven, and the first degree of the Longitude of any star beginneth at the first point of Aries, and endeth at the same point. Now of the lesser Circles there be four principal, that is, the two Tropiques, that is to say, the Tropic of Cancer towards the North pole, and the Tropic of Capricorn towards the South pole, betwixt which two Tropiques, the Sun continually goeth right under the foresaid Ecliptic line, never mounting higher than the Tropic of Cancer, nor descending lower than the Tropic of Capricorn. Of the other two lesser Circles, the one is called the Circle Arctique environing the North Pole, and the other the Circle Antarctique environing the South Pole: all which four Circles are called Parallels, that is to say, equally distant one from another, and by reason of these four Circles, the Globe is divided into five Zones, that is to say, two cold Zones, two temperate, and one extremely hot: of the cold Zones, the one lieth betwixt the Circle Antarctique, and the South pole, and the other betwixt the Circle Arctique and the North pole.  
And of the temperate Zones, the one lieth betwixt the Tropic of Cancer and the Circle Arctique, and the other lieth betwixt the Tropic of Capricorn and the Circle Antarctique. The hot Zone is that which lieth betwixt the two Tropiques, through the midst of which Zone passeth the foresaid Equinoctial line, right under which, they that devil, have no Latitude at all, and therefore their days and nights are always equal, (that is to say) twelve hours day, and twelve hours night. Also besides these four foresaid lesser Circles or Parallels, the Globe is traced with eight other Parallel Circles on each side of the Equinoctial towards either of the Poles, making on each side nine equal spaces, every space containing ten degrees of the Equinoctial, so as they do make in all 90. degrees, which is a quarter of the great Circle. Moreover there be certain Mariners Compasses divided into 32. circular lines, signifying the 32. winds, whereby Marmers do sail, and do direct their ships from port to port, which lines do diversely cross the other Circles before rehearsed. Besides these circular lines before described, there is fastened unto the two poles of the Globe, a Meridian of brass commonly called the movable Meridian within the which the Globe turneth about, & this Meridian is divided into four quarters every quarter containing 90. degrees, so as the whole circuit thereof is 360. degrees, one of which quarters towards the North pole is divided with circular lines into three several spaces, in the first and lowest whereof being next to the body of the Globe, are set down the numbers of degrees of Latitude. In the next space above that, are graven the Parallels and numbers of hours of the longest day in every Latitude, and the spaces of every hour are divided with little stréeks into four parts, signifying the four quarters of an hour: and these hours do increase till you come to 24. hours, and from thence the day incereaseth by months, from one to six, over which is graven this Latin word Menses, (that is to say) months, and the third and uppermost space containeth the Climes, beginning at one, and so proceedeth to 8. The like Climes, Parallels, and hours of the longest day, are to be accounted, also in the said Brazen meridian proceeding from the Equinoctial towards the South pole, though they be not there set down. And upon this Brazen meridian is placed at the North pole another little brazen Circle, together with his Index called the hour wheel, every half whereof containeth twelve hours, the Index of which wheel being set upon the North pole turneth about as the Globe turneth, and yet you must imagine the Pole itself to be immovable. Also there is a quarter Circle of thin brass plate, divided into 90. degrees fastened in such sort upon the brazen Meridian, as you may remove it too and fro, yea and also take it clean off if you will, which quarter Circle hath a square head of brass, signifying the Zenith or vertical point of any place, and this Circle is called the quarter of altitude, which quarter is divided into 90. degrees, proceeding from the Horizon upward to the Zenith, whereby may be described upon the Globe the 90. Almicanterathes or Circles of Altitude. And this quarter serveth for diverse purposes, as to find out thereby the Altitude of any star or point of the Ecliptic line, or of any other point in heaven, and the very edge thereof on the right hand, showeth the Azimuth or vertical Circle of any place, and in what coast and part of heaven any star is. Also how any place of the earth beareth one from another as you shall more plainly perceive hereafter by the examples of certain propositions thereto belonging. Than there is another Circle of brass plate somewhat thicker called the semicircle of position, which serveth chiefly for matters of Astronomy, as to find out the twelve houses of heaven, and is to be fastened on the Horizon, so as it may be removed and set upon which side of the Globe you william. Also there belongeth to the Globe a little round squire of brass, made with right Spherical Angles, the head or style whereof is to show the shadow of the Sun being set upon the Globe, in stead whereof a needle being set right up, will sometimes serve the turn. Finally to the Globe belongeth another Circle called the Horizon, which is a broad Circle of wood, having a foot of wood, within which Horizon, the Globe together with the brazen Meridian is to be set at what Altitude you list. And this Horizon is divided with divers circular lines into seven unequal spaces, whereof the first and narrowest space next unto the body of the Globe, containeth the degrees of the 12. signs of the zodiac, every sign containing 30. degrees. The second space containeth the Characters and names of the said signs, and also the number of the degrees, as 10. 20. 30. The third the days of every month. The fourth the names of every month, and of certain Festival days. The fift, the names of the 12. winds, which the ancient Greeks and Romans' were wont to observe, whereof I have already spoken in my Sphere. The sixth and seventh do show the 12. houses, together with their significations necessary for the setting of figures and calculating of nativities, and in the foot of the said Horizon is a little Compass with a needle to show the North and South, according to which the Globe is to be placed. But yet not before the said needle be truly rectified, according to the variation which it hath in that place where you are to use the Globe: for otherwise the needle may 'cause great error, whereof we shall treat in the next Chapter. Thus having briefly described the Terrestrial Globe, and all the parts thereof, I think it good now to show you how to place the Globe according to the four quarters of the world.   

This Treatise of the two Globes, containeth 50. Propositions as followeth.  
HOw to place the Globe truly, according to the four quarters of the world, and according to the Latitude of any Region. Proposition. 1.  
To know under what Clime any place or Region is, and of how many hours the longest day is there, and also what Latitude any place described in the Map hath. Proposition. 2.  
How to know what Longitude any place described in the Map hath. Prop. 3.  
How to know the distance betwixt any two places described in the Map. Prop. 4.  
To know how one place beareth from an other. prop. 5.  
How to found out by the Globe the place of the Sun, that is to say, the degree and minute of that sign wherein the sun is every day throughout the year. prop. 6.  
How to rectify the Index of the houre-wheele for every several day throughout the year. prop. 7.  
How to know every day at what hour the sun riseth or setteth. Proposition. 8.  
How to know in what part of the Horizon the sun riseth and setteth every day. Prop. 9  
How to know the length of every day and night throughout the year, aswell by help of the houre-wheele, as by counting the degrees upon the Horizon. Prop. 10.  
How to know by the Globe how much the sun declineth every day throughout the year from the Equinoctial. Proposition. 11.  
How to know by the Globe the Meridian altitude of the sun, that is to say, his height at noon tid● every day throughout the year, and how far he is then distant from your Zenith. Prop. 12  
How to know the altitude of the sun at every other hour of the day. Prop. 13.  
How to know the hour of the day by the Globe. Proposition. 14.  
To know how much the unequal hours otherwise called the Planetary hours, do differ from the artificial hours throughout the year, and how many minutes every unequal hour containeth. Prop. 15.  
How to know every day when the dawning of the day, and the twilight of the night beginneth and endeth, and the time of their continuance. Prop. 16.  
How to know the ascension of the sun, both right and obliqne. prop. 17.  
The 18. proposition, containing the description of the Celestial Globe, and showing wherein it is like or differing from the terrestrial Globe.  
The 19 proposition containing a particular description of the 48. Images of the fixed stars that are in the Celestial Globe, together with their sundry names, and also the names of so many stars as are named in the Globe. prop. 19  
How to find out in heaven any unknown star described in the Globe two manner of ways, that is, either by the help of some known star, or else by knowing the true hour of the night. prop. 20.  
How to know by the Globe the Meridian altitude which is the highest altitude of any star, and also how high or low he is at any other time. prop. 21.  
How to know by the Globe what stars are above the Horizon at any time of the day or night. prop. 22.  
How to know by the Globe at what time any star riseth above the Horizon, mounteth to the highest, and setteth, and with what degree of the Ecliptic he riseth, mounteth, and setteth: and also in what part of the Horizon he riseth and setteth. prop. 23.  
How to know in what part of the firmament any star is, and how many degrees it is distant from the Meridian at any hour, and being right under the meridian to know how far it is distant from your Zenith. Proposition. 24.  
How to find the hour of the night by the Globe. pro. 25.  
How to know the vertical stars in every latitude. Prop. 26.  
How to know the true place of any star, that is to say, in what sign and in what degree thereof any star is. Prop. 27.  
How to find the place and Longitude of any star by the Globe. Proposition. 28.  
How to found out the Latitude of any star. Prop. 29.  
How to found out by the Globe, the declination of any star: Proposition. 30.  
How to know the magnitude or greatness of any star, and his nature and quality by the Globe, and also the right ascension of the Arque of the Ecliptic, which accompanieth the right ascension of any star. Prop. 31.  
A Table of the fixed stars.  
How to found out the right and obliqne ascension of any star and also of the ascentionall difference. Prop. 32.  
To know in what quantity of time any whole sign or any other Arque of the Ecliptic doth rise or set. Prop. 33.  
How to know by the Globe what stars do rise or set every day Cosmically, Acronically, or Heliacally. Prop. 34.  
To know in what time of the year any star riseth or setteth, either Cosmically or Acronically. Prop. 35.  
Of the Horoscope and the rest of the 12. houses. Prop. 36.  
How to found out the Horoscope or ascendent at any time of the day or night by the Globe, and thereby to know the four principal Angles of heaven. Prop. 37.  
How to erect a figure by the Globe according to Regio Montanus his way, which is called the reasonable way, and is counted the best of all others. Prop. 38.  
How to know the Latitude of any place or region by any of the fixed stars described in the Globe. Prop. 39  
Another way to found the elevation of the pole. Prop. 40.  
A third way how to found out by 2. stars the elevation of the pole, not knowing their Meridian altitude. Prop. 41.  
A fourth way to found out the Latitude of any Region by any known fixed star or Planet that may be seen. Prop. 42.  
A brief description of the diurnal Table set down in Stadius his Ephemerideses, together with the use thereof. prop. 43.  
How to find out the place of any Planet by the Ephemerideses. prop. 44.  
A brief description of the Table of Stadius, set down in the 112. page of his Ephemerideses, to found out thereby the daily latitude of the Moon be it North or South, together with the Cannon or rule thereof, plainly declared by example. prop. 45.  
How to know the true place of the Sun or Moon or of any other Planet every hour of the day throughout the year. prop. 46.  
How to found out the place of the Moon by the Globe, when she is above the Horizon, without the help of any Ephemerideses or other table whatsoever. prop. 47.  
Another way to found out the place of the Moon without taking the altitude of any star. prop. 48.  
How to found out the longitude of any region. prop. 49.  
Another way to found out the unknown longitude of any place by the Globe. prop. 50.   

How to place the Globe truly according to the 4. quarters of the world, and according to the latitude of any region. The first Proposition.  
FIrst in bedding your Globe together with the brazen Meridian into the 2. nickes of the Horizon and also into the slit of the pin which standeth in the midst of the foot, so that the North pole of your Globe be answerable to the North quarter or north wind of the world, described upon the Horizon, and see that that part of the brazen Meridian, wherein are described the Climes, Parallels & hours of the longest day, may stand above the Horizon, and also that the one half of the brazen Meridian may justly and evenlie appear above the Horizon, and the other half under the Horizon: Again, you must see that the Equinoctial line of the Globe do meet just with the middle point or stréeke of the brazen Meridian whereas the first degree of latitude doth begin, and also that the body of the globe do not lean to the one side of the Horizon more than to the other, but to be equally distant from the same in all places, and in any wise to see that the Horizon stand always level, to which end some Globes have a plummet of lead hanging by a little chain or thread, which because it will move with every wind, I for my part do think it better for you to have such a little level made of purpose, as you may set the same upon any place of the Horizon where you list, and thereby make the Horizon to stand level on every side as you will yourself, and it may be made of a little piece of thick board, like a Triangle, thus.  
Than with your 2. hands laying hold of the 2. next pillars, turn the foot of the Globe until it stand right North and South, which is to be done thus. First found out the true Meridian of the place whereas you are to use the Globe by such means as M. Burrough teacheth in his discourse of the variation of the Compass, the 7. chapter. which undoubtedly is a most certain way, by which you shall find three things at one instant, that is, the true Meridian, the variation of the needle, and the true latitude of any place. But if you have not the instrument of variation, by help whereof this matter is to be accomplished, then in some open place betwixt 8. and 9 of the clock in the forenoon or sooner, upon a smooth table or plank, standing level, draw a good large circle with your Compasses, in the Centre whereof must be fixed a round and strait pin of Iron or Latton wire, in length a good deal shorter than the semidiameter of that circle, and to the intent that the pin or style may stand right up, without inclining on either side, it would be rectified by a true Squire: That done, wait diligently until the shadow of the pin head do justly touch the circumference of the circle, so as it neither pass beyond the circle, nor come short of the same, and there make a prick, and so let it stay until about three of the clock in the afternoon, about which time the shadow of the pin will begin to approach nigh unto the said circle, and so soon as it toucheth the same, make there another prick: that done, divide the Ark or portion of the circle, contained betwixt those 2. pricks into two equal parts, and in the midst thereof set another prick, them laying your ruler to the middle prick, and to the Centre of the circle, draw a right line through the Centre & also through the middle prick from the one side of the circle to the other, and beyond if you list: for you may make the said line of such length as you shall think most meet to serve your turn, and that line shall be the true Meridian for that place, showing the right North and South part of your Horizon, and by crossing the said live with another right line in the very midst with right angles, you shall have the true East and West, and to avoid long waiting, you may draw diverse circles one within another. Thus having found the true North and South, East, and West, place your Globe accordingly, so as the brazen Meridian of the Globe may answer the Meridian line already drawn upon the board or plank, and to the intent that afterward you may know at all times both day and night, how to place your Globe right North and South, it shall be necessary to draw a right line upon the foot of the Horizon, answerable to the foresaid Meridian first drawn upon the board or plank, and uponthat line to fasten a pretty handsome Compass, having a needle of an inch long at the lest, which is much more certain, than such a little needle as is wont to be set in the foot of every Globe: And when the needle standeth still, mark how much the North point thereof declineth either East or West from the true Meridian before fastened, and whereas you see the Needle to decline, be it East or West, there set fast a little pin of Latton to serve as a mark, whereto you may always direct the North point of the needle when you would have the Globe to stand right North and South, for there is no needle touched with the load stone, be it never so good a stone, but it will vary from the true Meridian line, either more or less, & therefore no trust is to be given to the needle until you know the true variation thereof, the finding of which variation, as I said before, is most truly taught by M. Borough in his book before mentioned, whereunto I once again refer you, and the rather for that it is written in our mother tongue. But Gemma Frisius teacheth to set the Globe right North and South thus, first go into an open place whereas the Sun shineth, and upon some table standing level, and also the Globe standing level, the Pole being elevated above the Horizon according to your latitude, fix a right needle in the degree of the sign, wherein the Sun is that day so as the nédle may stand right up without inclining any manner of way, and if it be in the forenoon, turn the East side of the Globe towards the Sun, moving both the Globe and his seat to and fro, until you see the needle to cast no shadow at all, for so shall the Globe stand right North and South. But if it be in the afternoon, you must turn the West side of the Globe towards the Sun, and then work as before, and by any of these ways before taught you shall not only place the Globe answerable to the four quarters of the world, but also you shall found that the Circles, Poles, and Axletrées of the Globe, are answerable to the Circles, Poles and Axletrees of the heavens. Now having placed your Globe answerable to the 4. quarters of the world, learn to know by some Table or modern Map, or else by such ways as are set down both in my Sphere, and also in the latter end of this Treatise, the latitude of the region wherein you devil: As for example, the latitude of these parts here about Norwich is 52. degrees, then taking hold of the brazen Meridian aswell with your left hand above the Horizon, as with your right hand beneath the Horizon, turn the same up and down in the nickes of the Horizon, until the North pole be elevated above the Horizon 52. degrees, the last of which degrees must meet even with the upper brim of the Horizon, that done, seek for England in the Globe, not leaving to turn the body of the Globe to and fro until you have brought that Meridian which passeth through England right under the brazen Meridian, and holding it still at that stay, draw the head of the brazen quarter of altitude right over the place, and over that Meridian under which you devil, so shall the head of your brazen quarter stand for your Zenith in the very midst of the Horizon and keeping all things thus at a stay, you may see how every region or country is situated, and how it beareth from you, and which be under your Climate and which be not, and to be sure that the brazen Zenith may stand in his right place, it shall be needful to set it so as the left square side thereof made with a long notch may touch the self same degree of latitude in the brazen Meridian, which it hath of altitude, for look how many degrees the Pole is elevated above the Horizon, which is called the altitude, so many degrees must the brazen Zenith be distant from the Equinoctial of the globe, which is called the latitude, and is ever all one with the altitude of the Pole. Thus having showed you how to place the Globe according to the four quarters of the world, I think good now to declare unto you, first the uses of the terrestrial globe, whereof some are common also to the celestial globe, as you shall perceive hereafter by the propositions following, which propositions may be very well divided into 3. sorts, whereof the first do properly belong to the description of the universal Map, wherewith the terrestrial globe is covered, which are but few in number. The second kind of propositions do chief belong to the Sun, and to his appearances, which are also common to the celestial globe, and may be found out by either of the globes. But the third kind do belong most properly to the fixed stars and to their appearances, and therefore I mind not to set down them until I come to describe the celestial globe, and to show the uses thereof.  
The chiefest propositions belonging to the universal Map, wherewith the terrestrial Globe is covered, are these following.   

To know under what Clime any place or region is, and of how many hours the longest day is there, and also what latitude any place described in the map hath. The 2. Proposition.  
Having set the Globe at your latitude, bring the place or region which you seek right under the brazen Meridian, and the upper space of the said Meridian will show the Clime. And the second or middle space will show the hours of the longest day. And the third space the latitude: As for example, if you would know under what Clime London is, then having found London, fix a long needle in the read spot next to the name of London (for all towns for the most part in the Globe are marked with read spots) that done, turn the Globe with your hand until the needle do touch the brazen Meridian, and by staying the Globe there with your hand, you shall found London to be under the 8. Clime, and that the longest day in the year is there 16. hours and 20. minutes, and the third and nethermost space doth show the latitude of the place, which is 51. degrees, 32′· Again, by observing this order without changing your latitude, you shall find Venice to be under the 6. Clime, and their longest day to be 15. hours and somewhat more, and the latitude of that City to be 44. degrees, 30′·S And you shall find jerusalem to be under the third Clime, and the longest day there to be 14. hours, and the latitude thereof to be 31. degrees. 30′·   

How to know what longitude any place described in the Map hath. Proposition. 3.  
THe Globe standing still at your own latitude, prick the needle in the place whereof you seek the longitude, and bring it as before, to the brazen Meridian, and staying it there, look at what number of degrees the brazen Meridian cutteth the Equinoctial, and that is the longitude of the place, counting the degrees from the first degree of the Equinoctial, which beginneth at the first point of Aries unto the place of the section: by doing thus, you shall find the longitude of London to be 19 degrees, and the longitude of Venice to be 36. degrees, and the longitude of jerusalem to be 67. degrees and 30. minutes.   

How to know the distance betwixt any two places described in the Map. Proposition. 4.  
OPen your Compasses so wide as you may set the one foot thereof just in the one place, and the other foot in the other place, and apply that wideness to the Equinoctial line, counting how many degrees of the Equinoctial are contained betwixt the two feet of your compasses, & by allowing to every degree 60. miles, you shall have the true distance of the places. Thus you shall find the distance betwixt London and Venice to be 13. degrees, 30′·S which being multiplied by 60. maketh 810. miles, and the distance betwixt London and jerusalem to be 40. degrees. 30′·S which being multiplied by 60. maketh 2430. miles. And it maketh no matter to what part soever of the Equinoctial you do apply the wideness of your compasses, so as you set the first foot at the very beginning of some one degree, and let the other foot fall out as it will, either at a whole degree at a half, or at a quarter of a degree, which small parts are to be counted by minutes by conjectural discretion. And note here that no 2. places can be distant one from another East and West more than 180. degrees which is just the one half the circuit of the Earth, beyond which half or on this side thereof the places must needs be nearer together by means of the roundness of the Earth, and if either of the 2. places whereof you would know the distance be not expressed in the Globe, then learn to found out the longitude and latitude thereof by some table, whereof the first meridian is supposed to pass through the islands of Canariae and mark upon the Globe where the said longitude and latitude do cross, for there aught that place to stand which is missing, to which place direct the one foot of your compasses & then work as before is taught. And by this means you may also know the distance betwixt any two stars contained in the Globe.   

To know how one place beareth from another. Proposition. 5.  
THough you may partly find out this by marking the direction of the lines, proceeding from the mariners Compass set down in the Globe: yet in mine opinion it is readier to do it by applying the Fly described in my little treatise of universal Maps, unto the Globe by setting the centre thereof upon the first place from whence you go, and by drawing a thread through that place whereto you would go, in such order as is there taught, so you shall find Venice to bear from London South East and by East, and jerusalem to bear from London East, South East, and two quarters more towards the South. But Gemma Frisius teacheth to know by the Globe how one place beareth from another thus, having set the Globe and also the Zenith of the quarter of altitude at such latitude as the first place or region hath, from whence you would go, bring that first place under the brazen Meridian, and there stay the Globe until you have brought the quarter of altitude to the second place, and the neither end of the quarter of altitude will show you upon the Horizon amongst the winds how the second place beareth from the first, so shall you found Hispaniola to bear from Spain right West, these be the chiefest propositions belonging to the universal Map, wherewith the terrestrial Globe is covered, and therefore I will now set down those that belong to the Sun, which may be done aswell by the one Globe as by the other, and first how to found out the place of the Sun.   

How to found out by the Globe the place of the Sun, that is to say, the degree and minute of that sign wherein the Sun is every day throughout the year. Proposition. 6.  
THough the surest way be to found it out by the Ephemerideses, which showeth the very minute, yet without having respect to the minutes, you may found it out by the Globe thus, seek out the day of the month upon the Horizon, and that will point you to the degree of the sign wherein the sun is that that day: As for example, the 6. day of May pointeth right to the 25. degree of Taurus the Globe standing level, and your rule being rightly laid upon the Horizon, but during the leap year you must add one degree more than the Horizon showeth every day from the beginning of the leap year throughout all that year.   

How to rectify the Index of the hour wheel for every several day throughout the year. Proposition. 7.  
Having placed the Globe at your latitude and also found out the degree of the Sun, as is before taught, bring that degree of the Sun to the brazen Meridian, and there staying it with the one hand, turn with your other hand the index of the hour wheel to the highest part of the said wheel, marked with the number of 12. setting the point of the index just with the stréeke of the wheel made to show the hour of 12. or neonetyde, and that will serve your turn for all that day, and thus must you do every day in which you have to use the help of the said hour wheel for any purpose.   

How to know every day at what hour the Sun rifeth or setteth. Proposition. 8.  
Having set the Globe at your latitude, and rectified the index of the hour wheel by the 7. proposition, turn the Globe to the East, so as the degree of the Sun may touch the Horizon, and then the index of the hour wheel will show you at what hour the Sun riseth. Again, if you bring the said degree of the Sun unto the West part of the Horizon, the index of the hour wheel will show you at what hour he goeth down. As for example, Anno 1590. the third of june, the Sun being in the 21. degree 33′· of Gemini. I bring that point to the very edge of the Horizon on the East part thereof, and there staying it, the index of the hour wheel showeth that the Sun riseth 8′· before 4. of the clock in the morning: and the said point of the Ecliptic being turned to the West part of the Horizon, the index showeth that he setteth 8′· after 8. of the clock at night.   

How to know in what part of the Horizon the Sun riseth and setteth every day. Proposition. 9  
Seek the degree of the Sun in the Ecliptic live, & turn it to the East part of the Horizon, than you shall see whether it riseth just East or not, and whether it inclineth towards the South, or towards the north, and likewise by bringing the said degree to the West part of the Horizon, you shall see in what part of the said Horizon he goeth down: As for example, in the last proposition the Sun being in the 21. degree, 33′· of Gemini, and brought to the East part of the Horizon, I find that the Sun did rise distant from the East towards the North 38. degrees, 30′· of the Horizon, which is three points and somewhat more of the Mariners Compass from East, towards the North, so as the Sun riseth North-east and by East, and a little more Northward, and the said place of the Sun being brought to the West part of the Horizon, I found that he setteth North West and by West, and somewhat more Northward.   

How to know the length of every day and night throughout the year, aswell by help of the hour wheel as by counting the degrees upon the Horizon. Proposition. 10.  
FIrst you must know when the Sun riseth and setteth by the 8. proposition, then look how many hours the index doth go from the Sun rising to the Sun setting, and that is the length of the day, which number if you take from 24. the night will appear: as for example, knowing by the 8. proposition, that the Sun being in the 21. degree, 33′· of Gemini, riseth 8′· before 4. in the morning, and goeth down 8′· after 8. of the clock at night, I found by counting the hours which the index of the hour wheel hath run, that the length of the day is 16. hours and 16′·S which you may know also by counting the degrees upon the Horizon from the place of the Suns rising, unto the South point of the said Horizon, which you shall found to be 128. degrees, which being doubled maketh 256. degrees, which if you divide by 15. it will make 16. hours and 16′·S as before.   

How to know by the Globe how much the Sun declineth every day throughout the year from the Equinoctial. Proposition. 11.  
Having found the place of the Sun, bring the same to the brazen Meridian, and by counting how many degrees are betwixt that place and the Equinoctial, you shall know what declination the Sun hath that day: As for example, supposing the Sun that day you seek to be in the first degree of Gemini, bring the said degree of the Ecliptic to the brazen Meridian, and you shall found upon the same Meridian the Sun to be declined from the Equinoctial Northward almost 20. degrees. Again, supposing the Sun to be in the first degree of Aquarius, if you bring the same degree of the Ecliptic unto the Meridian, you shall find the declination of the Sun to be almost 20. degrees Southward.   

How to know by the Globe the Meridian altitude of the Sun, that is to say, his height at noontyde every day throughout the year, and how far he is then distant from your Zenith. Proposition. 12.  
BRing the place of the Sun that day you seek to the brazen Meridian, and staying it there, count upon the said brazen Meridian how many degrees are contained betwixt the place of the Sun and the South point of the Horizon, and that is the Meridian altitude of the Sun for that day, which if you subtract from 90. the remainder will show how many degrees he is distant that day at noontide from your Zenith: as for example, supposing the Sun to be that day you seek in the first degree of Taurus, bring that degree of the Ecliptic to the brazen Meridian, and stay it there until you have counted how many degrees of the said Meridian are contained betwixt the place of the Sun, and the South point of the Horizon, and you shall found the number of degrees to be 50. which is the Meridian altitude of the Sun for that day, which 50. degrees being taken out of 90. there remaineth 40. and so many degrees the Sun is that day at noontide distant from your Zenith. The like order is to be observed in seeking to know the Meridian altitude of any Star, or any other point in heaven.   

How to know the altitude of the Sun at any other hour of the day. Proposition. 13.  
Having rectified the index of the hour wheel by the 7. proposition, if the hour which you seek be in the forenoon, turn the Globe so as the index of the hour wheel may touch that hour of the forenoon, at which you desire to know the altitude of the Sun, and there stay the Globe, until you have brought the quarter of altitude on the Eastside of the globe unto the place of the Sun, so shall you found upon the said quarter the altitude of the Sun at that hour. And if you desire to know the altitude of the Sun at any hour in the afternoon, then turn the Globe so as the index of the hour wheel may touch the hour of the afternoon, and there stay the Globe until you have brought the quarter of altitude on the West side of the Globe unto the place of the Sun, and the said quarter will show you the altitude of the Sun at that hour. As for example, I would know how high the Sun is at 8. of the clock in the morning, the Sun being in the first degree of Taurus: here having rectified the index of the hour wheel, I turn the Globe so as the index of the hour wheel may lie upon the 8. hour of the forenoon, and there I stay the globe until I have brought the quarter of altitude on the East side of the Globe unto the place of the Sun, whereby I find the altitude of the Sun at that hour to be almost 28. degrees. Again, if you will know how high the Sun is being in the same degree of the Ecliptic at 5. of the clock in the afternoon, then turn the index of the hour wheel, so as it may touch that hour, and stay it there until you have brought the quarter of altitude on the West side of the Globe unto the place of the Sun, so shall you found the altitude of the Sun to be at that hour 21. degrees.   

How to know the hour of the day by the Globe. Proposition. 14.  
THis is to be done two manner of ways, the first is thus, set the Globe in some open place whilst the Sun shineth, and you must see that it stand both level and also right North and South, as is taught in the first proposition, and that the index of the hour wheel be rectified according to the degree of the sign wherein the Sun is that day by the 7. proposition, that done, fix a needle in the place of the Sun, and turn the body of the Globe to and fro until the needle cast no shadow at all, and there staying the Globe, the index of the hour wheel will show you the hour of the day. But if you seek the hour in the forenoon, remember to turn the East side of the Globe towards the Sun, if in the afternoon, then turn the West side of the Globe towards the Sun. The second way is thus, having rectified the index of the hour wheel, take the altitude of the Sun with some Quadrant or Astrolabe, and having marked the same altitude upon the quarter of altitude, apply it to the degree of the Sun on the East side of the Globe if it be in the forenoon: but if it be in the afternoon apply the quarter of altitude to the degree of the Sun on the West side of the globe, and the index of the hour wheel will show you the hour which you seek. As for example, the 6. of june 1590. the Sun being in the 24. degree, 24′· of Gemini, I found by my Astrolabe the altitude of the Sun to be 48. degrees, which I marked upon the quarter of altitude, and because I took the altitude of the Sun in the forenoon, I brought the quarter of altitude marked with that degree to the place of the Sun on the East side of the Globe, and there staying the Globe I found that the index of the hour wheel did point to the 9 hour of the forenoon and somewhat past. The like is to be done to know any hour of the afternoon, so as you forget not to apply the quarter of altitude unto the place of the Sun on the West side of the Globe. And by taking the altitude of any known Star, and working in like manner as before, you shall know the hour of the north, as shall be taught hereafeer when we come to treat of the Stars.   

To know how much the unequal hours otherwise called the planetary hours do differ from the Artificial hours throughout the year, and how many minutes every unequal hour containeth. Proposition. 15.  
FIrst you must know by the 10. proposition the length of the day, that is to say, how many hours it is long, and reduce those hours into minutes, and divide the product by 12. and the quotient, together with the remainder (if there be any left after the division) will show you the quantity of the unequal hour of the day, that is to say, how many minutes it containeth: The like is to be done to know the unequal hour of the night, for having the length of the artificial night, work as before, and you shall have your desire: As for example, knowing by the 10. proposition the length of the day when the Sun is in the 21. degree. 33′· of Gemini to be 16. hours 16′· here by reducing those hours into minutes, and by dividing the product thereof by 12. you shall find the unequal hour of the day to contain 81. minutes. and ⅓ of a minute or 20. seconds, which is more than one whole artificial hour of the day by 21. minutes and 20. seconds. Again, knowing the length of the artificial night by the said proposition to be 7. hours, 44. minutes, if you do reduce the same into minutes, and divide by 12. you shall find thereby the unequal hour of the night to contain no more but 38. minutes, and 40. seconds, which is less than the artificial hour of the day by 21. minutes, and 40. seconds.   

How to know every day when the dawning of the day and the twilight of the night beginneth and endeth, and the time of their continuance. Proposition. 11.  
Having rectified the index of the hour wheel by the 7. proposition, first find out the opposite point to the degree of the Sun, and turn the Globe together with that opposite point, and also together with the quarter of altitude towards the West, so as the opposite point may meet even with the 18. degree of the quarter of altitude, and staying the Globe there, the index of the hour wheel will show at what hour the dawning beginneth. As for example, I would know at what hour the dawning of the day beginneth the 19 of April 1590. when as the Sun is in the 8. degree of Taurus, the opposite point whereof is the 8. degree of Scorpio, wherefore I turn the Globe together with the said point opposite and also together with the quarter of altitude towards the West, so as the said point opposite may meet even with the 18. degree of the quarter of altitude, and there staying the Globe, I find by the index of the hour wheel that the dawning of the day beginneth at 2. of the clock in the morning and 20. minutes after, which dawning always endeth when the Sun riseth, as in the former example, the Sun being in the 8. degree of Taurus doth rise 45. minutes after 4. of the clock in the morning, so as the continuance of the dawning is two hours and 25. minutes, for by taking two hours and 20. minutes out of 4. hours and 45. minutes, there remaineth 2. hours and 25 minutes. Again, the twilight beginneth when the Sun goeth down, which in the former example is at 7. of the clock and 15′· Now to know when the twilight endeth, you must do thus, turn the Globe and the quarter of altitude towards the East, so as the opposite point, which is the 8. degree of Scorpio may meet even with the 18. degree of the said quarter, and the index of the hour wheel will show you that the twilight endeth at 9 of the clock and 45′· after, so as the continuance of the twilight is two hours, and 30′ for by taking 7. hours & 15′·S out of 9 hours & 45′·S there remaineth 2. hours and 30′·S But you have to understand, that the dawning twilight is not always to be known throughout the year by the Globe, for from the 11. day of May to the 10. of julie, you shall found that the opposite point of the Sun will not agree just with the 18. degree of the quarter of altitude: because that no opposite point during that time will amount to above 16 or 17 degrees of the quarter of altitude at the most, because the Meridian altitude itself of any such opposite point is not above 17. degrees, for during all that time both dawning and twilight had need in this our latitude to be accounted as night, unless you will make no night at all.   

How to know the ascension of the Sun both right and obliqne. Proposition. 17.  
Having set the Globe at your latitude, bring the degree of the Sun to the brazen Meridian, and there staying it, mark at what number of degrees the said Meridian cutteth the Equinoctial, counting that number from the first point of the Vernal Equinoctial point to that section, and that is the right ascension, As for example, the Sun being in the first degree of Gemini and brought to the Meridian, you shall find that the meridian cutteth the Equinoctial in the 58. degree thereof, and that is his right ascension. Now if you would know his obliqne ascension, being in the first degree of Gemini, bring that degree to the East part of the Horizon, so as it may touch the upper edge thereof, and staying the Globe there, look what degree of the Equinoctial toucheth the Horizon at that instant, which you shall find to be the 30. degree of the Equinoctial, and that is his obliqne ascension. These are the chiefest propositions that belong to the Sun, and are to be found by either of the Globes, wherefore I will now proceed to those propositions that are to be known most properly by the celestial Globe, But first I will make a description of the said celestial Globe, whereby it shall plainly appear wherein the one Globe is like the other, and wherein the one differeth from the other.   

The 18. Proposition, containing the description of the celestial Globe, and showing wherein it is like or differing from the terrestrial Globe. Proposition. 18.  
THe Celestial globe is like to the terrestrial globe, in that it is round, having both like Axletrées, Poles, hourewhéele with his index, brazen Meridian, quarter of altitude of brass or Latton with his square head or Zenith, and a half circle of brass or Latton, called the semicircle of position, also a standing foot with an Horizon of wood divided into seven several spaces, containing in a manner the self same things that are before described in the Horizon of the terrestrial Globe, also in the body of the celestial globe are set down certain circles like unto the terrestrial Globe, that is to say, the Equinoctial and the Ecliptic line. Moreover the four lesser circles, that is to say, the two tropiques, the circle Arctique, and the circle antarctique. But the celestial Globe differeth from the terrestrial Globe in these four things following. First the celestial Globe hath one thin demicircle of brass or Latton more than the terrestrial Globe hath, which Demicircle is divided into two quarters, each quarter containing 90. degrees, made so at each end as it may be fastened when need is, upon the 2. Poles of the zodiac, to found thereby the longitude and latitude of every Star described in the Globe, and therefore may very well be called the Semicircle of longitude and latitude. The second difference is that whereas the terrestrial Globe is traced with 12. Meridian's, dividing the Equinoctial into 24. spaces, every space containing 15. degrees. The celestial Gobe is only traced with 6. Meridian's, dividing the Equinoctial into 12. spaces, every space containing 30. degrees. The third difference is that the celestial Globe hath not those 8. Parallels of latitude wherewith the celestial Globe is traced. The fourth difference is that whereas the terrestrial Globe is covered with an universal Map containing the four principal divisions of the earth, that is, Europa, Africa, Asia and America: the celestial Globe is covered with a Map, wherein are painted all the fixed Stars that were known to the ancient Astronomers divided into 48. Images, with which Images, to the intent you might be the better acquainted, and that you might the more readily found out any Star described in the Globe, I thought good to set down a particular description of the said 48. images as followeth.   

The 19 Proposition, containing a particular description of the 48. Images of the fixed Stars that are in the celestial Globe together with their sundry names and also the names of so many stars as are named in the Globe, of which 48. Images 21. are ascribed to the North part of the firmament, 12. to the zodiac, and 15. to the South part of the firmament. Proposition. 19  
THis description is divided into two parts according to the twofold declination of the fixed stars, that is to say, Northern and Southern, for those Stars are said to have North Declination, which are situated betwixt the Equinoctial and the North pole, and those to have South declinatition which are situated betwixt the Equinoctial, and the South pole, and because that six great Circles or Meridian's, passing through the poles of the world, do divide the Equinoctial into twelve equal spaces, every space containing 30. degrees, I will begin my description at the first point of Aries, which is the Uernal Equinoctial point, and so proceed towards the right hand round about the Globe, setting down all such Images, or parts of Images as are situated towards the north pole, and are contained in every several space betwixt two Meridian's, & having described all the north part, I will use like order in describing the South part. And you may behold all the Northern Images by turning the Globe about with your hand without taking the same out of his bed or seat, the pole being elevated above the Horizon 50. or 60. degrees, but to view the Southern Images, it shall be needful to take the Globe clean out of his seat, and to hold it so as the north pole may stand right up, so shall you see every Southern Image and Star at your pleasure. And yet to know how the Stars are situated in heaven, you had need to imagine yourself to be within the Globe, in the very centre thereof and not without the Globe for otherwise those stars that are situated in heaven on your right hand, if you have regard to the outside of the Globe, will seem to be on your left hand.  

The Northern Images contained in the first space, intercepted betwixt the first Meridian and the second Meridian.  
IN this space you shall first see next unto the Equinoctial the following Fish of the sign Pisces together with the bond, both Southern and Northern, called in Latin Linum australe & Septentrionale, also the knot of the bond, which is called Nodus, Syndesmon, and hipouraion, which is a fair Star of the third bigness.  
Item the first part of Aries with the two Stars in his right horn from the former Star, whereof the Astronomers do always make their computation.  
Item the whole Image of Andromeda, her head and right arm excepted, in whose girdle is a Star of the third bigness called Mirach, and in her left foot a Star of the third bigness called Alamac.  
Item the Triangle called Triangulus and Deltoton with his four stars.  
Item the whole Image of Cassiopeia saving her right arm, and the upper part of the back of her chair, in whose breast is a star of the third bigness called Shedar.   

The Northern Images contained in the second space.  
FIrst the head of the Whale called Cetus in whose snout is a star of the third bigness called Menkar.  
Item all the hinder parts of Aries called in Greek Chrios, in English the Ram.  
Item the right leg, neck, breast, right ear, and muzzle of the Bull, in whose right thigh towards the shoulder point is a star of the fourth bigness called Alfon and in his breast a star of the third bigness called Alfo, and in his Mozel lying upon his right leg, is another star of the 3. bigness called Alfon, and in his neck toward the Withers are 7. Stars of divers bigness called by these divers names, that is, Vigilie, Atlantides, Pleiades, and Athoratae, commonly called the 7. Stars.  
Item the whole head of Medusa, called Caput Medusae vel Gorgonis, and Ras Algol.  
Item the whole image of Perseus otherwise called Chelube, his right hand, sword, and right foot excepted.   

The Northern Images contained in the 3. space.  
FIrst the left leg of the Bull having 2. stars thereon: moreover his head, horns and most part of his right ear, on whose left eye is a Star of the first bigness, called Oculus Tauri, Palilicium, and Aldebaran, also in his face are certain lesser stars called succulae and Hyades.  
Item the upper part of Orion otherwise called Alguze, holding a club in his right hand, & a Lion's skin in his left hand on whose right shoulder is a Star of the first bigness called Bed Alguze, and on his left shoulder a star of the second bigness called Bellatrix  
Item the left foot of the former Gemini, containing two stars of the fourth bigness, whereof the one is called Propous.  
Item the whole Image of Auriga, otherwise called Ericthonius and Heniochos, holding a rain and a whip in his right hand and having a goat hanging on his back, which hath two little Goats sucking her behind, which be two stars of the second bigness, called by diverse names, as Hedi, heriphoi, and Sadateni, and in the flank of the Goat is a star of the first bigness called hircus, aix, holenie and Alhaiot.   

The Northern Images contained in the 4. space.  
FIrst the whole image of the little Dog, in whose left flank is a Star of the first bigness called Canis minor, Protion, Algomeisa, and Alsahere.  
Item the whole image of the two twins called Gemini or Didimoi, the left foot of the former twin only excepted, which former twin is called Apollo, Castor, Anhelar, and the other is called Pollux, and Abrachaleus, in whose left ear is a star of the second bigness called Ras Alguze.  
Item the tail and half body of Cancer.  
Item the Muzzle of the great Bear, whereon is a Star of the fourth bigness.   

The Northern Images contained in the 5. space.  
FIrst the head and neck of Hydrus.  
Item the fore part of Cancer, called in Greek Carchinos, upon whose right Clea is a Star of the fourth bigness called Acubene, and betwixt his head and his right Clea is a star called Presepe, Phatue, and Meelleph, and on his back are two stars called Aselli and Onoi.  
Item the fore part of Leo against whose heart is a star of the first bigness, called with these names, Cor Leonis, Regulus, Basileus, and Calb alezet.  
Item the fore part of the great Bear called Vrsa maior, Arctos Eliche, and Calisto.   

The Northern Images contained in the sixth space.  
FIst the hinder parts of Leo otherwise called Alezet, in whose tail is a star of the first bigness called Cauda Leonis, and Deneb Alezet.  
Item the head and shoulders of Virgo.  
Item the dark star of Bernices hair.  
Item the hinder parts of the great Bear, his tail excepted.  
Item the hinder parts of the Dragon's tail, containing two stars of the fourth bigness.   

The northern Images contained in the seventh space.  
FIrst, the most part of Virgo, who is otherwise called Parthenos, Erigone, Previndemiator, Protigiter, Almucedie, and Alaraph.  
Item the left leg and left arm of Bubulcus, otherwise called Boötes' vociferator, Arctophilax and Lanceator, betwixt whose legs is a star of the first bigness called Arcturus, Asimech, and Alramech.  
Item Bernices hair, called Cincinnus, Cesaries, plochamos, Berenice's crinis, and Trica.  
Item the tail of the great Bear containing 3. stars of the third bigness, wherein that next his rump is called Aliot, & that which is in the tip of his tail, is called Benenacz.  
Item a part of the Dragon's tail containing 2. Sarres standing nigh together nigh unto the Circle Arctique.   

The northern images contained in the eight space.  
FIrst the head and neck of the serpent called Anguis, serpens, Engchelis and Ophis.  
Item the crown of Ariadna, called Corona gnosia, Stephanos Ariadnis, and is commonly called Corona septentrionalis, that is, the Northern Crown, in which is a Star of the second bigness called Melfelcare, alpheta, and Muniir.  
Item the most part of the Image of Bubulcus, having a club in his right hand, in whose left shoulder is a Star of the fourth bigness called Ceginus, and there is another of the fourth bigness in his club, right against his face called Incalurus.  
Item the fore part of the little Bear called Vrsa minor, arctos and Cinosura.   

The northern images contained in the 9 space.  
FIrst the upper part of Serpentarius, otherwise called Ophioucos, and alangue in the Crown of whose head is a Star of the third bigness called iras alangue.  
Item the whole Image of Hercules with the lions skin hanging on his left arm, otherwise called Engonasi, algethi, Nessus, and ignotum idolum (his right hand holding the club, and his right leg excepted) in whose head is a star of the third bigness, called Ras Algethi, & this Image lieth groveling with his heels towards the North pole, and his head towards the Equinoctial, which meeteth almost with the head of Serpentarius.  
Item the head of the Dragon called Draco & Aben, in whose head is a Star of the third bigness called Ras Aben.  
Item a part of his tail containing 6. stars.  
Item the hinder part of the little Bear, containing 2. stars.   

The northern images contained in the tenth space.  
FIrst, the upper part of Antinous, having at each elbow a star of the third bigness.  
Item the last end of the Serpent's tail, in the tip whereof is a star of the fourth bigness.  
Item the whole image of the Eagle called Aquila, Vultur volans, Aetoes and Alcair.  
Item the whole image of the Shaft called Sagitta, Telum, and Hoistos.  
Item the whole image of the Harp called Lira and Alohore, that is to say, Vultur cadens, and Chelis, in the upper part whereof towards the left hand is a fair Star of the second bigness called Fidicula, Lira, Alangue, Vega, and Brineck.  
Item the head, neck, and left wing of the Swan, called avis, Cignus, Olor, Hornis, Adigege, and of some Gallina.  
Item the neck, body and fore part of the Dragon containing 11. Stars.  
Item part of the little bears tail, containing one Star next to his rump of the fourth bigness.   

The Northern Images contained in the 11. space.  
FIrst, the Crown of Aquarius his head, containing one Star of the fift bigness.  
Item the little Horse called Equus and Hippos, whose neck is enclosed with a cloud, and in his head are four little stars.  
Item the head and two fore feet of the winged horse called Pegasus, on whose right nostril is a Star of the third bigness called Emph alpharaz.  
Item the Dolphin called Delphinus, containing ten Stars, whereof one is of the third bigness.  
Item the body, legs, and the right wing of the Swan, which lieth on her back with her belly upward, in whose body towards the tail, is a fair Star of the second bigness called Deneb Adigege, and Arided.  
Item the right arm and right leg of Cepheus, on whose right shoulder is a star called Alderaimim.  
Item part of the little bears tail containing the middle star of his tail.   

The Northern images contained in the 12. space.  
FIrst, the most part of the former fish of the sign Piscis together with part of her band.  
Item the neck, body, and wing of Pegasus, otherwise called Equus Gorgoneus, and Alpharaz, rising out of a cloud, in which cloud is the head of Andromeda, having on the right side thereof a fair star of the second bigness, and in the right wing of Pegasus is a star of the second bigness called Marcab Alpharaz and on his right shoulder another Star of the second bigness called Scheat Alpharaz.  
Item the right arm and hand of Andromeda holding part of her chain, in the ring whereof is a Star of the fourth bigness.  
Item the left arm of Cepheus.  
Item the tip of the little bears tail, in which is the North star called Alrucuba of the third bigness.  
The names of the Images contained in the celestial globe betwixt the Equinoctial and the South pole together with so many stars as are named in that part of the globe, beginning as I did before in describing the northern Images at the Vernal Equinoctial point, and so proceed from space to space contained betwixt every two Meridian's towards the right hand.   

The Southern Images contained in the first space, beginning at the Vernal Equinoctial point.  
FIrst, the most part of the Whale called Cetus, Pistrix, and Balena (his head and fore part of his belly excepted) in the mid body whereof towards the back, is a star of the fourth bigness called Baten kaetoes, and in the lower part of his tail is another star of the third bigness, called Deneb Kaitos.   

The Southern Images contained in the second space.  
THe fore part of the Whale's belly and his ghilles, containing five Stars.  
Item the most part of the Flood Eridamus, called of some Nilus, and in Greek Potamos Eridanos, containing 22. Stars, whereof one is called Angetenar, which is about the midst of the Flood nigh unto the Whale's belly, and there is in the very end of the Flood another star of the first bigness, called Acarnar.   

The Southern Images contained in the third space.  
FIrst, the neither part of Orion or Alguze from the middle of his back downward, in whose girdle are three fair stars, whereof the middle star is of the second bigness called Orion or Alguze.  
Item another part of the flood Eridamus which seemeth to come from the left foot of Orion, which star in his left foot is called Algebar, Rigel, Alguze.  
Item the whole Image of the Hare, called Lepus and Lagos containing 12. little stars.  
Item the rest of Eridanus containing four Stars, whereof there is one called Beemum of the fift bigness.   

The Southern Images contained in the fourth space.  
FIrst, the whole image of the great Dog called Canis maior, and Syrios in whose mouth is a Star of the first bigness called Asceher and Alhabor.  
Item the fore part of the great ship Argos, with her two Oars having a scutcheon with 4. Stars, the greatest whereof being of the third bigness is called Markeb, and under the upper hatches in the fore part of the ship is a Star of the fift bigness called Alphard, and in the lest oar towards the South pole, is a fair Star of the first bigness, called Canopus and Suhel.   

The Southern Images contained in the fift space.  
FIrst, the mid part of the Serpent called Hydrus and Asuia, in the which is a fair Star of the 2. bigness called Alphard.  
Item the hinder part, mast and top of the ship Argos, which seemeth to come out of a cloud, containing divers Stars of divers bigness without name.   

The southern Images contained in the sixth space.  
ITem another part of Hydrus whereupon standeth the image of the cup or bowl called Crater, vas and patera, and also the crows head.  
Item the hinder part of Centaurus, in every part whereof are divers Stars without name.   

The southern Images contained in the 7. space.  
FIrst, the left wing of the sign Virgo and her left hand, holding an ear of wheat, whereon is a Star of the first bigness, called Spica Virgins, stachiss, Acimon, Alacel, and Azimech.  
Item the Crow called Coruus and Corax, his head & neck excepted, in whose left wing is a star of the fourth bigness called Algorab.  
Item the rest of Hydrus, whereon the Crow standeth, containing three stars without name.  
Item the rest of Centaurus or Chiron with his borestaffe trimmed with boughs, his right hand and right foot excepted.   

The southern Images contained in the eight space.  
THe whole image of Libra, the ring only excepted.  
Item the fore part of Scorpio, whose fore cleas do lie upon the two balances, that is to say, his right clea upon the North balance, and his left clea upon the South balance, having upon each clea a star of the second bigness.  
Item part of the Serpent called Anguis or Ophis, having one Star of the fourth bigness.  
Item the left hand of Serpentarius, holding part of the Serpent, upon which hand are two stars of the third bigness called Yedd.  
Item upon the head of Scorpio are 3. stars of the third bigness, standing all in a row and divers others as well upon his back as upon his left little clea without name.  
Item the whole Wolf called Fera, Lupus and Therion.  
Item the right hand of Centaurus holding the said Wolf by the belly in both which are divers stars without name.  
Item the right foot of Centaurus, in which is a fair star of the first bigness, and is called by the name of Centaurus.   

The southern images contained in the 9 space.  
FIrst, the neither part of Serpentarius, that is to say, from his mid back downward, having the serpent winding betwixt his legs and above his right arm, in both which are divers Stars without name.  
Item the hinder part of Scorpio from his mid body to the outermost end of his tail, who hath divers names, as Scorpius Nepa, Alatrab, in the midst of whose body is a fair Star of the first bigness called cor Scorpionis, Antares, and chalb Alatrab, and in the tip of his tail are two Stars of the third bigness, called Alascha and scomlec Alatrab.  
Item the most part of the Bow with the head of the shaft of Sagittarius and his right foot, in all which parts are divers stars without names  
Item the Altar with the flame and smoke called Ara, thuribulum, lar, sacrarium, thimiaterion.   

The southern images contained in the tenth space.  
THe lower part of Antinous from the breast downward kneeling upon an Altar containing four Stars without name.  
Item the forepart of Capricornus his head.  
Item the image of Sagittarius, otherwise called Crotus, and toxeuter (his bow, the end of his arrow, and his right foot excepted) having divers Stars without name.  
Item the Southern Crown, called Corona Australist, and Notios' stephanos, and of Aratus, it is called Dinotos Cyclos, that is to say, the Southern circle which Crown is placed betwixt the two fore legs of Sagittarius, and in the said Crown are divers Stars, amongst which there is one of the second bigness touching the left knee of Sagittarius called Corona Australis, who also hath on his left foot another Star of the second bigness without name.   

The southern Images contained in the 11. space.  
THe fore part of Aquarius otherwise called Ganymedes and Hydrochos, holding a handkerchief in his left hand, wherein are three Stars, and he hath divers Stars in his body without name.  
Item the whole Image of Capricornus, otherwise called Pan Aigoceros, and Algedi the fore part of his head only excepted in whose tail is a Star of the fourth bigness called Deneb Aldegi.  
Item the hinder part of the Southern Fish having divers stars without name.   

The Southern Images contained in the twelfth space.  
FIrst Aquarius his right hand holding the water pot called Vrna and Chalpi, out of the which he poureth the water down into the mouth of the Southern fish, which water is called Aqua and hydor in which are divers Stars without name.  
Item the lower belly part of the former fish of the sign Piscis wherein are two Stars without name.  
Item both the thighs and legs of Aquarius, upon the calf of whose right leg is a Star of the third bigness called Scheat Aquarii, and Crus Aquarii.  
Item the head of the southern fish called Piscis Meridionalis, and jothis' notios, in whose mouth is a fair Star of the first bigness, called Fomahant.  
But you have to understand that besides the 15. Southern Images before mentioned, there are lately found out by the Portugals and others that have sailed into the East and West Indies 4. other images towards the south Pole, as the cross or Crosier, the south triangle, Noah's dove or Pigeon, & another image made like a Philosopher called Polophilax, all which are set down in the celestial Globe, lately set forth, first at the great charges of M. Sanders, and now at the like charges of M. Molinax of Lambeth of whom I lately bought both the Globes, that is, the terrestrial and celestial, and I wish that the longitude, latitude and declination of every Star contained in the said 4. images were truly set down, for Plancius maketh some doubt thereof. Notwithstanding if you be desirous to know the longitude, latitude, and declination of the said Stars by help of the foresaid great Globe, than you must work as I do show you hereafter in seeking for any Star contained in Mercator his globe, so shall you have your desire.  
Moreover, to most of the Stars described in the Globe are annexed the Characters of some of the 7. Planets, to show the nature & quality of the Stars & some stars are also marked with some one letter or other, the more readily to found out thereby the foresaid characters, As for example to Cor Leonis are annexed the Characters of jupiter and Venus, & under the said Star is set the letter m. to show the Characters which are not always set hard by the Star, but sometime a good distance off, for where the characters are set nigh unto the star, there needeth no letter, as in the Star called spica Virgins, whereunto are annexed the characters of Mars and Venus, without any letter to signify the same, and where divers Stars be of one self quality, they are severally marked with letters of one self same shape, as about the Star Spica Virgins you shall find divers little Stars, each one marked with the letter h. signifying their nature to be all one, that is, to participate of Mars and Venus, to whose characters is also joined the letter h, signifying that they be of that nature and quality.  

The Characters are these here following set over every Planet's head.  ♄ ♃ ♂ ☉ ♀ ☿ ☽ Saturn. jupiter. Mars. Sol. Venus. Mercury. Luna.   

The nature of every one of the Planets here followeth.  
SAturne is cold and dry. jupiter is temperately hot and moist. Mars is extremely hot and dry. Sol hot and somewhat dry. Venus is temperately cold and moist. Mercury is of changeable nature, and pliant to the nature, be it good or bad of every other Planet or fixed Star whereto it is joined. Luna is cold and moist.  
Besides the images and stars both Northern and Southern above mentioned, there is also set down in the celestial Globe a certain impression called in Greek Galaxia, that is to say, the milk white way, the description whereof here followeth.   

A brief description of the milk white way, called in Greek Galaxia, and in Latin Via lactea.  
THis way as Garceus writeth, proceedeth from the sign Gemini, and so passeth through the legs and loins of: Ericthonius, and from thence through the right side of Perseus, and then through the whole image almost of Cassiopeia, and from thence through the left wing of the Swan called avis, Gallina and Cignus, and from thence through the Image called in Latin Telum in English a Dart, shaft, or quarrel, and from thence through the flying Eagle called in Latin Vulture volans, and from thence through the greatest part of Sagittarius his bow, & from thence through the Altar called Ara and Thuribulum, and from thence through the legs of Centaurus, and so to the ship called Argos, from whence rising again, and passing through part of the great Dog called Canis maior, it returneth again to Gemini.  
Thus having described unto you all the 48. Images, and showed the names of as many Stars as are named in the Globe, and also the milk white way, I mind now to proceed to the propositions belonging to the fixed Stars described in the Globe, as followeth.    

How to found out in heaven any unknown Star described in the Globe two manner of ways, that is, either by the help of some known star, or else by knowing the true hour of the night. Proposition. 20.  
THe first and surest way is thus, take with your Quadrant or Astrolabe the altitude of the known Star, marking therewith in what part of the heaven the same is situated East or West, North or South, and then having set your Globe right North and South, and at the true latitude of the place where you are, bring the quarter of altitude to the said Star, and therewith turn the Globe until you see that the said Star hath the like place and the like altitude in the Globe that it had in heaven, then keeping the globe still at that stay, seek in the globe the star that you would found out in the firmament, & mark well in what part of the Globe it is situated, and how it beareth from the known Star, either East, West, North or South, and bring the quarter of altitude to that Star, that you may know the altitude thereof by help of the said quarter, which altitude once taken, turn your eyes towards that part of the firmament, and having placed the Diopter of your Astrolabe, at that altitude, look what Star in that part of the firmament doth answer to such altitude, and that is the Star which you seek, whose name for the most part is set down in the Globe. The second way is thus, having learned the true hour of the night by some clock or watch, bring the degree of the Sun unto the brazen Meridian, & holding it there, set the index of the hour wheel just at the 12. hour, that done, turn the body of the Globe to or fro until the index of the hour whéel fall just upon the hour which you seek, and keeping the Globe at that stay, seek out the unknown Star in the Globe, and consider how it beareth from you in the Globe either East, West, North or South, then bring the quarter of altitude to the Star, that you may know thereby what altitude it hath in the Globe, which once found and having set the Diopter of your Astrolabe at that altitude, turn your eye towards that part of the firmament whereunto the place of the Star found in the Globe directeth you, and that Star which answereth to that altitude is the Star which you seek. But this way is not so sure as the other way first taught, unless you know the true hour indeed.   

How to know by the Globe the Meridian altitude which is the highest altitude of any star, and also how high or low he is at any other time. Proposition. 21.  
Having set the Globe at your latitude, bring the star to the brazen Meridian, and there staying it, number upon the brazen Meridian the degrees contained betwixt the said Star and the South or north point of the Horizon, according as the Star is situated either Northward or Southward, for so in the latitude 52 you shall found the Meridian or highest altitude of the Star Arcturus or Bubulcus to be 60. degrees. But if you would know his altitude at any other time, than you must rectify the index of the hour wheel by the 7. proposition, and having set the index at the hour wherein you seek, stay the Globe there, until you have brought the quarter of Altitude unto the Star, and that will show the altitude of the Star at that hour if it be above the Horizon.   

How to know by the Globe what Stars are above the Horizon at any time of the day or night. Proposition. 22.  
OF Stars according to divers latitudes some are always above the Horizon & some are always under the Horizon, & some do both rise and set, if you would know what Stars he above the Horizon in the day time, then having rectified the Index of the hour wheel by the 7. proposition, take the altitude of the Sun with your Astrolabe or quadrant, and therewith consider whether the Sun be in the East part or in the West part of the firmament. Than bring the quarter of altitude on the East or West side of the Globe according as you saw the Sun at that present to be in the firmament, and make the degree of altitude, marked in the quarter of altitude to meet even with the degree of the Ecliptic line wherein the Sun is that day, and there staying the Globe, you shall see all the stars that be above the Horizon at that present, as well on the East side as on the west side of the Globe, and the Index will show you at what hour you took the aforesad altitude. But if it be in the night season, and that the stars do appear, take with your Astrolabe the altitude of some known star, and by doing as is before taught, you shall have your desire. But you must not forget first of all to rectify the Index of the hour wheel by the 7. proposition.   

How to know by the Globe at what time any star riseth above the Horizon, mounteth to the highest, and setteth, and with what degree of the Ecliptic he riseth, mounteth, and setteth, and also in what part of the Horizon he riseth and setteth. Proposition. 23.  
Having rectified the Index of the hour wheel by the 7. proposition, bring the Star to the East part of the Horizon, so as it may touch the edge thereof, and the Index of the hour wheel will show at what hour he riseth and by looking at that instant to the Ecliptic line you shall see what degree of the Ecliptic riseth then with him. That done, bring the said star to the brazen Meridian, and the Index of the hour wheel will show at what hour he is at the highest, and there staying the Globe, mark what degree of the Ecliptic line doth fall right under the brazen Meridian at that instant, for that degree is said to accompany him when he is at his highest. Than bring the said Star to the West part of the Horizon, and you shall find by the Index of the hour wheel at what hour he setteth, and what degree of the Ecliptic doth accompany him at his setting. As for example, I would know the sixtéenth day of june 1590. the sun being in the fourth degree of Cancer, when Arcturus otherwise called Babulcus and Lanceator doth rise, mounteth to the highest, and setteth, here having rectified the Index of the houre-wheele by the seventh Proposition, I bring the said star of the East part and very edge of the Horizon, and I find that he riseth a little before twelve of the clock at noon, and that the 28. degree and 30′· of Virgo, riseth with him, and by looking amongst the winds upon the Horizon right against the place of his rising, I found that he riseth North-east and by East. Secondly by bringing the said star to the brazen Meridian, the Index of the hourewhéele showeth that he is at his highest half an hour after seven of the clock at night, and is then plain South, and that the 29. of Libra, doth then accompany him. Thirdly by bringing the said star to the West part of the Horizon, the Index of the hourewhéele showeth that he setteth or goeth down a quarter of an hour before four in the morning, and that the fourth degree of Capricornus doth accompany him at his setting: and by looking there upon the Horizon, I found amongst the winds that the said star setteth Northwest and by West.   

How to know in what part of the firmament any star is, and how many degrees it is distant from the Meridian at any hour, and being right under the Meridian to know how far it is distant from your Zenith. The 24. Proposition.  
Having rectified the Index of the hourewhéele by the seventh Proposition, turn the Globe until the Index touch the hour wherein you seek. And staying the Globe there, bring the quarter of Altitude to the star be it on the East or West side of the Globe, and the neither end of the said quarter will show upon the Horizon among the winds, in what part of the firmament the star is: now if you would know how far that star is distant from the Meridian do thus: look what degree of the Equator is at that instant under the Meridian, and there make a mark, & then turn the Globe until you have brought the said star under the brazen Meridian, and mark what degree of the Equinoctial the said Meridian cutteth at that instant, that done, count the degrees contained betwixt the two marks upon the Equinoctial, for so many degrees is the star distant at that time from the Meridian either towards the East or West, and by allowing 15. degrees to an hour, and 4′· to a degree, you shall know in what time the said star hath to approach to the Meridian, or how much he is past the Meridian, and having brought the said star right under the Meridian, you shall know how far it is distant from your Zenith, by counting the degrees that are contained in the Meridian betwixt the said star and your Zenith: As for example, the sixth of October 1591. (the sun being then in the 22. of Libra) I find the star Cor Leonis at three of the clock in the after noon to be West, Northwest, & there staying the Globe, I see that the Meridian cutteth the Equinoctial at the 246. degree, whereas I make a mark, that done, I bring the star Cor Leonis to the said Meridian at which instant the Meridian cutteth the Equinoctial in the 146. degree, which being taken out of 246. there remaineth 100 degrees, which is his distance from the Meridian, which being divided by 15. I found in the quotient 6. hours and 40′·S for so much as he is past the Meridian towards the West, and by bringing the foresaid star to the Meridian, I find him to be distant from our Zenith 39 degrees, and 30. minutes.   

How to found the hour of the night by the Globe. The 25. Proposition.  
Having set the Globe at your Latitude, rectify the Index of the hourewhéele by the 7. Proposition, then having taken the Altitude of some star that you know, and is in the Globe with your Astrolabe or Duadrant, bring the quarter of Altitude unto the star, be it in the East or West, according as you found the star to be in the firmament, not leaving to turn the Globe until you have made the star to have the like Altitude in the Globe upon the quarter of Altitude, and also the like situation that you found it to have in the firmament by your Astrolabe or Duadrant, & staying the Globe there, the Index of the hourewhéele will show the hour: As for example, in the year 1590. the first of januarie the sun being in the 22. of Aquarius, I having with my Astrolabe found that the star called Canis maior, that is to say the greater dog, was elevated above the Horizon in the East part of the firmament 20. degrees, I brought the quarter of Altitude to the East side of the Globe, not leaving to turn the Globe until I had made the star to meet even with the 20. degree of the quarter of Altitude, and there staying the Globe, I found by the Index of the hourewhéele that it was 8. of the clock at night, and a quarter past.   

How to know the vertical stars in every Latitude. The 26. Proposition.  
THe Globe and the brazen Zenith being set according to the Latitude of the place where you are, turn the Globe from East to West, and as many stars as pass right under your Zenith are said to be vertical as in the Latitude 52. you shall see the tail of the great Bear, the head of Perseus, and diverse others to pass through your Zenith in turning the Globe from East to West.   

How to know the true place of any star, that is to say, in what sign and in what degree there of any star is. The 27. Proposition.  
Having fastened the semicircle of Longitude and Latitude upon the two Poles of the zodiac, the North pole whereof is in the Circle Arctique not far from the right claw of the Dragon, and the South pole thereof is in the circle Antarctique right opposite to the other, then bring the said semicircle to the star, whose place you seek, and mark therewith what point of the Ecliptic the said semicircle cutteth, and that is the place of the star. As for example, by bringing the said semicircle to the star called Hircus, that is to say, the Goat, I find it to be in the 15. degree 30. minutes of Gemini, and that is his place.   

How to found the Longitude of any star by the Globe. The 28. Proposition  
I Have told you in my Sphere, that the Longitude of any star is that arch or portion of the Ecliptic line, which is contained betwixt the first point of Aries, and that Circle which passeth through the Poles of the zodiac, and also through the body of the star, which Circle, the semicircle of Longitude and Latitude here representeth: and by making the said semicircle to pass through the star called Hircus before mentioned, I find that the number of degrees of the Ecliptic, contained betwixt the said Circle and the first point of Aries to be 75. degrees & 30′·S which is the Longitude of the said star, and thereby maketh his place to be in the 15. degree 30′· of Gemini, as before is set down in the last Proposition.   

How to found out the Latitude of any star. The 29. Proposition.  
I Told you also in my Sphere, that the Latitude is none other thing, but the distance of any star from the Ecliptic line, either towards the North or South pole of the zodiac, which distance the semicircle of Longitude and Latitude made to pass through the body of the star, and cutting the Ecliptic line doth always show, as in the former example, in making the foresaid semicircle to pass through the foresaid star called hircus, I found by counting the degrees of the said semicircle contained betwixt the Ecliptic line and the body of the star, that the Latitude of that star is 22. degrees and 30. minutes towards the North, likewise by bringing the said semicircle to the star which is in the right shoulder of Orion called Bed Alguze, I found his Latitude to be 17. degrees towards the South.   

How to found out by the Globe the declination of any star. The 30. Proposition.  
THe declination is none other thing but the distance of any star from the Equinoctial, either towards the North pole or South pole of the world, which is to be found thus: First having brought the star right under the brazen Meridian, and there staying the Globe, count the degrees of the said Meridian contained betwixt the said star and the Equinoctial point or stréeke of the said Meridian, and that shall be the declination of the star: As for example, bring the star Hircus unto the Meridian, and you shall found the declination thereof to be 45. degrees towards the North pole of the world, and that the star which is in the foot of the Hare called Lepus, hath of South declination 23. degrees.   

How to know the magnitude or greatness of any star, and his nature and quality by the Globe, and also the right ascension of the Arch of the Ecliptic which accompaneth the right ascension of any star. The 31. Proposition.  
MArcator describeth the six magnitudes of the stars by making six figures or shapes of stars placed not far from the head of the great bear, whereof the first and greatest hath 16. points or beams, the second eight, the third six, the fourth five, the fift six, which indeed by that account would be but four: and the sixth hath five, which would have but three: but in mine opinion it had been better to have made the first magnitude with ten points, the second with nine, the third with eight, the fourth with seven, the fift with six, and the sixth with five: so should the magnitude of every star described in the Globe have been the more easily known. But to the intent that you might exercise yourself in finding out by the Globe, the place, Longitude, Latitude, and declination of any star that is described in the Globe: I have thought good to add hereunto the Table of Garceus, showing not only the Longitude, Latitude and declination of the most notable stars that are both Northward and Southward, but also the right ascension, magnitude or bigness, the quality or nature of every such star, and also the Arch of the Ecliptic line, which accompanieth the right ascension of every star, which Table though by the said Garceus was calculated out of the Astronomical Tables for the year of our Lord 1564. and not by the Globe, yet for your better exercise in matters of the Globe, I thought good to set down such Longitude, Latitude, declination, magnitude, and right ascension, and all other things contained in the said Table, according as they are to be found out by the Celestial Globe of Mercator, and not calculated by any of the Astronomical Tables.  
Though the Longitudes and declinations of the fixed stars set down in this Table to serve Mercator his Globe, do not altogether agree with the great Celestial Globe, lately set forth by M. Sanderson and by M. Molinaxe, by reason that the Longitudes and declinations of the said stars are mere lately calculated, that is to say, for the year of our Lord, 1592. yet it will serve to show you the way how to exercise yourself in the said Globe, and thereby you may correct this Table, and make it to agree in all points with the new great Globe, whereby you shall reap more pleasure than grief or pain. For I had this Table (as I have said before) out of Garceus, and did here set it down more for your exercise, and to acquaint you with the fixed stars that are described in the Celestial Globe, then for any other purpose.  

The use of the Table next ensuing.  
THis Table is divided into eight collums, The first whereof on the left hand, containeth the names of the stars, the second the degrees and minutes of Longitude, together with the Characters of the 12. signs: the third the degrees and minutes of Latitude both Northern and Southern, the North Latitude being marked with the letter N. and the South Latitude with the letter S. the fourth containeth the degrees and minutes of declination both Northern and Southern, marked with the letters N. and S. as before. The fift showeth the magnitude or greatness of the star, whether it be of the first, second or third bigness, etc. The sixth containeth the degrees and minutes of the right ascension of the foresaid fixed stars. The seventh containeth the Characters of the Planets, signifying the nature and quality of the stars. The eight containeth the degrees and minutes of the right ascension of the Ecliptic line and the signs of the zodiac.  
Here followeth the Table.  
The names of the stars. The Longitude & place. The signs. The Latitude North & South. Declination Nor. & South Magnitude or greatness The right Ascension Degr. Min. The nature of the stars showed by the charrect● of the Planets The right ascension of the arch Ecliptic acco●anin●●●● star ●he 〈◊〉 of the s●gnes. The first star of the Rams horn. 27 40 ♈ 7 40 Nor. 17 20 Nor. 3 23 0 ♂ ♄ 25 0 ♈ The right shoulder of Cepheus 8 0 ♈ 69 0 Nor. 61 0 Nor. 3 317 15 ♄ ♃ 15 0 ♒ The last star of Eridanus. 22 30 ♈ 53 30 Sou. 41 0 Sou. 1. 43 30 ♃ ♀ 15 50 ♉ The shoulder of Andromeda. 16 20 ♈ 24 30 Nor. 28 26 Nor. 3 5 0 ♀  5 0 ♈ The girdle of Andromeda. 24 50 ♈ 26 30 Nor. 33 0 Nor. 3 12 10 ♀  13 0 ♈ The right side of Perseus 25 0 ♉ 30 0 Nor. 48 0 Nor. 2 43 0 ♄ ♃ 15 29 ♉ The head Algol or Meduza. 20 40 ♉ 23 0 Nor. 39 30 Nor. 2 40 20 ♄ ♃ 13 0 ♉ The South star of Pleiades. 23 29 ♉ 4 20 Nor. 22 0 Nor. 5 50 10 ♂ ☽ 23 10 ♉ The North star of Pleiades. 23 10 ♉ 5 0 Nor. 23 10 Nor. 5 50 0 ♂ ☽ 22 20 ♉ The bulls eye. 2 50 ♊ 5 0 Sou. 15 30 Nor. 1 63 0 ♂  4 20 ♊ The right shoulder of Orion. 23 0 ♊ 17 0 Sou. 6 20 Nor. 1 83 20 ♂ ☿ 23 50 ♊ The left shoulder of Orion. 12 0 ♊ 17 0 Sou. 4 40 Nor. 2 72 0 ♂ ☿ 13 0 ♊ The right shoulder of Auriga 24 0 ♊ 20 0 Nor. 43 0 Nor. 2 82 0 ♂ ☿ 22 50 ♊ The Goat, or Hireus'. 16 0 ♊ 22 30 Nor. 45 0 Nor. 1 71 30 ♂ ☿ 13 0 ♊ The former of the Kids 13 0 ♊ 18 0 Nor. 40 0 Nor. 4 68 50 ♂ ☿ 10 28 ♊ The latter of the Kids 13 30 ♊ 18 0 Nor. 39 40 Nor. 4 69 0 ♂ ☿ 11 0 ♊ The left foot of Orion. 10 10 ♊ 31 30 Sou. 9 0 Sou. 1 73 0 ♃ ♄ 14 20 ♊ The tail of the little dear 21 10 ♊ 66 0 Nor. 86 0 Nor. 3 4 30 ♄ ♀ 4 30 ♈ The nuddle star of Orion's girdle. 18 0 ♊ 24 30 Sou. 1 50 Sou. 2 79 25 ♃ ♄ 20 20 ♊ Canopus in the ship Argo. 8 30 ♋ 75 0 Sou. 52 0 Sou. 1 94 24 ♄ ♃ 3 30 ♋ The great dog. 9 0 ♋ 39 10 Sou, 16 0 Sou. 1 97 0 ♂ ♃ 6 30 ♋ The little dog. 20 30 ♋ 16 0 Sou. 6 10 Nor. 1 109 30 ♂  18 0 ♋ The former head of Gemini. 14 20 ♋ 9 40 Nor. 32 0 Nor. 2 106 47 ☿  15 28 ♋ The following head of Gemini 17 40 ♋ 6 30 Nor. 28 30 Nor. 2 110 5 ♂  18 20 ♋ The North Asellus. 1 30 ♌ 3 0 Nor. 22 30 Nor. 4 124 0 ♂ ☉ 1 40 ♌ The South Asellus in Cancer. 2 20 ♌ 0 10 Sou. 19 20 Nor. 4 124 50 ♂ ☽ 2 0 ♌ Praesepe. the Manger. ●n Cancer. 1 20 ♌ 0 40 Nor. 20 0 Nor. neb. 124 0 ♂ ☽ 1 30 ♌ The shoulder of the great dear. 11 40 ♌ 49 0 Nor. 62 30 Nor. 2 161 0 ♂  9 30 ♍ ●he bright star of Hydrus. 21 0 ♌ 20 30 Sou. 5 0 Sou. 2 137 0 ♄ ♂ 14 30 ♌ The lions heart 23 30 ♌ 0 10 Nor. 13 30 Nor. 1 145 93 ♃ ♂ 23 30 ♌ The lions neck. 23 10 ♌ 8 30 Nor. 21 40 Nor. 2 148 40 ♄  26 0 ♌ The lions tail. 15 30 ♍ 12 0 Nor. 16 20 Nor. 1 171 10 ♄ ♀ 20 30 ♍ The middle star of the great bears tail. 9 50 ♍ 56 0 Nor. 56 55 Nor. 2 196 0 ♂  17 0 ♎ The right wing of Virgo. 3 10 ♎ 15 10 Nor. 12 30 Nor. 3 189 0 ♄ ☿ 9 40 ♎ The left shoulder of Bubulcus. 10 0 ♎ 49 30 Nor. 40 0 Nor. 3 213 0 ♄ ☿ 5 0 ♏ The Ranens bill. 6 30 ♎ 21 40 Sou. 22 30 Sou. 3 176 43 ♄  26 20 ♍ Arcturus the great star betwixt Bubulcus his legs 18 0 ♎ 32 0 Nor. 22 0 Nor. 1 209 0 ♃ ♂ 1 0 ♏ Spica virgins. 17 40 ♎ 12 0 Sou. 10 0 Sou. 1 195 10 ♂ ♀ 16 30 ♎ The middle star in the front of Scorpio 26 40 ♏ 1 30 Sou. 21 0 Sou. 3 234 0 ♂ ♄ 26 30 ♏ The South balance of Libra. 9 0 ♏ 0 40 Nor. 14 0 Sou. 2 217 0 ♄ ♂ 9 20 ♏ The North star of the balance Libra. 13 10 ♏ 9 0 Nor: 8 0 Sou. 2 223 10 ♃ ☿ 15 46 ♏ The left hand of Serpentarius. 27 0 ♏ 17 0 Nor. 3 30 Sou. 3 238 30 ♄ ♀ 30 0 ♏ The bright star of Ariadne's crown. 5 30 ♏ 44 30 Nor. 28 0 Nor. 2 229 0 ♀ ☿ 21 20 ♏ The heart of the Scorpion. 3 30 ♐ 3 30 Sou. 25 0 Sou. 2 241 0 ♃ ♂ 3 0 ♐ The head of Hercules. 8 40 ♐ 37 30 Nor. 15 0 Nor. 3 252 35 ☿  14 0 ♐ The head of Serpentacius. 15 30 ♐ 36 0 Nor. 12 50 Nor. 3 258 30 ♂ ☿ 19 10 ♐ The head of the Dragon 20 20 ♐ 76 0 Nor. 52 0 Nor. 3 266 10 ♄ ♂ 26 30 ♐ The bright star of Lyra 9 0 ♑ 62 0 Nor. 38 20 Nor. 1 275 20 ♀ ☿ 5 0 ♑ The Eagle, alias vulture volans. 24 50 ♑ 29 20 Nor. 7 30 Nor. 2 210 40 ♄ ♂ 20 4 ♑ The first star of the tail of Capricorn. 16 0 ♒ 2 0 Sou. 18 0 Sou. 3 319 0 ♃ ♄ 17 0 ♒ The following star thereof. 17 10 ♒ 2 0 Sou. 17 30 Sou. 3 320 10 ♃ ♄ 18 0 ♒ The tail of the Dolphin. 8 10 ♒ 29 30 Nor. 10 0 Nor. 3 304 0 ♄ ♂ 1 10 ♒ The Fomahant. 28 0 ♒ 23 0 Sou. 33 10 Sou. 1 339 0 ♀ ☿ 7 0 ♓ The point of the dart. 1 10 ♒ 39 30 Nor. 18 10 Nor. 4 295 0 ♂ ♀ 23 8 ♑ The tail of the Swan, 0 20 ♓ 60 0 Nor. 43 30 Nor. 2 307 10 ♂ ☿ 5 0 ♒ The right thigh of Pegasus. 22 50 ♓ 31 0 Nor. 25 0 Nor. 2 341 0 ♃ ♂ 9 30 ♓ The tail of the Whale. 27 0 ♓ 20 20 Sou. 19 50 Sou. 3 5 35 ♄  6 0 ♈    

How to find out the right and obliqne ascension of any star, and also of the ascentionall difference. The 32. Proposition.  
THe right ascension of any star is an arch or portion of the Equator, to be counted from the first point of Aries according to the succession of the signs, with which portion in a right Sphere any star both riseth mounteth to the Meridian, and setteth: and in an obliqne Sphere it is a portion of the Equator, wherewith the star is mounted to the Meridian: as for example, in a great Sphere the star called Cor Leonis, that is to say, the hart of the Lion, both rises, mounteth and setteth with the 145. degr. 30′· of the Equinoctial. But the right ascension of the said star in an obliqne Sphere, is to be found only by bringing the said star to the Meridian, and you shall found it to be all one with the right ascension in a right Sphere, for by bringing the star called Cor Leonis to the Meridian in an obliqne Sphere, you shall find the right ascension thereof to be all one with that which it had in a right Sphere, that is 145. degrees 30′· of the Equinoctial. But if you would know the obliqne ascension of any star, then having set the Globe at your Latitude, bring the star to the East part of the Horizon, & mark what degree of the Equinoctial riseth therewith and that is the obliqne ascension: As for example, the Globe standing at the Latitude 52. bring the star, Cor Leonis to the East part of the Horizon, & by staying the Globe there, you shall found the 127. degree of the Equinoctial to rise with that star. Now if you would know the ascentionall difference, that is to lay, the difference betwixt the right and obliqne ascension, you have no more to do but to subtract the lesser out of the greater, and the remainder shall be the ascentionall difference, as in the former example, take 127 degrees out of 145. degrees and 30′ and there shall remain 18. degrees 30′· and that is the ascentionall difference, by help whereof you may know the increase and decrease of the artificial day and night throughout the year in any Latitude, if you observe that order which I have already set down in the first part of my Sphere, the 50. Chapter.   

To know in what quantity of time any whole sign or any other Arch of the Ecliptic doth rise or set. The 33. Proposition.  
Having set the Globe at your Latitude, and rectified the Index of the hour wheel by the seventh Proposition, bring the beginning of any Arch or sign to the East part of the Horizon, and mark what degree of the Equinoctial riseth therewith, that done, bring the end of the said sign or arch to the East part of the Horizon, and mark there also what degree of the Equinoctial toucheth the Horizon, and staying the Globe there, look how many hours or parts of hours the Index of the hourewhéele hath run betwixt the beginning and ending of the said sign or arch, and so you shall know the quantity of that time, you may know it also by the number of degrees of the Equinoctial contained betwixt the beginning and ending of the said sign or arch by allowing 15. degrees to an hour, and 4′· to a degree. As for example, supposing the sun to be in the first degree of Taurus, you shall find by working according to the rule before set down, that the whole sign of Taurus doth spend in rising one hour 8′· Now if you will know how much time he spendeth in descending, bring the first degree of Taurus to the West part of the Horizon, marking what degree of the Equinoctial toucheth the Horizon at that instant, and also to what hour the Index pointeth, then turn the Globe still Westward until the last degree of Taurus meeteth even with the edge of the Horizon, and then mark again as well the degrees of the Equinoctial that toucheth the Horizon, as also to what hour the Index pointeth, and you shall found the number of degrees of the Equinoctist to be 42. degrees, which maketh two hours 48. minutes, which is answerable to the hour of the hourewhéele, and so much time the whole sign of Taurus spendeth in his descension or going down.   

How to know by the Globe what stars do rise or set every day Cosmically, Acronically, or Helically. The 34. Proposition.  
AS for the threefold Poetical rising and setting of the stars, you shall found them plainly defined in the first part of my Sphere, Chap. 35. but to find out the same by the Globe, you must do thus. First having set your Globe at your Latitude, and sought out the place of the sun for that day, bring the degree of the sun to the East part of the Horizon, and stay the Globe there, that you may see what stars do rise a little before the sun, and which rise together with the sun. For those that rise a little before the sun, are said to rise Helically, and those that rise together with the sun, are said to rise Cosmically, and those stars that are in the very West part of the Horizon at the rising of the sun, are said to set Cosmically. Again those stars that rise immediately after the sun, do set Helically, that done, turn the degree of the sun unto the West part of the Horizon, and staying the Globe there mark what stars are ready to go down with him, for those are said to set Acronically, ans staying the Globe still there in the West, mark what stars at that present do rise in the East part of the Horizon, for those are said to rise Acronically.   

To know in what time of the year any star riseth or setteth, either Cosmically, or Acronically. The 35. Proposition.  
HEre having set the Globe at your Latitude, and knowing the degree of the sun, bring the star to the East part of the Horizon, and therewith consider what degree of the Ecliptic the Horizon cutteth at that present, that done, find out the self same degree upon the Horizon in the narrow space of degrees next unto the body of the Globe, and right against that degree, you shall find in what day and month that star doth rise Cosmically, As for example, I would know at what time of the year Cor Leonis riseth Cosmically in the Latitude 52. I bring the star Cor Leonis to the East part of the Horizon, and I find that the Horizon cutteth the Ecliptic in the 23. degree 30′· of Leo, which degree being found again upon the Horizon, pointeth to the sixth day of August, so as I conclude that Cor Leonis doth rise that present day Cosmically, for then both he and the sun are in a manner in one self degree of the Ecliptic: now to know the Cosmical setting of the said star, turn the same star to the West part of the Horizon, and mark what degree of the Ecliptic doth then rise in the East, and you shall find the same to be the 23. degree of Aquarius, which degree being found again upon the Horizon in the narrow space of degrees next to the body of the Globe, containing the degrees of the zodiac, will point to the 31. day of januarie, at which time the sun is opposite to the said star, and therefore it is said to go down Cosmically, because it goeth down when the sun riseth. Now to know the Acronical rising of any star at any time, bring the star to the East part of the Horizon, and mark therewith what degree of the Ecliptic goeth down in the West at that instant, for the sun being in that degree is opposite to the star: As for example, by bringing the star Cor Leonis to the East part of the Horizon, you shall found that the 23. degree of Aquarius goeth down at that instant, which degree being found again upon the Horizon, will show the day and month when the star riseth Acronically, and so you shall found the star Cor Leonis to rise Acronically the 31. day of january. Contrariwise if you would know when the said star setteth Acronically, bring the said star Cor Leonis, to the West part of the Horizon, & therewith mark what degree of the Ecliptic than setteth in the West, which degree being found again upon the Horizon, will show that the said star setteth Acronically the 6. of August.   

Of the Horoscope and the rest of the twelve houses. The 36. Proposition.  
THis word Horoscope doth not only signify the degree of the Ecliptic, otherwise called the ascendent which riseth above the Horizon in the beginning of any thing that is to be sought or known, but also sometimes the whole figure of heaven containing the 12. houses, and doth show the very secrets of nature, so that there is nothing that chanceth to the inferior bodies, but some cause thereof doth appear by mean of the Horoscope in heaven, and therefore the Astrologians have divided the whole heaven into 12. houses, which are numbered from the Horoscope, which is the East Angle, and so forth according to the succession of the signs, of which 12. houses the four principal are four points of the zodiac whereof two do fall upon the Horizon, and the other two upon the Meridian, and are called principal points, poles, or Angles, that is the beginning of the first house, of the fourth house, of the seventh house, and of the tenth house, and those that do follow next any of these principal Angles, are called succeeding houses, in Latin Succedentes, as the second, the fifth, the eight, and the eleventh house. And those that go next before any of the four principal Angles, are called falling houses, in Latin Cadentes, as the 12. the third, the sixth, and the ninth: and such houses as have no familiarity with the Horoscope or ascendent, as the second, the sixth, the eight, and the eleventh houses are said to be slow and deject, all which things this Table here following doth show, containing the number and names of the houses, and also their significations.  
The 12. houses. The names of the houses. The significations of the houses. 1 Angle East life 2 succeeding the lower gate gain 3 falling the Goddess brethren 4 Angle the bottom of heaven parents 5 succeeding good fortune children 6 falling evil fortune health 7 Angle the West wife 8 succeeding the higher gate death 9 falling God religion 10 Angle the middle of heaven kingdom 11 succeeding the good spirit benefactor 12 falling the evil spirit prison  
A general figure of the 12. houses of Heaven, according to the judicial of Astrology.  


1 the horoscope or ascendent. ♈  
2 the house succedent ♉  
3 the house cadent. ♊  
4 the angle of the earth. ♋  
5 house succedent. ♌  
6 house cadent. ♍  
7 the angle of the occident. ♎  
8 house succedent. ♏  
9 house cadent. ♐  
10 the angle meridional. ♑  
11 house succedent. ♒  
12 the house cadent. ♓     

How to find out the Horoscope or ascendent at any time of the day or night by the globe, and thereby to know the 4. principal angles of heaven. Proposition. 37.  
FIrst having set the Globe at your latitude, and rectified the index of the hour wheel according to the degree of the sign where in the Sun is that day you seek by the 7. proposition, if it be in the day time, take with your Astrolabe or Quadrant. the altitude of the Sun. But if it be in the night, take the altitude of some known Star, that thereby you may know the hour of the day or night in which you seek the ascendent. But if it happen that neither Sun nor star is to be seen that day or night, then learn by some true clock or watch what hour it is, and having set the Index of the hour wheel at that hour, stay the Globe there, and therewith mark what degree of the Ecliptic riseth in the East part of the Globe above the horizon at that instant, and that degree is the Horoscope or ascendent for that hour. As for example, you would found out the ascendent the 16. of june 1590. at 8. of the clock in the forenoon, at which day the Sun is in the 4. degree of Cancer, here having first set the Index of the hour wheel at that hour by staying the Globe there, you shall find that the 21. degree 30′· of Leo is the ascendent, which is the East angle or first house, whereby you may also at that instant found out the other three angles, that is, the west, South, and North angles, for the opposite point of the zodiac to the ascendent is the West angle or seventh house. And that degree of the zodiac, which is at that instant right under the Meridian above head in the South angle or 10. house, and the opposite point to that beneath, is the north angle or fourth house, for having found the ascendent, which is the East angle or first house to be the 21. degree, 30. minutes of Leo, the West angle must needs be the 21. degree. 30. minutes of A quarius. Again ●he south angle or 10. house is the 8. degree of Taurus, and the opposite point to that is the North Angle or fourth house which is ●he 7. of Scorpio.   

How to erect a figure by the Globe according to Regio Montanus his way which is called the reasonable way, and is counted the best of all others. Proposition. 38.  
FIrst you must found out the degree of the ascendent as is taught in the last chapter, which is always the first house, then staying the Globe with some pretty wedges of wood being thrust betwixt the Horizon and the body of the Globe, mark there what degree of the equinoctial doth touch the Horizon at that instant, and number from thence upward upon the said Equinoctial 30. degrees, to the end of which 30. degrees, bring the semicircle of position being first fastened in his due place upon the East side of the Horizon, and look what degree and sign of the zodiac the circle of position cutteth at that present, and that degree shall be the twelfth house: then number again other 30. degrees upon the Equinoctial upwards towards the brazen Meridian, and to that bring the semicircle of position, marking what degree of the zodiac the said semicircle cutteth, and that shall be the 11. house, that done, look what degree of the zodiac is right under the brazen meridian above head, and that shall be the 10. house. Than having set the semicircle of position upon the West side of the Globe, number from the brazen Meridian westward upon the Equinoctial other 30. degrees, to the end whereof bring the semicircle of position, and mark what degree of the zodiac the circle of position cutteth, and that shall be the 9 house, then from thence downward number upon the Equinoctial other 30. degrees, to the end whereof bring the semicircle of position, marking there what sign and degree of the zodiac the said semicircle cutteth, and tha● shall be the 8. house. Now having these six houses, the opposite points of the said six houses will show you the other six houses, and if you will know which houses, and also which signs are opposite one to another, mark well this Table following.  
The houses opposite are these, 1 to 7 The Signs opposite are these, ♈ to ♎ 12 6 ♉ ♏ 11 5 ♊ ♐ 10 4 ♋ ♑ 9 3 ♌ ♒ 8 2 ♍ ♓  
And to make all these things the more plain unto you, I thought good to set down this example following. Suppose that the 16. of june, Anno 1590. and at 8. of the clock in the morning you would erect a figure to know how the 12. houses of heaven are situated at that present, first having drawn such a square figure as this here following, representing the 12. houses, learn by the last proposition, who is the ascendent at that instant, and you shall find find it to be the 21. degree 30′· of Leo, which must be set in the first house, and the 28. degree of Cancer in the 12. house and the 22. degree of Gemini in the 11. house, and the 9 degree of Taurus in the 10. house, and the 5. degree of Aries to be in the 9 house, and the 12. degree of Pisces in the 8. house. Now the opposite house to the first house or ascendent is the 7. house which is the 21. degree. 30′· of Aquarius, for the opposite sign must always have like number of degrees, than the opposite to the 12. house is the 6. house, which the 28. deg of Capricorn, and the opposite to the 11. is the 5. house, which is the 22. degree of Sagittarius, and the opposite to the 10. house is the 4. house, which is the 9 degree of Scorpio, and the opposite to the 9 house in the 3. house, which is the 5. degree of Libra, and the opposite to the 8. house is the second house which is the 12. degree of Virgo, all which things this figure here following doth plainly show. And if you would know what Planets should be placed in every house, you must learn that out of the Ephemerideses, or out of some of the Astronomical Tables.  
16. junii, Anno 1590. the Sun being in the 4. degr. of Cancer, in the latitude 52.  
Now because most men that do show the use of the Globes, do also teach therein how to find out the latitude and longitude of any region, I thought good therefore to set down here some of their manifold ways touching the finding out of the same, notwithstanding that I have already written something thereof in the second part of my Sphere, in the 8. 9 10. & 11. chapters.   

How to know the latitude of any place or region by any of the fixed stars described in the Globe. Proposition. 39  
TAke in the night season with your Astrolabe the Meridian altitude of some known star that is to be found in the Globe, then having brought the star under the brazen Meridian, turn the Meridian up and down in the nickes of the horizon, until the same Star have the said altitude in the brazen Meridian, which you found it to have in the firmament by your Astrolabe, that done, number the degrees of the Meridian contained betwixt the Pole and the Horizon, and that is the latitude of that place.   

Another way to found the elevation of the Pole. Proposition. 40.  
Having brought the Globe into an open place where the Sun shineth at nonnetide, and placed the same right North and south as is taught in the first chapter, set a needle in the degree of the Sun, and bring the same to the brazen Meridian, not leaving to turn the same Meridian up and down in the nicks of the Horizon, until the needle cast no shadow at all, and there staying the Globe, look how many degrees the pole is elevated above the Horizon, for that is the latitude of that place.   

A third way to find out the latitude of any place without taking the Meridian altitude of any Star. Proposition. 41.  
TAke at any hour the altitude of 2 known stars, and such as are to be found in the Globe at one self instant whereof the one must be situated towards the East, and the other towards the West, then turn the Globe together with the Meridian up and down in the nicks of the Horizon, until you find by help of the quarter of altitude each Star to have the self same altitude in the Globe, that it had in heaven, which being done, look how many degrees the North pole is elevated above the Horizon, and that is the latitude of that place. But if you would know the latitude of any place that is towards the South Pole, than you must first place the South pole above the Horizon, and then work as before.   

A fourth way to found out the latitude of any region by any known fixed star or Planet that may be seen. Proposition. 42.  
FIrst take his Meridian altitude, and then learn to know either by the Globe or by some table his declination, which if it be Northern you must subtract the same from the Meridian altitude, if Southrens add it to the Meridian altitude so shall you have the elevation of the Equinoctial, which being subtracted from 90. the remainder shall be the latitude of that place as for example. I found the Meridian altitude of Oculus Tauri to be 53. degrees, 50′· 32″· and his declination Northward to be 15. degrees, 50′· 32″·S which declination being subtracted from the Meridian altitude, there remaineth 38. degrees, and that is the elevation of the Equinoctial, which being subtracted from 90. there remaineth 52. for the latitude of that place.  
Hitherto I have set down the chiefest propositions that are to be done by the Globe, touching the Sun and the fixed stars: & lastly, showed how to found out the latitude of any place, wherefore now I think good to show you how to found out the place of the Moon, and of every one of the rest of the Planets in the Globe by help of the Ephemerideses, and thereby to know when every planet riseth and setteth, and first of the Moon.   

A brief description of the diurnal table set down in Stadius his Ephemerideses, together with the use thereof. Proposition. 43.  
But for as much as the diurnal Table of the Ephemerideses showing the daily motion of the Planets is very needful to serve divers turns I think it not amiss here briefly to describe the same, and specially that of johannes Stadius whose diurnal table or Almanac beginning at the 202. page of his book, and at the year of our Lord 1583. continueth to the year 1606.  
Of which table every page on the left hand is divided into 9 Collums. In the first collum whereof on the left hand are set down the days of the month, first the Gregorian days according to the Roman account, and next to that the days of the month according to our English account, then in the front of every other collum are set down the characters first of the Sun, and then of other the six Planets that is to say, of the Moon, Saturn, jupiter, Mars, Venus, Mercury, and last of all, the head of the Dragon figured thus ♌ And right under these seven Planets, and also under the head of the Dragon are set down the signs and degrees wherein every of these is every day of the month throughout the year at noontyde, and in the foot of the said Table is set down the latitude of every one of the 5. Planets, proceeding by the days of the month divided into three parts. And in the margin of every left page are set down the chiefest feasts and Saints days that fall in every month throughout the year. Moreover, there is a Table on the right hand right against the left Table, in which are set down first the days of the month, and then what conjunction or any other aspect the Moon hath with any of the other six Planets, that is, with the Sun, with Saturn, with jupiter, with Mars, with Venus and with Mercury, which Planets are set down in the front of the said Table, and under them the characters of such aspects as the Moon hath that day with any of the other Planets. The characters of which aspects are these here following.  
☌ ☍ △ □ ⚹  
Whereof the first signifieth a conjunction: the second an opposition, the third a trine aspect, the fourth a quadrat aspect, and the fift a sextile aspect.  
Two Planets are said to be in a conjunction when they are both in one self sign. And to be in an opposition when they are in two several signs opposite one to another. For than they be distant one from another 6. signs. And they are said to be in a trine aspect when they be distant one from another by four signs. And to be in a quadrat aspect when they are distant one from another by three signs. And to be in a sextile aspect when they are distant but two signs one from another.   

How to find out the place of any Planet by the Ephemerideses. Proposition. 44.  
NOw to found out the place of any Planet, or of the head of the Dragon by this diurnal Table, you must first seek out the day of the month in the fist collum of the left table, right against that on your right hand in the said left Table, you shall find in the common angle right under the Planet or Dragon's head (which so ever of them you seek) the sign and degree wherein the said Planet or Dragon's head is the said day at noontyde.  
And to found out the aspects which the Moon hath with any of the Planets the same day, you must resort to the other table on the right hand, observing like order as before.  

The example.  
As for example the 21. of April 1592. which is the first of May, according to the Roman account, I found by the table on the left hand, the Sun to be in the 10. degree 50′· of Taurus, the Moon to be in the 30. degree, 47′· of Capricorn, Saturn, to be in the 8. degree, 10′· of Cancer, jupiter to be in the 18. degree. 28′· of Sagittarius, Mars to be in the 12. degree, 6′· of Gemini, Venus to be in the 2. degree, 0′· of Aries, and Mercury to be in the 6. degree, 20′· of Taurus, and the head of the Dragon to be in the 29. degree, 45′· of Gemini,  
And right against this in the Table on the right hand you shall found the Moon to be in a trine aspect with the Sun, to be in an opposition with Saturn, to be in a trine aspect with Mercury.  
Thus having briefly showed you the use of the diurnal Table, I will show you now how to found out the latitude of the Moon as well North as South, by help of Stadius his Table, set down in the 112. page of his Ephemerideses, and first I will briefly describe the said Table.    

A brief description of the Table of Stadius set down in the 112. page of his Ephemerideses to find out thereby the daily latitude of the Moon be it North or South together with the Canon or rule thereof plainly declared by example. Proposition. 45.  
THis Table is divided into 8. collums, whereof the first on the left hand containeth the degrees of every sign set down in the front of the Table, which degrees are to be counted descending from one to 30. for so many degrees there be in every sign: and the last collum on the right hand containeth the like number of degrees belonging to the signs set down in the base or foot of the said Table, and this number ascendeth upward from 1. to 30. and for that purpose it would not have been amiss to have set over each head of those 2. collums this word gradus, next under the word Signa. And of the other six collums the first three on the left hand do contain the degrees, minutes and seconds of the North latitude, and the other three towards the right hand do contain the degrees, minutes, and seconds of South latitude. Moreover, the 12. Signs are to be numbered in the front from the third collum on the left hand from 1. to 5. forward toward the right hand, and at the foot from 6. to 11. backward towards the left hand, set down in arithmetical figures.  

The rule or Canon together with a plain example showing the use of the Table.  
FIrst knowing the day of the month, resort unto the diurnal Table of motion of the Planets in the Ephemerideses, and having there found out the motion or place of the Moon, and also of the Dragon's head answerable to the day wherein you seek, subtract the place of the Dragon's head from the place of the Moon, which is easily done so often as the ark of the Moon is greater, that is to say, containeth more signs and degrees than the ark of the Dragon's head, beginning your account in both arks from the first point of Aries. But if the ark of the Moon be lesser than the ark of the Dragon's head, so as you cannot make your subtraction, than you must add to the place of the Moon 12. signs, which is 360. degrees, and you must add also thereunto the number of so many signs as are contained betwixt the first point of Aries, and the first point of that sign wherein the Moon is at that present, which sign itself is not to be numbered, and when you come to take out of that whole sum the place of the Dragon's head, you must first add to the said place of the Dragon's head the number of so many signs as are contained betwixt the first point of Aries, and the first point of that sign wherein the Dragon's head is at that instant, but not the sign wherein it is, and then having made your subtraction, remember always to take out of that remainder 90. degrees, which is three signs, so often as you have need to add 12. signs to the place of the Moon, and not otherwise, and with that remainder you must resort to the foresaid table of the moons latitude, as for example.   

The Example.  
Suppose that you would know what latitude the Moon had the first of November 1590. here resorting to the diurnal Table of the Ephemerideses, you found according to the day propounded, the place of the Moon to be in the 16. degree 49′· of Taurus, and the place of the Dragon's head to be in the 28. degree 14′· of Cancer. Now according to the rule before given, you must take the place of the dragon here, which is 28. degrees, 14′· of Cancer out of the 16. deg. 49′· of Taurus, which is the place of the Moon, and because you cannot take the greater sum out of the lesser, you must add to the lesser sum 12. signs, which make 360. degrees, and also one sign, for Aries going next before Taurus, in which sign the Moon is, so shall you make the whole sum to be 13. signs 16 degrees 49′· out of which sum you must subtract the Dragon's head, which with the signs that go next before Cancer, counting from the first point of Aries, do make 3. signs 28. degrees, 14′·S which being subtracted out of 13. signs, 16. degrees, 49′ there remaineth 9 signs, 18 deg. 35′· out of which you must also subtract 90. deg. which is 3. whole signs, & so you found the remainder to be 6. signs, 18. degrees 35′· with which last remainder you have to enter into the table of the moons latitude, in the foot whereof you shall found 6. signs, and in the last collum on the right hand 18. & in the next collum towards the left hand, & in the common angle answerable aswell to the said 18. degree. as also to the 6 signs, you shall found the latitude of the Moon to be 4. deg. 45′·S & 17″· (which seconds may be very well omitted) and her latitude to be south. But now because there are 35′· more annexed to the 18. degrees of the foresaid remainder, you must found out a proportional part answerable to those minutes which is to be done thus.  
Take out of the table the whole latitude answerable to 6 signs & 19 deg. which is one degree more, so as now the latitude of the Moon is 4. deg. 43′· omitting the seconds. Than subtract 4. deg. 43′· out of 4. degrees 45′· & there remaineth 2′· Now to found out a proportional part answerable to the former 35′· you must say thus. If 60′· require 2′· what shall 35′· require? and the quotient yieldeth 1′· 10″·S which being subtracted out of 4. degrees 45′· there will remain 4. degrees. 44′·S and so much was the south latitude of the Moon at that present day.    

How to know the true place of the Sun or Moon, or of any other planet every hour of the day throughout the year. Proposition. 46.  
But now because that to find out the true place of the Moon or of any other Planet in the Globe to work thereby certainly and truly, it is not enough to know their places in the zodiac at noontide, unless you know the same at the very hour in which you seek. Stadius teacheth a brief rule to found out the true place of any Planet at what hour so ever you desire by help of a short Table set down in the 109. page of his Ephemerideses, which rule and Table he borrowed of Reinholdus. And this Table consisteth of 3. collums, in every front whereof are set first degrees, minutes, seconds, and thirds, and under them minutes, seconds, thirds, and fourths. And note that the first row of numbers on the left hand signifieth sometime degrees, and sometime minutes, either of which do extend in this Table to 60. & to 61. right against which row in every collum is set down on the right hand the proportional part for one hour, the rule is thus.  

The Rule.  
First find out by the Ephemerideses the place of the Planet wherein it is at noontyde the same day that you seek, and also the place of the same Planet, wherein it is the next day following at noon, and subtract the lesser out of the greater, if the two places at noontide be still in one self sign, but if the two places be in two several signs, than you must count how many degrees of the Ecliptic the one place is distant from the other, and that shall be the difference, with which difference you must resort to the Table, and right against that on the right hand in the same collum wherein you found the difference, you shall also found the proportional part for one hour, as by this example you shall more plainly understand.   

The Example.  
Suppose then that you desire to know the true place of the Sun or of the Moon, or of any other Planet at five of the clock in the after noon the 30. of December 1591. at which day you found the Sun at noontyde to be in the 18. degree, and 7. minutes of Capricorn, and the next day at noon to be in the 19 degree and 8′· of the same sign, the difference whereof you find by subtraction to be one degree and one minute, with which difference you must enter into the table, and seeking for one degree in the first collum on the left hand, you find next unto that on the right hand 2′· 30″· then for one minute which before was named one degree, you found next unto it 2″· 30‴·S which being added to the former sum last found, maketh 2′· 32″· 30‴· which is the proportional part for one hour. Than having multiplied that by 5. hours to serve 5. of the clock in the afternoon, you shall found the product to be 12′· 42″· 30‴· which being added to the first Meridian place, which is 18. degrees and 7. minutes of Capricorn, maketh in all 18. degrees 19′· 42″· 30‴· And remember that if your difference be only minutes, than the body of the Table doth show the proportional part. But if the difference doth contain degrees, than the fourth's must be made thirds, and the thirds seconds, the seconds minutes, and the minute's degrees. And this table serveth for the 7. Planets for ever. Notwithstanding Stadius setteth down another Table to find out thereby the proportional moving of the Moon feruing as well for hours as for minutes of hours, which table beginneth at the 144. page, and endeth at the 184. page of his book.  
In the front of which table in every page are set down the differences of the two places of the Planets which you have found by subtracting the lesser out of the greater, if the planet be at both noonetydes in one self sign, but if she be at the two noontides in two several signs, than such difference is to be accounted upon the Ecliptic line, to know how many degrees the one place is distant from the other, and as for the minutes, if there be any, you may know the difference thereof by subtracting the lesser number out of the greater. And in the outermost collum on the left hand are set down the hours and minutes marked in the foot with the letter H. signifying hours, and under that with the letter M signifying minutes, whereof the hours proceed but to 24. but the minutes extend to 60. which make one hour proceeding by even numbers in this sort. 1. 2. 4. 6. 8. etc. and therefore not finding that hour or minute which you seek in the said collum, you must take the next number which is lesser by one, and so make it up by adding the first and only odd one unto it, and in each common angle you shall found the proportional part that is to be added to the first place of the Planet answering to the day of the month wherein you sought. All which things you shall more plainly understand by this one example.   

The example.  
Suppose then that you would know the true place of the Moon at 5. of the clock. 17′· in the afternoon the 28. of December 1591. and looking in the Ephemerideses you found the place of the Moon to be the same day at noontide in the 26. degree 12′· of Libra. and the next day at noontide to be in the 8. deg. 17′· of Scorpio. here by numbering how many degrees are contained in the Ecliptic line betwixt those two places, and by subtracting the lesser number of minutes out of the greater, you shall find the difference to be 12. degrees 5′· with which difference you must resort to the table of proportional parts, and there having found the said difference in the front of the 13. page of the said table, look in the first collum on the left hand, for the hours and minutes according to the rule before set down. As here in this example, because you cannot found 5. hours, you take four, right against which on the right hand in the common angle you shall found 2. degrees 0′· 50″· then for one hour to make the 5. hours, you shall found in the common angle 30′· 12″·S that done, seek out the 17′· in the first collum, which number not being there you must take 16. against which in the common angle you shall found 8′· 3″· 20‴· and for the one minute which is to be added to 16. you shall found in the common angle against that one minute 30″· 12‴· and by adding all these sums together, you shall found the proportional part to be 2. degrees, 39′· 35″· 32‴· which being added to the place of the Moon at noontyde first found in the Ephemerideses, maketh in all 28. degrees, 51′· 35″· 32‴· of Libra, which is the true place of the Moon for that hour. But you have to note that the denominations of the numbers contained in the common angles of proportional parts be not always like, for when they are to answer hours then the first number on the left hand in every angle signifieth degrees, and the rest minutes & seconds, but if they have to answer minutes of hours, than the first numbers do signify minutes, and the rest seconds and thirds, as you may easily perceive by examining the former example. But now to return to my first intention, which was to show you how to found out the true place of every Planet in the Globe, you have to understand that having found out the true place of any Planet at the day and hour wherein you seek, by such means as is before taught, then resort to the celestial Globe, and having set the same at your latitude, and also rectified the hour wheel according to the day wherein you seek. Suppose that the 26. of May 1592. you would know in what part of the Globe the Moon is to be found at 5. of the clock in the afternoon. Here first you must know her place at noon the same day, which is the 14. degree, 28′· of Aries, and also her place the next day at noon, which is the 29. degree 0′· of the same sign, Now by taking 14. degrees 28′· out of 29. degrees, you shall find the remainder to be 14. degrees, 32′· with which difference you must resort to the table set down in the 109. page. And by working as before is taught, you shall find her place or longitude to be at five of the clock in the 17. degree 23′· 6″· 43‴· of Aries, which being counted upon the Ecliptic line of the Globe, lay your semicircle of longitude and latitude to that point, and having learned her latitude by help of the Table of latitude before mentioned, by which you shall found her latitude to be at that time 4. deg. 46′· to the Southward, then having counted that latitude upon the semicircle of longitude & latitude, make there a mark upon the Globe, for that is her very place at that instant that is to say, at 5. of the clock in the afternoon the 26. day of May 1592.  
Now if you would know when she riseth and setteth, you have no more to do but to bring her place before marked on the Globe unto the Horizon on the East part, and the index of the hour wheel being rectified as is before said, will show the hour of her rising above the Horizon, and by bringing her place to the west part of the Horizon, the Index will show the hour of her setting, and by bringing her place to the brazen Meridian, you shall know at what hour she is full South. The like Tables you shall find also in Stadius to found out the true place of every other Planet at any hour, as of Saturn, jupiter, Mars, Venus, and Mercury, which Tables do begin at the 128. page, and end at the 143. page of his book, the order of working by which Tables is like in every respect unto that of the Moon last taught. By which Tables you may find out the true place of any other Planet in the Globe at any hour of the day or night, and thereby to know the hour of his rising and setting, if you rightly observe the rules before taught touching the use of the foresaid Tables. And note that you may find out the place of the Moon by the Globe without the help of any Ephemerideses, by such ways as here do follow.    

How to find out the place of the Moon by the Globe, when she is above the Horizon, without the help of any Ephemerideses or other Table what soever. Proposition. 42.  
Having set the Globe at your latitude, and placed it so as it may rightly answer the four quarters of the world, take the altitude of the Moon with your Astrolabe or quadrant, and mark therewith whether she be at that present East, or West, that is to say, on this side of the Meridian, or beyond the Meridian, that done, take the altitude of some fixed Star, which you know, and is to be found in the Globe, and also at that time is above the Horizon, marking therewith in what part of the firmament the said Star is, and with your cross staff take the distance betwixt the moon and that star. Than having these three things, that is, the altitude of the Moon, the altitude of the Star, and also the distance betwixt the Moon and the known star. Move the globe together with the quarter of altitude to and fro until you have made the Moon to have the same altitude and like place in the Globe, that you found it to have in the firmament, and there make a mark upon the Globe. Than bring the quarter of altitude towards the Star, that the Star may have the like altitude and like place in the Globe that it had in the firmament, and there having stayed the Globe that it may not move, take with your Compass upon the Equinoctial the distance betwixt the Moon and the star before found by the cross staff, and keeping your compasses at that wideness, put the firm foot of your Compasses in the fixed Star, and cease not to move the other foot together with the quarter of altitude towards the mark of the moons altitude until you make them to meet, for there is the place of the moon.   

Another way to found out the place of the Moon without taking the latitude of any without. Proposition. 48.  
Seek by your cross staff to know the distance betwixt the Moon and any two stars that you know & are to be found in the globe. That done, draw upon the globe on either star an obscure circle according to the distance of the Moon from either of those two stars, which two circles will cut or cross one another but in 2. points, the one whereof is the place of the Moon, and which of those places it should be, your eye will easily tell you. And by this means it were no hard thing (as Gemma Frisius satih) for our sea men in these days to find out all the stars that be in the neither Hemisphere and were unknown to Ptolemy and to the other ancient Astronomers, and to 'cause them to be set down in the Globe, yea and by this art the places of the rest of the Planets may be found out as well according to their Latitude as Longitude.  
Moreover Gemma Frisius saith, that by knowing the true place of the Moon in the zodiac, you may also find out thereby the unknown Longitude of any Region.   

How to find out the Longitude of any Region. The 49. Proposition.  
Having found out the place of the Moon in the zodiac, you must first know the very hour of her being in that place, and then learn by some Ephemerideses or by the Tables of Alfonsus, at what hour the Moon doth enter into the self same degree of the zodiac in some other Region or Town whose Longitude you already know, and having reduced the hours to 24. take the lesser number of hours out of the greater, the remainder whereof must be reduced out of hours and minutes into degrees thus: Multiply the hours by 15. and the minutes of hours by 4. so shall you have the degrees of the Equator contained betwixt the two Meridian's. And such distance so intercepted is called the difference of Longitude, which difference you must add to the known Longitude if the hours in that place were more in number, but if the hours were less in number, than you must subtract the foresaid difference from the known Longitude, so shall you collect the unknown Longitude of that place or Region which you seek, and how far it is distant from the fortunate Iles.   

Another way to found out the unknown Longitude of any place by the Globe. The 50. Proposition.  
Having set the Globe at your Latitude and rectified the hourewhéele by the seventh Proposition, bring the place whose Longitude you know to the brazen Meridian, and direct the Index of the hourewhéele to that hour in which the Moon doth occupy the former defined place in that Region or Town: that done, leave not to turn the Globe until the Index of the hourewhéele come to that hour in which you sought the unknown place of the Moon, and the degrees of the Equator which the brazen Meridian cutteth, will show the unknown Longitude of the place which you seek.  
To the end of this Treatise I have thought good to add a brief description of the two great Globes lately set forth first by M. Sanderson, and then by M. Molineux, and therewith to set down a brief description of Sir Frances Drake his first voyage into the West and East Indieses, and also the voyage of M. Tho. Candish both whose voyages and what course they held are to be seen by help of two lines drawn on the terrestrial Globe, of which lines the one is red showing the voyage of Sir Frances both outward and homeward. And the other line is blue showing in like manner the voyage of M. Candish.   

A brief description of the two great Globes lately set forth first by M. Sanderson, and then by M. Molineux.  
THese Globes do differ in a manner nothing at all from the Globes of Mercator touching the circles before described, but only in the Horizon, for the Horizon of these great Globes is divided into 13. spaces as followeth.  
Whereof the first narrow & innermost space next unto the body of the Globe, containeth the degrees of the zodiac.  
The second containeth the numbers of the said degrees proceeding from 10. to 30. in every sign.  
In the third space are set down the names and the characters of the 12. signs, and also the characters of the Planets that govern the said signs.  
In the fourth space are set down the letters of the days of the week.  
In the 5. and 6. are set down the numbers of the days and names of the months according to the ancient Calendar.  
In the seventh the festival days.  
In the 8. and 9 space the number of the days and names of the months according to the new Roman Calendar.  
In the 10. & 11. are set down the numbers of the days & names of the months according to the true Calendar lately calculated by a most excellent Mathematician and mine old acquaintance M. Dee of Mortlake, as I conjecture by the letters I D. set down upon the said Horizon.  
In the 12. are set down the English names of the 32. Rombes or winds of the Mariner's compass.  
In the 13. and outermost space are set down the Latin names of the said 32. winds.  
But the Map which covereth M. Molineux his Terrestrial Globe, differeth greatly from Mercator his terrestrial Globe by reason that there are found out divers new places aswell towards the North pole, as in the East & West Indies which were unknown to Mercator. They differ also greatly in the names, Longitudes, latitudes & distances of such places as have been heretofore set down not only in Mercators' Globe which was made many years since, but also in divers Maps more lately made, but who goeth nighest the truth I dare not judge, because I was never in those places. But as touching the Map of stars which covereth the celestial Globe of M. Molineux, I do not find it greatly to differ from that of Mercator, saving that M. Molineux hath added to his celestial Globe certain Southern images, as the Cross, the southern Triangle, and certain other stars, whereof some do signify noah's Dove, & others do signify the image called Polophilax, the images whereof are not here set down, but you shall found them described in Plancius his Map, made in the year 1592. whose longitudes, latitudes, and declinations, how truly they are set down in the said Globe or Map I am not able to judge. He setteth down also two clouds nigh unto the South pole, but not the use thereof. Moreover to this brief description of M. Molineux his two Globes, I thought good to add the first voyage of Sir Frances Drake, and of M. Thomas Candish, set forth by two lines, the one red, and the other blew described in the Terrestrial Globe of the said M. Molineux, and also how far Sir Martin Furbosher sailed Northward as followeth.   

The first voyage of Sir Frances Drake by sea unto the West and East Indies both outward and homeward.  
IN the great terrestrial Globe lately put forth by M. Sanderson and by M. Molineux, the voyage aswell of Sir. Fr. Drake, as of M. Th. Candish is set down, & showed by help of two lines, the one read, & the other blue, whereof the read line proceeding first from Plymouth, doth show what course Sir Frances observed in all his voyage, aswell outward as homeward, and the blue line proceeding also from Plymouth, showeth in like manner the voyage of Master Candish, and in that Globe is also set down how far Sir Martin Furbosher discovered towards the North parts. But first I will describe unto you the voyage of Sir Fr. Drake that wrothie Knight & most Noble Neptune, according as that read line directeth in the said Globe.  
First parting from Plymouth he sailed with a North Northeast wind to an isle called Mogodore, upon the coast of Maroccho, which place is not named in M. Molineux his Globe, and that place having in North Latitude 32. degrees, is distant from Plymouth according to that course, which the red line showeth 780. leagues. In this isle he built a little Pinnace or shallop, and from thence he sailed to Cape Dalguere which is further Southward, and having in North Latitude 30. degrees, is distant from Mogodore about 40. leagues, and from thence he sailed to the Isles Canariae, which are somewhat more Westward, and having 27. degrees in North Latitude, are distant from the Cape Dalguere about 100 leagues, and from thence he sailed to Capo Blanco, which is more Westernly, and having in North Latitude 21. degrees, is distant from the Canaries 120. leagues and somewhat more, and from thence he sailed to the Isles of Capo Verde, which having in North Latitude about 14. degrees, are distant from Capo Blanco about 140. leagues. And from thence to the great Cape of S. Augustine, which having in South Latitude about 8. deg. is distant from the Canaries 500 leagues, and from thence he sailed more Westernly unto the mouth of the River called Rio de Platta, which having in South Latitude 36. degrees, is distant from Cape Saint Augustine 740. leagues, and from thence to the Port Saint juliano, having in South Latitude about 50. degrees, whereas one doughty was executed for conspiracy, and this Port is distant from Rio de Platta 380. leagues. And from thence he sailed to the Cape Virgin Maria, which having in South Latitude 52. degrees 30′· is distant from Port S. julian 50. leagues, and from thence stréeking in betwixt the isle, whose North-east Cape is called the isle of the name of jesus, and the Port Famine, he entered into the strait Magellane, which having in South Latitude 53. degrees 30′· is distant from the Cape Virgin Maria 50. leagues, and from thence he passed through the Magellane straits to the Cape de Sancto spirito, which is a Cape of the South land, having in South Latitude 52. degrees 20′· and is distant from Cape Virgin Maria about 150. leagues, from thence he sailed somewhat Westernly about 20. leagues, & there fetching a turn about certain islands called Las Anegadas, he took his course Northward alongst the West coast of America unto the isle Lima, which having in South Latitude 12. degrees, is distant from the islands called Las anegadas 800. leagues, and from thence he sailed still Northward unto Cape Guija, which having in South Latitude 1. degree 30′· is distant from the isle Lima 160. leagues, from thence still Northward he sailed to Cape S. Francisco, which having in North Latitude 1. degree 30. minutes, is distant from Cape Guija 140. leagues, from thence he sailed still Northernly to the Cape Mondecino, which is in the land called Quivira, and this Cape having in North Latitude 40. degrees is distant by that course from S. Francisco 1740. leagues, from thence he sailed still Northward unto a certain Bay in the West part of Quivira, which he named Nova Albion (that is to say) new England having in North Latitude 46. degrees. And this was the furthest part of his voyage outward, in which voyage he sailed in all 6050. leagues, and from this Bay Sir Frances himself (as I have heard) was of very good will to have sailed still more Northward, hoping to found passage through the narrow sea Anian, which sea is not set down by Master Molineux in his Terrestrial Globe as a strait, but rather as a main Sea, bearing in breadth 400. leagues, and so from thence to have taken his course North-east, and so to return by the Isle's Crocklande and Groynlande into England, but his Mariners finding the coast of Nova Albion to be very cold, had no good will to sail any further Northward, wherefore Sir Frances was feign to come back again Southward to Mondecino, which (as hath been said before) is distant from the foresaid Bay of Nova Albion 140. leagues. From thence he sailed in a manner right Southeast to the Isles Moluccas and touched at the Isle's Terenate, Tidori, Machian, and Motill, which are nigh unto the isle Gilolo, which is right under the Equinoctial, amongst which Isles he remained a certain time, of which islands the isle called Terenate having about one degree of North Latitude, is distant by that course from the Cape Mondecino 1180. leagues, and from thence he sailed South-west until he came to the West end of the Isle java maior, which having in South Latitude nine degrees 30. minutes, is distant from the isle Terenate by that course 530. leagues, and from thence he sailed still South-west to the Cape di Buona Speranza, which having in South Latitude 35. degrees, is distant from java maior 1630. leagues, then from Capo di Buona Speranza he making his course Northwest, sailed to an Island called Serra Liona, which is upon the coast of afric, and having in North Latitude 7 degr. 30′· is distant from Capo di Buona Speranza 1090. leagues, then from thence he sailed towards the Isles of Capo verde until he came to the 12. degree of North Latitude right under the first Meridian, which point is distant from the Island Serra Liona according to that course 300. leagues, & from thence he sailed Northward nigh to the Isles called Azores on the West side thereof, which having of North Latitude 40. degr. are distant by that course from the Isles of Capo verde 600. leagues from whence he directed his course North-east to Plymouth, which having in North Latitude 51. degr. is distant by that course from the Azores 490. leagues, so as in his return from Nova Albion to Plymouth he sailed in all 5960. leagues, which if you add to the number of leagues of his outward voyage before set down, which is 6050. leagues, you shall find the total sum of the leagues to be 12010. leagues, which is almost twice so much as the compass of the whole world, which if you measure upon the Globe by the Equinoctial line containing 360. degrees, and do allow for every degree thereof 60. Italian miles, you shall find the number of such miles to amount to 21600. miles, which by allowing three miles to a league do make no more but 7200. leagues. But if it might please Sir Frances to writ a perfect Diary of his whole voyage, showing home much he sailed in a day, and what watering places he found, and where he touched, and how long he rested in any place, and what good Ports and Havens he found, and what anchorage good or bad, and what manner of people, what trade of living, and what kind of building and government they used, in what air they lived, and whether the ground were fertile or barren, dry or well watered with floods and fountains, what mountains, what mines, what woods or forests, what beasts, fowls, or fishes, fruits, herbs, plants, or other commodities he found therein, and in what manner of seas he sailed, and what winds and currents were most rife in every place: Also what rocks, sands, sholdes, and all other places of danger and peril, and by what marks such places are to be shunned. And finally what Moon doth make a full sea in every Port where he arrived, and what winds do altar any tide or Current, and all other necessary accidents most meet for sea men to know. In thus doing the said Sir Frances I say should greatly profit his country men, and thereby deserve immortal fame, of all which things, I doubt not but that he hath already written, and will publish the same when he shall think most meet.   

The voyage of M. Candish unto the West and East Indieses, described on the Terrestrial Globe by the blue line.  
THis blue line as you see taketh his beginning from Plymouth like as the red line doth, whereby you may plainly see, that M. Candish did not greatly differ in his course from that which Sir Frances held, saving that Master Candish having once passed the strait Magellane Westward, sailed not so far Northward as Sir Frances did, for he sailed no further Northward but to the Port Saint Lucas, which is almost under the Tropic of Cancer, for that Port hath in North Latitude 22. degrees 30′· keeping always a nigher course unto the main, than Sir Frances did, and from the Port S. Lucas he sailed in a manner Southeast about 320. leagues, and then he directed his course Eastward towards the Moluccas, & before he came to the Moluccas, he fetched about the isle Catana●●, which is more Northward than Sir Frances went in those parts, and from thence he sailed full South to the Moluccas, from whence he directed his course Southwest, and passed betwixt the Isles java maior and java minor, which two Isles in the said great Globe are made to have one self Latitude (that is to say about 10. degr. of South Latitude, counting from the Equinoctial to the Parallel which passeth through both the Iles. But in all other Globes and Maps java minor standeth 10. degrees more to the South, and is placed behind the South promontory called the land of Beach. Than having passed betwixt the foresaid two Isles, he held in a manner the like course that Sir Frances did in his return to Plymouth, which the blue line doth show so plainly as it needeth none other description. Now as touching Sir Martin Furboshers voyage, because his own description thereof is in print, nothing is set down in this Globe, but only the outermost end of his voyage named here Furbushers straits, having in North Latitude about 63. degrees, in the mouth of which straits is a little Island called Hales Island on the right hand, and on the left hand another little long Island called Lecesters' point. Truly this knight for his valorous venture, aswell in this voyage as in divers other his worthy services done upon the sea deserveth great commendation, and I wish with all my heart that he and such like might be much made of, and rewarded according to their desert. And thus I leave to speak any further of M. Molineux his Globes, the use whereof is to be learned by those Propositions which I have heretofore set down, showing the use of Mercator his two Globes, for the practice and manner of working both by his Globes and by these Globes is all one.   

A plain and full description of Petrus Plancius his universal Map, serving both for sea and land, and by him lately put forth in the year of our Lord, 1592. In which Map are set down many more places, aswell of both the Indieses as of afric, together with their true Longitudes and Latitudes than are to be found either in Mercator his Map, or in any other modern Map whatsoever, and this Map doth show what riches, power or commodities, as what kinds of beasts both wild and tame, what plants, fruits, or mines any Region hath, and what kinds of merchandises do come from every Region. Also the divers qualities and manners of the people, and to whom they are subject. Also who be the most Mighty and greatest Princes of the world: A Map meet to adorn the house of any Gentleman or Merchant that delighteth in Geography, and therewith this Book is also meet to be bought, for that it plainly expoundeth every thing contained in the said Map, Written in our mother tongue by M. Blundevill Anno Domini. 1594.  
Imprinted at London by john Windet.   

IN DOCTISSIMI VEREQVE GENEROSISSIMI THO. BLUNDEVILI, IN PETRI PLANCII TABULAM GEOGRAPHICAM ET HYDROGRAPHICAM ELVcidationes, Gualteri Hawghi 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.  
PLancius in tabula terras descripsit & undas,  
Quaecunque & toto condidit orb Deus:  
Multiplices horum partes, Regumque per illas  
Sceptra superba, simul totius orbis opes.  
Sed solis Gallis haec Plancius, atque Latinis  
Inter & hos, Doctis scripserat, haud alijs,  
Tu Generose tuis patrio haec Idiomate clarè  
Blundevile refers, Cunctaque nota facis.  
Tu simul haec auges, lucemque hijs addis, & horum  
Vsum, multiplici non sine fruge doces.  
Nec solum doctis, sed & omnibus haec Idiotis  
Perspicua ut pateant, tu brevitate facis.  
Plancius haec alijs, tu nobis omnia tradis,  
Tuque tuis Anglis Plancius alter eris.   

A plain and full description of Plancius his universal Map, set forth in the year of our Lord, 1592. written in our mother tongue by M. Blundevill.  
IN this Map, as you see are drawn with red ink two lines or Diameters crossing one an other with right Angles in the very midst of the Map, whereof the perpendicular Diameter passing through the Isle's Azores, and also through the Isles of Capo verde, signifieth the first Meridian and Axletree of the world, at the upper end whereof is set the North pole, and at the neither end the South pole. And the other overthwart Diameter signifieth the Equinoctial, & also the line of East & West, that is to say, East on the right hand, and West on the left hand. And in this line are set down the degrees of Longitude, which are to be counted from the foresaid first Meridian upon the said Equinoctial towards your right hand from one degree to 180. set down in Arithmetical figures thus, 10. 20. 30. and so forth until you come to 180. which is the East Longitude of the world, and the West Longitude beginneth on your left hand whereas is set down 190. and then 200. and so forth until you come to 360. degrees, which is the whole circuit of the Equinoctial, and of the whole earth. And in the first Meridian are set down the degrees of Latitude, which do proceed from the Equinoctial to either of the Poles from one degree to 90. written in Arithmetical figures thus, 5. 10. 15. and so forth to 90. which degrees do proceed with equal distances from the Equinoctial to either Pole, and the like degrees of Latitude are also set down upon the two outermost Meridian's, aswell on the right hand as on the left. And on each side of the Equinoctial are drawn with red ink the two Tropiques, that is to say, the Tropic of Cancer, and the Tropic of Capricorn, each one being distant from the Equinoctial 23. degrees 28′· which is the greatest declination of the sun, and towards each Pole are also drawn with red ink the two Circles both Arctique & Antarctique, whereof the circle Arctique passeth through the Northernly part of Island, having in Latitude 66. degrees 30′· and the Circle Antarctique passeth through the like degrees of Latitude towards the South pole, and the distance of each of those Circles from either Pole is equal to the greatest declination of the sun. And these four Circles are Parallels to the Equinoctial, bounding the 5. Zones that is, the two cold, the two temperate, & the hot Zone which lieth in the midst of the world betwixt the two Tropiques. And on the left hand or West part of the Map are set down from the Equinoctial upward towards the North pole, what number of miles and seconds of miles do belong to every degree of the North Latitude, proceeding from the Equinoctial towards the North pole: for though that every degree of the Equinoctial being a great Circle containeth 60. miles, yet the further that you go from the Equinoctial towards either of the Poles, the lesser and lesser are your Parallel Circles in compass, and therefore one degree of every such Parallel must needs contain the fewer miles, but because the sea men do commonly make their account on the sea by leagues, and not by miles: Plancius here right under the former Table of miles, setteth down the number of leagues incident to every degree or Parallel, in descending from the Equinoctial to the South pole, appointing three miles to a league, and therefore he setteth down upon the very Equinoctial 20. leagues, which is 60. miles, and next to that he setteth down 19 leagues 59′· and so proceedeth forth diminishing still the quantity of the leagues, even till you come to the very South pole, and on the right hand of the Map hard by the outmost Meridian are set down the nine Climes, and the longest day in every degree of Latitude, proceeding aswell from the Equinoctial to the North pole, as also from the Equinoctall to the South pole, from 12. hours to 187. days and 7. hours, which maketh the longest day to those that devil right under the North Pole to be half a year and the night as much. Besides the Circles and lines before mentioned, there are set down in this Map certain flies of the Mariner's Compass, each one containing two and thirty lines, which do signify the two and thirty rombes or winds of the Mariner's Compass to know thereby how one place beareth from another, and by what wind the Mariner hath to sail to any place whereunto he would go: In every which fly the line of North and South may serve in stead of a Meridian, and the line of East and West may serve as a Parallel, by help whereof you may the more readily take with your Compasses the Longitude and Latitude of any place contained in the Map in such manner as is taught in my first Treatise of Universal Maps. Also in the very front of his Map he setteth down the numbers of four and twenty hours, every hour containing fifteen degrees of the Equinoctial, which hours do begin on the right hand, and so proceed to the left, whereof the twelfth hour is placed at the end of that Meridian which passeth through the Fortunate Iles. Now betwixt the 72. and 86. degrees of North Latitude he setteth down two long Islands extending from the West towards the East somewhat beyond the first Meridian, and from the said Meridian more Eastward he setteth down other two long Islands, affirming that the North Ocean sea breaking in betwixt these Islands with nineteen gates or entrances, maketh four straitss and is continually carried under the North Pole, and there is swallowed up into the bowels of the earth, and he saith further that right under the North pole there is a certain black and most high rock which hath in circuit thirty and three leagues, which is ninety and nine miles, and that the long Island next to the Pole on the West is the best and most healthful of all the North parts. Next to the foresaid Islands more Southward he setteth down the Islands of Crockland and Groynelande making them to have a far longer and more slender shape than all other maps do: At the East end of the third long Island, is a strait having five gates or entrances, which by reason of their narrowness and swift course of stream are never frozen, if this be true, I marvel how any ship durst enter through any of those straitss to discover the North sides of any of those Islands, and how and where it came out again. Moreover at the East end of the last Island somewhat to the southward, he placeth the Pole of the loadstone which is called in Latin Magnes, even as Mercator doth in his Map who supposing the first Meridian to pass through Saint Marie or Saint Machaell, which are two of the outermost islands of the Azores Eastward, placeth the Pole of the stone in the seventy five degree of Latitude, but supposing the first Meridian to pass through the isle Coruo, which is the furthest Isle of the Azores Westward, he placeth the Pole of the loadstone in the seventy seven degree of Latitude, I thought good to sever the islands last before mentioned from the body of the Map as parts belonging rather to the North pole, then to Europe, Asia, Africa, or America: for if Virgil did not let to say that England was Penitus exclusa ab orb, me thinks that I may much more rightly say the like of these Islands, notwithstanding Plancius maketh the first two long Islands, and also Groynelande and Crockelande to be part of Mexicana, which me thinks is not meet, sith they be divided by the North Sea, likewise he maketh the land which is under the South pole, and is not as yet discovered, to be part of Magellanica, which (in mine opinion) aught not to be so, sith it is not only divided from Magellanica, by the strait Magellanicum, but also from afric, and from the East Indies by the great Southern Ocean, and I believe that when the South land shall be all discovered, it will contain twice so much land as Magellanica doth, and then I doubt not but that the Geographers will give it some other name, and make many divisions thereof: In the mean time I will follow Plancius his own division of the world, which greatly differeth from that of Mercator, and of all other modern Geographers, for they do divide the whole earth but into four parts, that is to say, Europe, afric, Asia, and America, but Plancius by dividing America into three parts that is, into Mexicana, Peruana and Magellanica, divideth the whole earth into six parts, that is to say, Europe, Asia, afric, and into the three parts of America last mentioned, according to which division, he describeth the earth in the french tongue in sixteen pages, set down at the foot of his Map, the four last pages whereof do only contain the interpretation of the seventy one little Tables or Inscriptions written in the Latin tongue, dispersed throughout the whole Map, expressing therein such things as he thought most meet to be noted in diverse parts of the world, all which Tables or inscriptions, I have here also set down in our mother tongue: And although that to the foresaid Tables or Inscriptions Plancius hath attributed certain numbers for the more easy finding out of the said inscriptions, yet not easy enough by reason that one self number is set down in diverse Tables, and therefore to the intent that you might the more readily find out every Table that is proper to the matter whereof it maketh mention, I have here following joined to the number of every such Table, his proper Longitude and Latitude, which with your Compass you may quickly find out, and more certainly, then by his numbers, if you remember the order thereof set down in my Treatise of universal Maps. But now I will declare the contents of the foresaid sixteen pages in order as followeth.  

The title of the first page is thus.  
A brief declaration of the division, form or shape, and of the particularities of the world.  
The Contents of the first page.  
THat the earth and the water do make both together one round body, which the Cosmographers do environ with five Circles, that is the Equinoctial, the two Tropiques, and the two Polar Circles, and thereby do divide the world into five Zones, two cold, two temperate, and one extreme hot: and though that the ancient Geographers do affirm that three of those Zones were unhabitable, the one for extremity of heat, and the other two for extremity of cold, yet within these hundred years last past it is known by good experience that those three Zones are well inhabited as the manifold Countries therein placed and greatly replenished with people of sundry languages do well testify, of all which things I have written at large in my Sphere. And therefore I make this page the shorter.   

The Contents of the second Page.  
TO know the true situation of the Provinces and places contained in this Map, it is necessary first to know their Longitudes and Latitudes. The degrees of Latitude or of the elevation of the Pole, which is all one thing, are counted from the Equinoctial to either Pole, which is 90. degrees, and the degrees of Longitude are counted upon the Equinoctial from the Isles of Capo Verde towards the East, and so round about the earth until you come to the number of 360. degrees. The Provinces and Towns that are situated under one degree of Longitude, have at one self time like hours of the day, but those that are situate under diverse degrees of Longitude, do differ in number of hours, for when it is in one Town noontide, it is in an other Town that is distant from thence towards the East 30. degrees two of the clock in the afternoon, and so consequently for every 15. degree of distance they differ one hour. Likewise they that devil under one self degree of Latitude have equal quantity of days and nights, but yet so as they which devil on the South side of the Equinoctial have the shortest day when we have the longest, and have Winter when we have Summer. But those that are situate under diverse degrees of Latitude, have inequality of days and nights, for the nigher that any place is situate towards any of the Poles the more hours the longest day of the year in that place containeth. But those that devil right under the Equinoctial have always their days and nights of like quantity, and I understand here by the day, the space betwixt the sun rise and the sun set, and you shall find the quantity of the longest day of the year in every degree of Latitude set down in the North-east part of this Card. As for example to those that have 30 degrees of latitude, the longest day is 13. hours 57′· and so the nigher that you go to the Pole the longer is the day, in so much as to those that devil right under the Pole, the year is but a day and a night, that is to say, they have 6. months day, & 6. months' night. Moreover the Geographers do divide the earth into 9 climes for to distinguish thereby the provinces and regions by the quantity of the longest day, the middlemost parallel of every clime increasing by half an hour, and you have to consider that the degrees of latitude are in all places of like bigness every degree containing 15. Almain leagues, or 60. Italian miles, but the degrees of longitude proceeding from the Equinoctial towards any of the 2. Poles are unequal, that is to say, every one containing lesser leagues or miles than other, but the degrees of the Equinoctial itself are equal to the degrees of latitude, every one containing 15. German leagues or 60. Italian miles, as you may plainly see in the Table set down in the Northwest part of this Map. And you have to note that one Almain league doth contain 4. Italian miles, and we have described the degrees of longitude in the South-west part of this Card by the hours of the ships way, every one decreasing less than other from the Equinoctial to the Pole, whereby you may conceive that two ships being right under the Equinoctial 150. degrees distant one from another, and are to sail with like gate towards the North pole: when they shall come to the 60. degree of latitude, their distance shall be no more but 75. leagues. And the further they go towards the Pole the less distant they shall be one from another, in so much as when they be right under the Pole itself, they shall both meet, as you may see in the 2. round figures containing the description of the earth, and set down in the 2. neither corners of the Map. This matter is to be considered of the Mariners that they may thereby the better perceive the imperfections of their sea Cards. Moreover, in the second page Plancius setteth down the division of the earth as well according to the ancient as modern Geographers, making first three general continents or firm lands, whereof the first is so much as was known to Ptolomey and to the ancient Astronomers, as Enrope, afric, and Asia, the second conti●ent is called America, and the third continent is the South part of the world, not yet fully discovered, called of Plancius Magellanica, and he divideth the second continent called America into three parts, that is, Mexicana, Pervana, and Magellanica, and by adding those three parts to Europe, afric, and Asia, he divideth the earth into six parts, and first he setteth down the description of Europe together with her bounds or limits, and then the commodities thereof, as followeth.  

Of Europe.  
EVrope is far less than all the rest, and yet exceedeth all others in nobleness, in magnificency, in multitude of people, in might, puissance, and renown, the which in times past hath commanded both Asia and afric as Queen, by reason of the Monarchies of the Greeks and of the Romans, and at this day is of great force by the power of the Turks and Muscovites. Moreover it commandeth many provinces in Mexicana and Pervana by the power of the Spaniards and Portugals, and of other Christian Princes.  
Europe is severed from Asia and afric by the sea Mediterraneun●, and by the sea called Marmagior, and by the Marish or sea called Palus Meotis, and by the Flood Tanais and Dwina. The chiefest provinces of Europe are these, Almanie, Italy France, Spain, Denmark, Norway, Swethland, Muscovia, Polonia, Hungaria, Sclavonia, and Greece. The chiefest Islands of Europe are these, England with Scotland, Ireland, Sardinia, Corsica, Sicilia, Candia, Nigro Ponte sometime called Euboia, and Stalimene sometime called Lemnos. And in this second page he setteth down also the description of Almanie thus.  
Almanie is reputed to be the greatest province in all Europe, and is situated in the midst thereof, which is bounded on the East with Polonia and Hungaria, on the South with Dalmacia, and Italia, and on the West with France, and on the North with the North Sea, and with the Sea called Mare Balticum. The inhabitants of this Country warre● in old time with the Romans for their liberty, and since many hundred years past it hath holden the imperial Sceptre. About the time of Christ his birth, it was a rude country, as Cornelius Tacitus saith, full of Wood, bushes, and marshes, but at this day it is so adorned with great magnificent Towns and well fortified, and is furnished with such a number of Castles, and Villages, and with such a number of people, and with such politic government, as it is to be compared to any province whatsoever in all the world. The soil thereof is very fruitful both for corn and Wine, and hath many navigable Floods stored with plenty of Fish. It hath most excellent Fountains, and hot baths, great mines of Gold, of Silver, copper, Tin, Led, and Iron. The inhabitants do exercise as well now as they have done in times past the Art military, and it hath many learned men very skilful in all sciences, and in Mechanical arts, they were the inventors of Artillery, of Gunpowder, and of the noble Art of printing, and of making artificial dials and horologies.  
The chief merchandizes that are transported out of Almanie into other countries are these, Gold, Silver, Copper, Tin, Lead, vitriol, Alum, Quicksilver, Colours of divers sorts, Slates to cover houses, Wheat, Wine, Fish, woollen cloth, Linen cloth, Bombasine, Fustian, Suile, Armour, all sorts of works made of Iron, or brass, and other merceries.    

The Contents of the third page.  
IN the third Page he describeth the 17. Provinces of the low Countries, also Italy, France, Spain and Denmark, as followeth.  
THe 17. Provinces of the low Countries are counted a part of Almanie, by reason that the most part of the inhabitants have as well their original as their language from the Almains. But those of Artoys, of Henough Namures, and part of the inhabitants of Brabant, Flaunders, Lymburgh, and Liezenburgh do speak the French tongue. These provinces are situated partly within the ancient limits of Almany, that is to say, beyond the Oriental part of the Flood Rhenus, & partly in Gallia Belgica, as also are the provinces of the 4. princes electors and of many other provinces of the Empire. In these provinces are many navigable Floods and rich in fish, namely the Rheane, Mosella, Mosa, and the Escaut, and there is great abundance of all sorts of corn and cattle meet for man's use, there is also a number of great Towns rich, mighty, and well peoplished, also of fortresses well fortified, fair Villages, but chiefly of brave and commodious ports and havens, and an incredible number of ships, and the continual wars that they have had and have at this day do witness to all the world their great force, might, and riches. The inhabitants in time of the Roman Monarchy were and also are at this present greatly renowned for their skill in the art military, and besides that, they are most excellent and industrious in all sciences and mechanical arts, and they have a great number of Mariners and Pilots well practised in the art of Navigation, and to these provinces is attributed the invention of the Mariner's Compass, according to the opinion of many learned men, which truly is one of the noblest inventions that ever was found out since the world began. In the town of Brughesse was invented the Art of Painting with Colours tempered with Oil.  
The Provinces of Belgia do send unto other Provinces all sorts of clothes made of Wool and Flax, linen cloth of Cambria, Skarlettes interlaced with gold, Silver, and Silk, Taffatas, Borattas, Grograines, single Buffin, Says of Leyden, and Howscot, Worsteds, and half Worsteds, Fustianapes of Vellures, and of Wool, Bayes, silk, parchment lace, Sarsenet and inkle, all manner of twisted thread silk ready dressed, purified Sugar, buff, Chamois, striped Marokines, painted Pictures, Books, Armour, Cables, Ropes, and other Munitions belonging to Ships, Knives, pings, and all sorts of mercery or Haberdash ware, and Fish dried and salted.  
ITalie being the mother of eloquence and of all Latin erudition doth extend itself like an arm towards the Southeast, lying betwixt the Tuscan sea and the gulf of Venice, & is bounded on the west part with France, & on the north part with Almany being separated from the said two provinces by the Flood Varo and the Alps, and all the rest is environed with the sea, at the time of the nativity of Christ and since, she most flourished, being adorned with the fourth Monarchy, and with the most mighty town of Rome, which at that time was Queen of many provinces, of Europe, Asia, and afric, which city in the time of the Emperor Vespasian, had in circuit 13. Italian miles and 200. paces, as Pliny writeth in his third book and fift chapter, Flavius Vopiscus reciteth that this town was enlarged by the Emperor Aurelius to 30. Italian miles, which is 10. hours of way or gate, allowing three mile for an hour. This Province hath brought forth, as it doth at this present, inhabitants of great industry and wit, and it containeth many noble cities and of great renown, as Rome, which hath been sometime the head of the world, Venice the rich, Ravenna the ancient, Naples the gentle, Florence the beautiful, Genua the proud, and Milan the great. In a town called Amalphe situate upon the sea, betwixt Naples and Salerno, the Mariner's Compass was invented, in the year 1300. according to the opinion of some, by one john Goia, citizen of the same town: notwithstanding john Gorop Becanus doth attribute that invention to Flaunders which seemeth the more likely, for so much as all the Pilat's & Mariners of France and Spain and other places do name the 32. winds or rombes of the Compass by the Belgic names. The chief merchandises that are sent out of Italy into other countries are these, Rice, Silk, Velvet, Satin, Taffatas, fine pieces of Linen, Grograines, Rash, Stamin, Bombasins, Fustians, Feltes to make riding cloaks, plenty of rich armour, Wiar of gold and silver, Alum, Galls, drinking glasses and looking glasses of Venice.  
FRance hath been always esteemed to be the chiefest realm of all Europe, whose soil is most fertile, and bringeth forth all kind of Grain, and every other thing that is necessary for man's sustenance, there is great store of wine, and great plenty thereof is distributed to other Provinces nigh adjoining. The Province doth abound in oil Olive, and in Coral, and in many other noble fruits. In France are many great towns well walled, as Paris, Rouen, Amiens, Orliens, Tours, Nantes, Poicters, Burghes, Tholous, Lion, Na●bona, & Marcelles. It hath 15. Archbishoprickes, and 108. bishoprics, and a great number of towns and villages, and 132000 parishes, it is greatly peopled, and hath not such deserts or heaths as are in other provinces of Europe, the French men have been, and are at this present renowned in the Arte militare, and there be many learned men in all faculties and sciences.  
The chief Merchandizes that are carried out of France into other provinces are these, Wheat, Rye, Beefs, Hogs Swine, and other cattle, Salt, wines, wild Olive, Chessnuts, Almonds, Prunes, Coral, Dyer's wadde, Clotheses, linen, Canvas and Skins.  
Spain is environed round about with the Sea, saving that on one side it is separated from France by the mountains Perenei. This country was sometime divided into three Provinces or kingdoms, that is, Taraconensis, Lusitania and Betica, but now it is subdivided into many Realms, that is to say, Castilia, Arragon, Portugal, Gallicea, Lion, Navarra, Toledo, Valentia, Murcia, Granado, Cordoa, and Algarbia, the which Realms if they had been reduced to one body of a Realm, as France is, and as they be at this hour subject to one only king and Lord, it should be without doubt one of the most mighty and puissant kingdoms of Europe. The inhabitants of Spain have been and are at this present much renowned in the art militare, and in feats of war, and it hath brought forth in times past many great clerk, as Seneca, Quintilian, Lucan and Martial, and in our time it had johannes Lodovicus vives, johannes Osorius, and Benedictus Arias Montanus. The provinces of Spain are become very rich and mighty, by reason of their navigation into America, Africa, Arabia, Persia, India, the Isles Moluecas and China, in which Provinces (China excepted) the king of Spain possesseth many countries that be rich and of great power, and many towns and fortresses in a manner round about the earth.  
The chief Merchandizes that grow in Spain, and are carried into other Countries are these, wines, Oils, Rice, all sorts of fruits of Spain, Licorice, Silk, great quantity of wool, Lamb skins, Cork, Rosin, Steel, Iron and Armour.  
DEnmarke and Norway are very great Regions, and are as large as the countries of Almany, bordering upon Almany towards the South, they extend towards the North to 71. degree 30′· of North latitude, and towards the East they border upon Swethland, and on the west and North side they are environed with the sea. These two Realms are at this day under the government of one only king, who also is lord of Island and of the Isles of Fero, Hitland, and Gothland. juthland was sometime the habitation of the Cimbres, who in times past made cruel wars against the Romans.  
The Merchandizes sent from the two foresaid realms into other provinces are these, Oxen, Barley, Malt, stockfish, tallow, nuts and filberts, hides of Oxen, and Buck skins, Masts for ships, planks, and the tops of Wainscot, Solives, and firewood to burn, pitch and tar, Sulphur and such other things.   

The Contents of the 4. page, wherein he describeth Swethland. Polonia, Hungaria, Sclavonia, Greece, England, and Scotland as followeth.  
SWethland is a great and mighty realm, bordering towards the East upon Russia, and towards the South upon the East sea, called Mare Balticum, dividing Swethland from Almany and Pomerania, and towards the west upon Norway and Denmark, and towards the North upon Finmarke. Stockholme is the Metropolitan city in this realm, wherein the king keepeth his court. From this realm is transported into other provinces these Merchandizes, that is to say, Copper, Iron, lead costly Furs, hides or skins of Elkes, of Oxen, of Bucks, of goats, tallow, tar, barley, malt, nuts and Filbirds, and such like.  
The description of Muscovia which should follow next, is set down in the third table or inscription, which standeth in the very front of the table, written in Latin, the interpretation whereof hereafter followeth in his place.  
THe kingdom of Polonia containeth Lituania, Podolia, the lesser Russia, Volhinia, Massovia, Samogitia, Prussia, and in a manner all Livonia, which two last Provinces did belong not long since to Almania. Polonia is bounded on the East with Muscovia and with the Tartaries, Perocopsiques, and on the South with Moldania, and Hungaria and towards the West with Almania, and towards the North with the sea baltic and Muscovia.  
The chief merchandizes that go out of this realm into other Provinces are these, Wheat, Rye, and other grain, Spruce or Dansk Bear, yellow Amber, Wax honey, a certain drink made of Honey which we call Mead, hides of Oxen dried and salted, Flax, Hemp, Pitch, and Tar, Ashes, Clavellees, wood Mazier, and of Cuvelier, and other such like merchandizes.  
HVngarie is a very fruitful Realm, rich and mighty, and it is bounded on the East with Moldavia, and Valachia, and on the South with Bosnia and Croacia, & on the West with Almania, and on the North with Polonia, it hath many navigable rivers, wherein are great store of fish, that is to say, Danubius, Dravus, Savus, and Tibistus. The chief towns are these, Buda, Gran, Weissenburgh, Rab, Prezburgh, Agria, Colocza, and Belgrada. The inhabitants of this country are warlike and hardy, and have been long time heretofore a most faithful Rampire and Bulwark to all Christendom, but in the end by reason of their civil wars, the better part of them have been subdued in our time, and are made most miserable slaves to the Turk.  
The Merchandizes which go out of Hungary into other Provinces are these, Gold, Silver, Copper, and divers sorts of Colours, Salt, Wine, Wheat, Beefs, and fresh fish of the river salted.  
Sclavonie is bounded on the East with Bulgaria, and Greece, and on the South side with the Gulf of Venice, and on the West, with the North part of Italy, and on the North side with Almanie and Hungary. This Region containeth many particular Provinces as Liburnia, Croacia Bosnia, and Balmatia, the chief towns whereof are these, Raguza, Salona, Sabenica, and Zara. Sclavonie at this time is divided into many jurisdictions, for one great part thereof is subject to the Turk, another part to the Emperor of Almany, and the rest situated upon the sea coast is subject to the signory of Venice. At this day there is no tongue (the Arabia tongue excepted) that extendeth further than the Sclavonie tongue, for as it is the Vulgar tongue of Sclavonie, so is it familiar to them of Histria, Bohemia, Moravia, Sileucia, Polonia, and to the large Provinces of the great Duke of Muscovia, Circassia, Perihoka, Georgiana, Mengrelia, Moldavia, Valachia, Bulgaria, Russia, Servia, Albania, and to part of Hungary, that is also familiar in the Court of the great Turk, and among his soldiers that serve in Asia and afric.  
GReece sometime the mother of all science and erudition, is on the East, South and West side environed with the sea, but on the North side it is bounded with Servia and Bulgaria, it hath in times passed valiantly fought with and beaten the Monarchy of Persia for the liberty of their country, and finally by Alexander the great hath triumphed over the same, and thereby erected the third Monarchy, by means whereof it came to pass that the Greek tongue was made common throughout Asia, Syria, and Egypt, until such time as the Saracens and the great Turk did corrupt and change the same. The Emperors did rule in Greece from the time of Constantine the great unto the year 1542. in which year Mahomet the great Turk forced the town of Constantinople, and abolished the Empire of Greece in such sort, as ever since this magnificent and strong imperial town of the Christians, hath been the seat of the Emperor of Turkey, and all the country made slaves to the mahometans.  
The chief merchandizes that come from this country to other Provinces are these, Gold, Silver, Copper, Vitriol, divers sorts of colours, wines, Oil, Velvets, Damasks, Grograins, Turquesques and Wood  
ENgland together with Scotland making both but one Island is the greatest and mightiest of all Europe. And England is environed on all sides with the sea, saving on the North side, which bordereth upon Scotland. The air according to the situation is indifferent temperate, for though it be more Northward than Flaunders, yet it is not subject to such hard frosts and cold winters. The soil is very fruitful, bringing forth great plenty of wheat and of other corn, it hath great plenty of fruit trees, and there be many large and fair woods, sweet fountains, floods and rivers full of fish, and a number of good havens, also it hath many rich Ours, as of Gold, Silver, Led, Iron, and chief of fine Tin, wherefore it may be worthily counted amongst the most puissant and richest Islands of the world. This Island nourisheth also a great number of cattle meet for man's use, and chief of sheep, which yieldeth fine and good wool, in which partly consisteth the profit and riches of the country, in such sort, as the golden Fleece aught to have been sought for in this Island, and not at Cholcos'. The inhabitants most commonly are tall of stature, beautiful and white of visage, courageous and meet for the war, also they are ingenious and studious in the Art of navigation, in so much as in these days they have traffic into very far countries, as into Greece, Anatolia, Syria, Egypt, Barbary, Muscovia, and into many other provinces. London being situated upon the Thames is the Metropolitan and chief town of this Realm, and the Staple of the trade of Merchandizes, and the Court royal, but Cambridge and Oxford are Universities.  
The Merchandizes sent from England into other provinces are these, broad clothes, Carsies, Stamines, Bays, Says, Saffron, Tin, Lead, Wheat, Barley, Malt, Bear, read Hearing, sea Cole and wood.  
SCotland is the North part of this Island, and is likewise environed round about with the sea, saving on that side with which it bordereth upon England. This Country is not so fruitful as England, notwithstanding it is sufficiently provided of all things that is needful for man's nutriment, it is watered with divers arms of the Sea, and is endued with many mountains full of grass, which serveth to feed their cattle. Edinburgh is the Metropolitan city of this realm, wherein the keepeth his court. The Scottishmen are good Soldiers, which can endure scarcity and the injuries of the air, and are very desirous to win honour. The inhabitants of the South part thereof do speak the English tongue: but those of the North, and those of the Isles Hebrides do use the Irish tongue, and those of the Orcadeses do use the Norway tongue.  
The Merchandizes which Scotland sendeth to other countries are these, course clotheses, Kerseys, Stamins, Friezes, Wool, Barley, Malt, Fish, hides, leaden Hour, and Smiths coal.   

The contents of the fift page.  
IN this page he describeth Ireland, the Isle's Azores, Corsica, Sardinia, Sicilia, Candia, nigro Ponte, Stalimene, all which Islands do belong to Europe, and in the latter end of this page he beginneth to describe Asia.  
IReland is nigh unto England and Scotland, and is very rich in meadow ground, and hath great plenty of cattle as well tame as wild, and fish as well of the sea, as of fresh rivers, and great quantity of foul and birds, but it hath scarcity of corn by reason of the great moistness of the air. This I'll is free from all venomous beasts, the inhabitants are wild people, great and strong, and swift in running, and by little and little they wax every day tamer than other, under the government of the English men.  
THe Isles of Azores are called of the Flemish Pilots & Mariners the Flemish Isles, because those of Burghes were the first that discovered those islands, & albeit that at this present the inhabitants thereof are Portugals, there is yet a remnant of Flemish families, as of the Bruin's, of the Vltrickts & others. These Isles are fruitful, and be 9 in number, that is to say, the isle of S. Marry, S. Michael, Tercera, Graciosa, S. George, Pico, Fayal, Flores, and Coruo.  
Tercera amongst all the rest is the strongest, & bringeth forth dyer's Wad. The I'll of S. Michael bringeth forth Sugar, and great abundance of good Dyer's Wad.  
COrsica is situate in the sea Mediterranean, and bringeth forth most excellent wines, rough Horses, and great hunting dogs: and this isle is governed by the Genueses.  
SArdinia is a very fruitful I'll, and chief of Wheat, which is transported from thence into Italy and into Spain, likewise it hath very good Wine, both read and white, and very good Salt, it hath also certain mines of silver, but not of so profitable yield as in times past. The inhabitants are strong, and able to endure great labour and travel. In great towns they speak the Spanish tongue of Arragon, but in small towns they speak the vulgar tongue of the isle.  
SIcilia hath been always famous, and is called of Diodorus the Paragon of Isles, also the Greeks and the Latins have greatly celebrated this isle in their writings. This Island hath great abundance of Wheat and of all other grain, also of wine, Sugar, wax, Honey, Saffron, Silk, and of all things else appertaining to the use of man. Wherhfore this Isle, together with Egypt was sometime called the Grange of the Romans. In this Island is the hill Aetna, which always burneth, and in the sea of Sicill nigh unto Drepano, as Pliny writeth in his 32. book and second chapter, there groweth very fair red Coral, in shape like to such a tree or bush as is here figured, which while it is under the water is green and tender, but so soon as it cometh into the air, it waxeth hard like a stone, and is read, there is found thereof also nigh unto the sea coast of Province, also in Italy nigh unto Monte Alto, and to Naples, likewise in the read sea, and in the Gulf of Persia, and there be three sorts of Coral, that is, red, black, and white.  
CAndia sometime called Creta, was in old time enriched with the famous Labyrinth, and with a hundred cities, it had also a great number of good ships and expert Pilots, this isle together with the others, as the isle of Zante, Cephalonia, Corfue, and divers others, be at this present governed by the Senate of Venice.  
The Merchandizes transported out of Candia into other provinces are these, noble wines, as Malmsey, muscadine, Corrants grain of Scarlet, Sugar, Crystal of the mountain, Cotton, and Buckeskins.  
NIgro Ponte, sometime called Euboia, is a very fruitful I'll in Wheat, Oil and Wine.  
STalimene sometime called Lemnos, is an Isle which hath abundance of wheat, and most excellent Wines. In this isle they dig out in the month of August a certain medicinable earth called of the Physicians Terra sigillata. There be many other Isles besides these in Europe, as the Isles of Denmark, the Isles of Zealand in Flanders, the isle Frumentera, juica Maiorica and Minorica, and a number of Isles that are in Sclavonie and Greece.  
ASia is separated from Europe by the floods Tanais & Dwina & from afric by the narrow part of land, which is nigh to Egypt, betwixt the Mediterrane sea, & the read sea. Asia far exceedeth in greatness both Europe, afric, and Pervana, and also in riches, as in pearls of great price, and precious stones and spices it exceedeth all the other countries of the world.  
This region hath been always renowned by the first and second Monarchy of the world, obtained by the Syrians & the Persians', as also it is at this day by the mighty Princes of China, and of Persia, and by the puissancie of the Tartarians.  
In this part of the world man was created of God, placed in Paradise, seduced by Satan, and redeemed by our Saviour jesus Christ, and in this region, were done in a manner all the histories and acts mentioned in the old Testament, and a great part of those of the new Testament. The most celebrated provinces of Asia are those that belong to the great Duke of Muscovia, also Tartary and China, the rich province of India, as Guzarette, Corasan, Sigistan, Chirmania, Parthia, Persia, Media, Assyria, Armenia, Anatolia Syria, and Arabia. The principal Isles of Asia are these, japan, Luconia, Mindanao, Borneo, Sumatra, Ceilan, and Cypress, for as for the Isles of Gilolo, Moluccas Banda & Celebes, they belong to that part of the world which is called Magellanica.  
The most mighty Potentates of Asia are these, the king of China, the king of Persia, the great Turk, and the Emperor of Russia, otherwise called the great Duke of Muscovia, according to which Seniories all Asia is divided into six parts, that is to say, the Asiaticall provinces, belonging to the great Duke of Muscovia 1. Tartary 2. China 3. the Indieses 4. the Provinces of the king of Persia 5. and those of the great Turk 6. And as touching the Asiatical Provinces of the Emperor of Russia, and of the provinces of Tartary, we shall make mention thereof hereafter, when we come to translate the Tables or Inscriptions written in Latin, marked with the numbers 3. and 4.   

The Contents of the 6. page, wherein he describeth China, and the plant of pepper there growing, with the shape thereof.  
CHina or Sina is the third part of Asia, sometime called of Ptolemy Sinacum regio, which on the East side is environed with the sea called of the ancient Geographers Oceanns Sicicus, or the east Ocean, and on the West it is bounded with the Indieses and with Brumas, & on the North with Tartary. This country is for many causes esteemed to be the most ample, the richest & most mighty Realm of all the world, for it extendeth from the 18. degree to the 55. degree of North latitude, and it containeth in longitude 450. leagues of Almanie, and it is divided into 15. great provinces or Realms, that is, Quincii, otherwise called Paquin, Xanton, Xiancii, Sancii, Suchuan, Honao, Nanquii, Chequiam, Foquiem, Cantam, Quancii, Suinam, or Huinam, Quiecheu, Fuquam, or Hucquam, and Quiancii. This Realm is adorned with many navigable Floods, and full of Fish, it is very fruitful, and bringeth forth great abundance of all kind of grain and amongst the rest, of Rice, every year three or four times in a year. It hath goodly woods & forests, wherein do keep a number of wild Boars, Foxes, Hares, Coneys, Sables, and Martins. The monntaines are full of grass, serving to feed infinite herds or troops of cattle, both great and small. There be also many mines of precious stones, of Gold, silver, Copper, Steel, and Iron, and a great number of pearls, but not very round, and great abundance of silk. The towns there are very great, fortified, and well peoplished, which is easily known by the greatness of Cantan, which is one of the lest Metropolitan cities of the Realm, and yet it containeth in circuit 12. Italian miles, and 350. Geometrical paces, which is more than four hours journey, not reckoning the suburbs, which are very large and full of people. The principal Metropolitan town where the King keepeth his court, is named Paquine, or Suntie, that is to say in their tongue, the celestial or heavenly city, touching the greatness whereof, the Portugals and the Castilians do writ many incredible things, and according to the opinion of many, that is the self same town which Marcus Paulus Venetus calleth Quinzay, as that which hath divers names in divers languages. The like may be said of a town in Flaunders, which the French men call Lile, the flemings Russill, and in Latin it is called Insulae. In these provinces be many good Ports and Havens upon the Sea, and a great number of ships: by reason whereof the Inhabitants are moved to say, that amongst them, there are as many that devil in Ships upon the Sea, as be of them that devil in houses upon the land, and that their King might easily make a bridge to pass from China to the town of Malaccha, which is distant from them 300. Almain leagues. But above all there is one thing worthy of great admiration, and that is a wall which hath in length 400. Spanish leagues, which the King of China caused to be built, to defend the country against the invasion of the Tartarians, of which thing if the ancient men had had any knowledge, they would have counted this work amongst the seven wonders of the world. The inhabitants are men of Spirit, and given to labour. There was also invented by them such a kind of writing, that every man of what nation so ever he were, being some what exercised therein might pronounce in his mother tongue, even as it were ciphered: They invented also certain Chariots, wherein they might sail by the Wind upon plain ground, as they do in ships upon the sea. There are also men amongst them that are well learned in all Sciences and especially in Architecture, wherein they excel all others, they are great lovers of learning, and those that do excel others therein are promoted to the most honourable estates: they have good municipal laws, and will suffer no Stews, and they forbidden that any man shall marry any woman with whom he hath lived before in adultery, and they grievously punish all offences, & do forbidden idleness as the mother of many evils, yea they constrain blind men to get their living, by turning with their hands, mills made to grind corn, or any other things, and in their wars against the Tartarians, they get the victory more by fine policy and stratagems, and by multitude of people, than by prowess or feats of Arms. The Portugals do report that the King bringeth to the field 300/000. footmen, and 200/000. horsemen. Now as touching their religion they be Paynims and superstitious Idolaters, saving that there are in many places some Christians, as Marcus Paulus Venetus testifieth.  
The chiefest Merchandizes transported out of China into other provinces are these, Gold, precious stones, Pearls, Musk, Rhubarb, the medicinable root, China, Purslane, abundance of silk, Sugar, Rice, and all sorts of grain. The plant of Pepper is sown at the roots of other trees, but specially at the root of that Indian tree, which is called Faufell, and at the root of the Date tree, to the tops whereof it climbeth up, much like as ivy doth upon a tree, or like to that which is called in Latin Clematis in English Perwinkle, which will wind about every herb that groweth nigh it, the root is but small, and his leaves thinlike unto Citron leaves, but somewhat less, and sharp pointed, green and biting in the taste, the grains do grow nigh one to another, like the long grape, and are always green until they be through ripe and dry, there be two kinds thereof, that is, white and black, but the plants of both are much like to our white and read Vines. Pepper groweth in places near the sea side of Malacha, and in the Isles of Sunda and Cuda, situated nigh unto the Isle java maior, but the best kind of Pepper most plentifully groweth in the province of Malabar, betwixt the Cape Comori and the cape Canonar, but long Pepper is found in the realm of Bengala, and is another kind of plant, altogether unlike to this.   

The Contents of the seventh page: wherein he first describeth the read fig tree of India, and setteth down the shape thereof, than he describeth the East Indies, and last of all he showeth the nature of the Elephant, whose shape he setteth down in the same page.  
THe Indian fig tree groweth round about Goa, the body thereof is high and great, and extending his branches in a round form, which like yellow or golden fillets do stoop down towards the earth, and so soon as it toucheth the earth, it bringeth forth a new generation of trees, which differeth nothing at all from the mother but only in thickness, des Souches, and the branches of those do bring forth new trees in like manner, in so much as the mother with her offspring will in short time spread as much ground as containeth an Italian mile in circuit, the fruits are small Figs, and read as blood, as well without as within.  
India took his name from the Flood Indus, which bordereth towards the East upon the Realm of China, and towards the South upon the great Ocean of India, and towards the West upon the sea of Arabia, and also upon the Flood Indus, and towards the North upon the sea Mare Euxinum, or Mar mayor, and upon Bramas.  
This country is judged at this day, as it hath been long since to be the noblest and richest country in all the whole world, and it is divided by the Flood Ganges into 2. parts, whereof the West part is called Indostan, or India intra Gangem, and the East part is called India extra Gangem, and it containeth many provinces and Realms, as Cambaiar, Delli, Decan, Bisnagar, Malabar, Narsingar, Orixa, Bengala, Sanga, Mogores, Tipura, Gouros, Avarice, Pegua, Aurea, Chersonesus, Sina, Camboia, and Campaa. These provinces are watered with a number of goodly rivers, amongst the which Indus and Ganges are the most renowned rivers of the world. Moreover, these provinces do abound in all things that may grow either within the earth or upon the earth, except it be copper and lead, as Pliny affirmeth, also all manner of Plants that grow there are very great, brave and excellent good India exéedeth all other countries in precious stones and in spices, furnishing therewith almost all the world. It hath many rich mines of Gold, and great store of fair pearl, also great multitude of all manner of cattle, horses only excepted, which are brought thither out of Persia and Arabia. It is not long since that Callicute was the chief town of Merchandise in India, but at this present Goa is the chief: there is also great traffic used at Dio, at Cananor, at Cochin, at Bengala, at Pegu, at Malacha, and at Sian.  

Of the Elephant.  
THe Elephant amongst all other four footed beasts is the greatest save the Dragon and the Crocodyle, he is very ingenious, in so much, as it is incredible that which the ancient men writ of him, and also the modern, which have sought more diligently to know his nature and disposition, he is of force incredible, and meet to draw ships and boats both out of the water and into the water, and to draw artillery and ordinance, he is also meet for the war, his teeth that shoot out of his mouth are ivory, there is great number of them found in the Indieses, and in afric, but the greatest and fittest for war, are found in the isle of Ceiland, nigh to Calicute.    

The Contents of the 8. page, in which he describeth the beast called Rhinoceros, and setteth down his shape, and he describeth the Cinnamon tree, showing the shape of the trunk and of the leaf thereof, and also the Musk Cat, with her shape, and in the latter end of the page he describeth the Realm of Persia.  
THe beast called Rhinoceros, is as long and as large as the Elephant, but not so high, for his legs are shorter, he is armed not like a Tortoise (as Plancius saith) for that is covered a● over with one shell, whereas this beast is armed with manifold strong, hard, & thick scales, whic● are yellow and spotted with purple, he hath a strong horn or bon● upon his nose, whereof he taketh his name, and he hath another little horn upon his back, and he is a great enemy by nature unto the Elephant, he is found in the Realms of Cambaia, and of Bengala.  

The Cinnamon tree.  
THe Cinnamon tree, is as big as the Olive tree, the branches and griftes whereof are very right, his leaves in colour are like to those of the Laurel tree, but in shape like to those of the Citron, his Flowers are white, and the fruits thereof are black and round, like a hazel Nut, the Cinnamon itself is no other but the bark of the said tree, which groweth in the Province of Malabar, and in the Isles of java and Mindanao, but the best is found in the isle of Ceiland.   

The Musk Cat.  
THe Musk Cat, is like in shape to a common Cat, but she is greater than either Cat or Fox, her muzzle is somewhat long and armed with sharp teeth, and with harsh hair, which hairs (being angry) she will set up as a Swine doth his bristles, she is in colour like a Wolf, but that she is spotted with black spots, the neither part of her muzzle and the hairs of her beard are white, her feet are black, her flanks are whitish, and do wax whiter and whiter towards her belly, and next to her genitories, she hath a little bag like to a bladder or purse, into the which doth fall the precious grease or humour, which they call Civet and Zibeth, which Civet is gathered out from thence with a spoon, if she be in man's keeping, but when she is abroad and at her own liberty, her bag being full, she will void that Civet of herself, and it will yield such a sweet savour, as all they that sail by that coast may smell it a far off, as I have heard. These musk Cats are brought from the Realms of Pegu and Tarnassary.   

The description of Persia.  
AS those in Persia have enjoyed in times past the second Monarchy of the world, so at this present they be still very mighty, for the king of Persia is one of the greatest Potentates in the whole world, as he which commandeth all the great provinces that do border towards the East upon the Flood Indus, and towards the South upon the sea called Mare Caspium, and upon the Flood Oxo, within which limits, are comprehended all the greatest Realms and lands, which the ancient Geographers were wont to call by these names, Assyria, Media, Susiana, Persia, Parthia, Hyrcania, Margiana, Bactriana, Pa●opanisa, Aria, Drangiana, Arachosia, Caramania, and part of Armenia mayor, the which at this present are called by other names as you may see in the Map.  
The Persians' are a hardy and warlike people, and thought to be the best riders or horsinen in all the world, they have very hard wars with the Turks, they be of most free and gentle nature, lovers of civility, they make great account of learning and Sciences, they honour Nobility, wherein they greatly differ from the Turks. Now as touching their religion, they be Mahometists, and yet in such sort, as both they and the Turks do count each one the other as Heretics in that religion.  
From the provinces of Persia are transported into other parts of the world these Merchandizes, stones called Turquesses, very fair and excellent pearls, great quantity of silk, Velvet, Damask armour, and a great number of most excellent horses.    

The Contents of the 9 page.  
IN this page he describeth and setteth down the shape of the precious stone named Bezoar, and the Dominions of the great Turk in Asia, and the city Aden in Arabia, also he describeth the beast called Cameleopardalis, and setteth down his shape, also he setteth down the shape of three of the greatest Pyramids that are in Egypt.  

Of the stone Bezoar.  
THe stone Bezoar or rather Pazar (for that is his right name) groweth in Persia in manner de Boucz, named Pazan, which are of divers colours, but most commonly red.  
This stone Bezoar groweth in a concavity in manner of a girdle about two handful long and three inches broad, it is medicinable, and of great efficacy against all manner of poisons and venoms, and many other maladies, there is to be found of them in the entry of Malacha and also in Pegu, but the best of them are in Persia.   

Of the dominions which the great Turk hath in Asia.  
THe great Turk doth possess in Asia, Anatolia, sometime called Asia minor, and almost all Armenia, Mesopotamia, called at this present Diarbech, or Diarbekir, Syria, and a great part of Arabia, the most notable Merchant towns of this country are these, Trapezunda, Alepp. and a port upon the sea called Tripoli, also Aman, Damascus, with his port Barutti and Mecha.  
The Merchandizes that are sent from these Provinces into other countries are these, great quantity of silk, Velvet, Damask Turkey Carpets, Cotton, and grain of Scarlet.   

The City Aden.  
ADen is the chief Merchant town of the upper part of Arabia, which is governed by divers Kings, and this town sendeth into other Provinces of the fairest Pearls, the true Balm, Frankenscence, Myrrh, and Horses.   

The Beast called Camelopardalis.  
THis beast is called of the Arabians, Gyraffa, but the name Camelopardalis is compounded of Camel and Pardale, which is a leopard, he hath a very long neck like unto the Camel, and is spotted with many spots, as is the Pardale or Leopard, he is a fair beast, and of gentle nature, as the sheep, his head is like unto the head of a Hart or Stag, but greater, his horns are small topped, and covered with hair, and are about a handful and a half long, he hath ears tongue and feet like to an Ox, his forelegges are long and tall, and his hinder legs are short, whereby he seemeth always to stand right up, his head is somewhat higher than the Camel, and this beast is to be found in Arabia, Aethiopia, and India.   

The Pyramids.  
IN Egypt are many Pyramids, whereof the two greatest are counted amongst the seven wonders of the world, the greatest of them (as witnesseth Peter Belon) who most diligently viewed the same, is at the foot four square, and every square containeth in length 324. paces, and in height 250. degrees or steps, and every step hath in breadth 45. inches, which is three foot and 9 inches, but he setteth not down what depth every step hath, which must not be over deep: for then how can any man easily mount up to the top thereof, for he saith it is plain in the top, and so large as 50. persons may stand thereon. It is found by writing that 360/000 men wrought 20. years in building this Pyramids.  
The second great Pyramids is somewhat less, and smooth on the outsides without any degrees or steps, and the top thereof is sharp pointed.    

The contents of the 10. page.  
IN this page he describeth the Crocodyle, and setteth down the shape thereof, secondly, he showeth whereof the Mummy is bred. Thirdly, he describeth the Unicorn, and setteth down his shape.  

The Crocodyle.  
THe Crocodyle is found in Egypt, in the flood Nilus, and in India in the flood Ganges, and in the two Provinces Mexicana and Pervana in many rivers. This is a four footed beast, which hath a horrible head, sharp teeth, a very small tongue, and a thick tail, and his skin is hard and armed with hard scales, the neither part of his mouth is immovable, and the upper part movable, contrary to all beasts, he doth devour both men and beasts, and doth keep more in the water than on the land, and that which is greatly to be wondered at, he is engendered of an Egg, as great as a Goose egg, and he groweth by little and little until he come to the length of 18. cubits, or as some say to 22. cubits, which maketh 33. foot.   

Of Mummy.  
MVmmie is made of bodies embalmed, which they bring from Egypt, whereas many such embalmed bodies were buried, about four hours journey beyond Cayre, whereas was sometime the great city of Memphis, for before the nativity of jesus Christ the Egyptians being Paynims did spare no cost to keep the bodies of their Parents from putrefaction, and therefore they are great palpable lies, whereby fools are persuaded, that the Mummy proceedeth of those bodies which do perish in the sands that be in the deserts of Arabia, as though it were possible that those bodies could be preserved in those sands without stench or putrefaction.   

Of the Unicorn.  
THe Unicorn, as Lewes Vartiman testifieth, who saw two of them in the town of Mecha, is of the height of a young horse or colt of 30. months old, which is two years and a half old, he hath the head of a Hart, and in his forehead he hath a sharp pointed horn three cubits long, he hath a long neck, and a mane hanging down on the one side of his neck, his legs are slender, as the legs of a Goat, and his feet are cloven much like to the Goat, his hinder feet are hairy, and his hair in colour is like to a bay horse. This beast in countenance is cruel and wild, and yet notwithstanding mixed with a certain sweetness or amiableness. His horn is of a marvelous great force and virtue against Venom and poison. The Unicorn is found in Aethiopia, like as the Indian Ass is found in India, which hath likewise one only horn in his forehead,    

The Contents of the 11. Page, wherein he first describeth afric, and then certain fruits and spices, as Nutmegs, Mace, & cloves, & setteth down the shape of them, than he showeth which be the mightiest Princes in afric, and thirdly he describeth Mexicana, which is the first north part of America.  
afric being the third part of the World, is separated from Europe and Asia by the sea Mediterraneum and the read Sea, and by the land strait which is betwixt Egypt and Palestina. The chief Provinces of afric are these, Egypt, Barbary, Biledulgarid, Sarra, Aethiopia, Nubia, the large Provinces of the Abassines, falsely called the land of Prester john, and also Monomotopa. The most renowned Isles belonging to afric are these, Socotora, Madagascar, S. Thomas, the Isles of Capo Verde, and the Isles of Canary and Madera.  

The Nutmeg tree.  
THe Nutmeg tree groweth in the isle of Bada, and differeth not much from the Peach tree, saving that the leaves of the Peach tree are shorter and rounder: The fruit is covered with a thick bark or husk, which when it is ripe cleaveth in sunder, and showeth the Nut together with the shell, which is covered with Mace, the which at the first view is as read as Scarlet, and pleasant to behold, but when the Nut waxeth dry, the Mace do sever from the Nut, and losing by little and little their Scarlet colour, do wax nigh unto the colour of an Orange.   

Of the Clove tree.  
THe Clove tree groweth in the Isles of Moluccas, which in greatness and shape is like unto the Laurel tree, saving that the leaf thereof is somewhat narrower. It hath many branches, and a great number of flowers, first white, afterward green, and then read, but being dried, they become black. The cloves do grow upon the outermost ends of the branches, one hard by another and whilst the flowers are green they excel all other flowers in sweet odour.   

The chiefest Princes of Africa  
THe most puissant Princes of afric are these, the Emperor of the Moors or Ethiopians, which of the Arabians and of the Mahometistes is called Aticlabassi, and of his own subjects, he is called Acegue and Neguz of the Abassines, that is to say, Emperor and King of the Abassines and Moors. Than the king of Monomotapa, the king of Morocho, the king of Fez and Sus. The great Turk also possesseth many provinces in afric.  
The chiefest Merchandizes that come from afric into Europe are these, Gold, ivory, wood of Ebony, Aloes, Balm of Egypt, Mummy, Myrrh, Anil feathers, Sugar, Ginger, Dates and wines of Madera, and of the isle of Canary.   

Mexicana.  
MExicana which is the fourth part of the world, is on all sides environed with the sea, saving that nigh unto Nombre de Dios it is joined by a land straight to Peruana. The chief provinces of Mexicana are these, the province of Mexico, otherwise called nova Hispania, terra Florida, Norum Bega, nova Francia, Estotiland, Saguenay, Chilaga, Toconteae, Marata, California, Tolm, Quivira, Agama, and Anian. The chiefest Isles lying on the North & Northeast part of Mexicana are these, Groynland, Crockland, Island, Freezland, Bacalaos, and Cuba.  
The chief Merchandizes that come from Mexicana into Europe are these, Gold, Silver, Pearls, Cochenilles, to die with, Balm, Salsaparillia, the root Mechoicana, Brimston, hides of Oxen and Molue.    

The contents of the 12. page.  
IN this Page he first describeth the beast called in that tongue Aiotochli, in Spanish Armadillio. Than he describeth the 2. Provinces Peruana, and Magellanica, than he showeth which he the most mighty Princes of the world, and finally the divers qualities of the people inhabiting the world.  
THe beast Armadillio is found in the Realm of Mexico, and he is no bigger than a cat, he is headed like a Swine, and hath the feet of a Herison, and a long tail, he is armed with scales, whereof he taketh his name, he keepeth for the most part within the ground, and as some suppose, doth live by the earth, by reason that he is never seen to eat abroad out of his den, the bones of his tail are medicinable, and do remedy the pain and deaffnes of the ears.  
Though Plancius saith that this beast is armed with scales, yet my countryman William Greenway, who is a proper servitor both by sea and land, and hath been in the West Indies, and hath eaten of this beast, affirmeth his flesh to be white & very delicate, and that he hath no scales, but that his skin is white and smooth like to a pig new scalded, and that sometime he will shrink up the skin upon his back into divers plates, and specially towards his fore parts and hinder parts, in such sort, as he will make them almost to meet, and the former plates do hung down upon his shoulders like unto two Poldrons, and his hair is white and short, growing thin here and there one, and he is eared and tailed like a rat, even as he is here portrayed, saving that he is throughout of one self colour, and without scales.  

Peruana.  
PErvana being the fift part of the world, is also environed on all sides with the sea, save whereas the foresaid landstrait doth join the same to Mexicana, and the chief provinces which it containeth are these, Brasilia, Tisnada, Caribana, Carthagena, Peru, Charchas, Chili, Chica, & the land of the Patagones'▪ The most renowned Isles are these, Hispaniola, otherwise called S. Domingo, Boriquen, & Margarita, which is the isle of pearls.  
The Merchandizes which are transported out of Pervana into Europe are these, gold, silver, Emeralds, Pearls, the medicinable stone called Bezoar, Balm, Ginger, Sugar, wood of Brasill, wood of Guaicum, called Lignum vitae, long Pepper, Pepper of Brasill, Cassia solutiva and hides of Oxen.   

Magellanica.  
THis is the sixth part of the world, which as yet is but little known, in such sort as we cannot writ any thing touching the provinces of the same, notwithstanding it is thought the province of Beach is very rich, and hath abundance of Gold, the chief Isles of Magellanica are these, java maior, and java minor, Timor, Banda, the Molucques, Romeros, the Isles of Solomon.  
From the Isle Timor doth come into Europe, the white and pale medicinable simple called Sandalum.  
From the Isles Banda doth come Nutmegs and Maces.  
And from the Isle's Molucques Cloves.   

Which be the great Princes of the World?  
THe most mighty Princes of the world are these five, that is, the King of China, otherwise called the great Cham 1. the king of Persia 2. the great Turk 3. the Emperor of Aethiopia 4. the Emperor of Russia, otherwise called the great Duke of Muscovia 5. amongst which the king of China is a Pagan or Heathen: and the great Turk and the king of Persia are Mahometists: but the Emperor of Aethiopia and the great Duke of Muscovia do make profession of the Christian religion. Now as touching the king of Spain, his puissance should be much greater than it is, if his provinces were not so separated, and so far distant on from another.   

The qualities of divers people in the World.  
AS touching the qualities of peoples, though God almighty hath created all men of one self blood, and that all do take their beginning from the Ark of Noah, and that all men be of one self quality and shape of body, yet they differ in greatness, in proportion of members, and in colour: for the Patagones' do exceed all other creatures in greatness. Again, the men of China have most commonly broad faces, little eyes, flat noses, and little beards, and those that have smallest feet, are counted amongst them to be most beautiful, those of Africa have grosser and thicker lips than other people, the inhabitants of Agysimba, and of Guinea, and specially of the lands that be nigh unto Cape de bona esperanza, are black, from whom the Oriental Indians do not much differ.  
The Abassines or Moors of Egypt, be of a duskish colour like to the Olive, the Inhabitants of Barbary, be called white Moors, and those that devil betwixt them and the Nigrites or black Moors be of a yellowish colour, the Spaniards have not found either in Mexicana, or in Pervana any Nigrites or Blackmoores, but only in certain villages nigh unto Carque, the other nations under the hot Zone, be of colour brown bay, like a chessenut, and the nigher that they devil to either of the Poles Arctique or Antarctique, the whiter most commonly they be, and as touching the rest all are like in qualities, shape, and fashion of body, as hath been said before, wherefore they are mere lies that are wont to be told of the Pigmeans, in that they should be but a foot and a half high, and likewise that which hath been spoken of people, that should have their heads their noses, their mouths, and their eyes in their breasts, or of those that are headed like a dog, or of those that have but one eye, and that in their forehead, or of those that have but one foot and that so great, as that it covereth and shadoweth all their body, or of those that have great ears hanging down to the ground. All these are mere lies, invented by vain men to bring fools into admiration, for monsters are as well borne in Europe, as in other parts of the world.    

Now in the four pages following, he setteth down an interpretation of the Latin inscriptions dispersed throughout the Map, every which inscription hath his number added, which I do also hear set down in the same order, joining to every number his longitude and latitude, to the intent you may the more easily found out the said inscriptions, of which inscriptions there be in all 71. divided into five parts, whereof the first part containeth 21. inscriptions, belonging partly to Europe, but most to Asia, the second part containeth 12. inscriptions belonging to afric, the third part containeth 11. inscriptions, belonging to Mexicana, the fourth part containeth 6. inscriptions belonging to Peruana, and the fift part containeth 21 inscriptions, belonging to Magellanica.  

The 21. inscriptions belonging to Europe, but most to Asia.  
The number of inscrip. Longitude Latitude North or South. 1 5. 85. 30′· North. THis arm of the sea doth take his course through 3. places running continually towards the North pole, and is frozen 3. months of the year, and it containeth in breadth about 37. leagues. 2 38 0′· 86. 30′· North. It is said that this Country is inhabited of Owarttes called in Latin Pigmei, being in height four foot as those be of Groinland, which are called Serelinges. 3 81. 0′· 77. 0′· North. Muscovia is bounded on the North side with the sea Puzorique, called of the ancient Geographers Mare Glaciale, that is, the frozen sea, and towards the East it bordereth upon Tartary, and towards the South upon the sea called Mare Caspium and also upon the Turks, and Tartaries Perocopsiques, and towards the West, it bordereth upon Lituania, Livonia, and upon the Realm of Swethland: as touching their religion they observe the Faith and ceremonies of the Greek Church, all their Bishops called in their language Vladiques, and also their Metropolitan are under the obedience of the Patriarch of Constantinople. The people is wise and subtle, and yet loving servitude more than liberty or freedom, and they all confess themselves to be the servants and slaves of the great Duke, having seldom or never peace, for either they have wars with those of Lituania, or with those of Livonia, or with those of Swethland, or with the Tartarians, or if they have no wars, than they lie in garrison nigh unto the 2. floods Tanais and Dwina, to defend their bounds from the depredations and invasions of the Tartaries, they sharply punish robbers and stealers, and yet privy theft and murder is seldom punished with death: their silver money is not round, but hath the form of an egg, their country is every where full of woods, and they have great abundance of rich furs, they sand into all countries of Europe very good Flax and hemp to make Cables and ropes, and a great number of hides, as well of Oxen as of Elkes, great balls of Wax, much salted Fish, and Whale's grease. The great Duke, called of his servants the Emperor of Russia, is rightly accounted amongst the most mighty monarchs of the world. 4 98. 0′· 82. 0′· North. This arm of the sea hath 5. mouths or entries, and by reason son that it is so strait and hath so violent a course, it is never frozen. 5. 123. 30′· 58. 0′· North. Under the name of Tartary at this day are comprehended all the provinces that border towards the East upon the sea of China & towards the South, are limited with the provinces of China, of India, with the flood Oxio, with the sea Mare Caspium, & with the lake or marish called Palus Meotis, & towards the West they are bounded with the flood Boristenes, & with the limits of Muscovia: for the Tartarians have conquered all the countries which they possess at this present, so as Tartary comprehendeth all that country which the ancient men were wont to call Sarmasia, Asiatica, and also the Scythias, that is, intra Imaum and extra Imaum, they began first to be renowned in Europe, in the year of Christ 1212. The Tartarians are divided into certain commonalties, and Colonies, called of them Hordes, but for so much as they devil in divers provinces, that do extend far, and be far distant one from another, they differ also in their manners and trade of life, they be men of a square stature, having broad and gross faces, their eyes hollow sunk into their heads, and looking somewhat a squint, and thick beards, they be strong of body and hardy, they eat horses and all other beasts how so ever they are slain, saving hogs, from which they abstain, they are able to endure hunger, thirst, watch, and all discommodities, and when they are distressed in their voyages with hunger and thirst, they let their horse's blood, and with that blood quench their hunger and thirst, which kind of meat they call in their language Besermannen, they call their Emperor Cham, that is to say, Prince, and therefore Cambalu is interpreted to be the seat or town of the prince. 6 125. 40′· 50. 0′· North. This round lake in the province of Sancii, took his first original and beginning in the year 1557. by reason of an inundation or flood, which carried away 7. towns, besides villages and other places nigh adjoining, & a great multitude of people, whereof none were saved but only one infant, sitting upon a tree. 7 124 0′· 37. 0′· North. The inhabitants of China, are of good spirit and ingenious, insomuch as they have invented certain kind of carts, wherein they may sail upon even ground, having wind, and sail as they do in ships upon the sea. 8 171. 30′· 80. 30′· North. 
Plancius in this inscription setteth down the opinion of Mercator, touching the beginning of longitude, and touching the Adamant stone, otherwise called the loadstone, in Latin Magnes.  
Frances of Diep, a most skilful Pilot, doth witness that the needle of the Mariner's compass, doth turn directly to the North Pole, being in the islands of Capo Verde, that is to say, the isle of Sal, the isle of Bonavista, and the isle of Mayo, whereunto those do agree very nigh, which do say that the needle doth the like in the Isles of Tercera, and of S. Marry, which are part of the Flemish Isles, otherwise called the Azores: but some others do affirm, that the needle showeth the North pole best, being in the isle of Corvo, which is the furthest Isle westward of the said Azores, and because the longitude of places by most lively reasons, aught to take his beginning from the common Meridian of the world, and from the rock or Pole of the Adamant stone, we here following the opinion of those that are most skilful in this matter, have set down the first Meridian betwixt the Isles of Capo Verde, and the Azores, and because the needle in all other places, declineth either more or less from the Pole of the world, there must needs be another Pole in some one place whereunto the needle doth incline from all coasts of the world, and I have found by the declination of the needle observed at Ratisbone, otherwise called Regensberg, that is the same place, which I have set down in the Map, and I have likewise marked in this Card, the situation of the Pole of the stone, in respect of the isle Corvo, to the intent that according to the outermost places limited by the first Meridian, the outermost bounds, betwixt which, this Pole aught to be found, might be known, until the diligent and curious consideration of the Pilots shall bring us something of more certainty. Thus far Mercator.  9 162. 0′· 71. 0′· North. These are the plain fields of Bargu, whereof the inhabitants are called Mecriti. 10 154. 30′· 62. 0′· North. The Mount Askai, in which are to be seen the sepulchres of the Kings of Tartary. 11 171. 30′· 70. 0′· North. Upon this Mountain are set by the Tartarians two trumpeters of brass, for a perpetual memory of their freedom gotten. 12. 175. 0′· 67. 0′· North. The Province Vng, which of our men is called Gogg. 13 170. 30′· 65. 20′· North. The Realm of Tendus, which in the time of Marcus Paulus Venetus, which was in the year 1290. was governed by those Christians, which descended from the king Vncham. 14 165. 30′· 65. 0′· North. The Castle of King Vncham was builded in this place, against the invasion of the Tartarians. 15 157. 0′· 65. 30′· North. The Province Mongul, which of our men is called Magogg. 16 159. 20′· 66. 0′· North. The desert Belgian, which is very great, all sandy and barren. 17 159. 0′· 61. 30′· North. In this Country of Cergutha, is found the best sort of musk, which groweth like as an imposthume or bag, nigh unto the navel of a certain beast. 18 165. 30′· 58. 0′· North. Campion is the Metropolitan Town of Tanguth, whereas the inhabitants are partly Christians, partly Idolaters, and partly Mahometists. 19 158 30′· 53 0′· North. This wall hath in length 400. Spanish leagues, and was built betwixt the mountains by the King of China, against the invasions and excursions of the Tartarians. 20 176. 0′· 40. 0′· North. japan hath three islands much renowned, separated one from another by a strait of the sea, whereof the first and greatest is divided into 53. Provinces or Realms, of which Meaco is the Metropolitan city, the second is called Xima, which hath nine Provinces or Realms, the third is called Xicoca which containeth but four Provinces or Realms. 21 71 0′· 29. 0′· North. Medina Talnabi, is the town wherein is to be seen the Sepulchre of Mahomet.   

The 12. Inscriptions belonging to afric.  
The number of inscrip. Longitude Latitude North or South. 1 67. 0′· 26. 0′· North. Coptos is a trim Merchant Town, whereto are brought the Merchandises of India, and of Arabia, in which town are dwelling many Christians. 2 58. 0′· 21. 0′· North. The flood Nilus, which by his inundations doth yearly water and fat the Country of Egypt, and maketh it marvelous fruitful. 3 55 0′· 17 0′· North. The flood Nubia, taketh his original from the lake Nuba, as Ptolemy saith. 4 50. 0′· 15. 0′· North. The flood Niger, here taketh his course, running under the ground 50. leagues. 5 62 0′· 5. 0′· North. In this place is the ample jurisdiction of the Emperor of Aethiopia, wrongfully called of those of Europe, the land of Prester john, the Arabs and the Moors do call him Aticlabassi, but his own subjects do call him Acegue, and Neguz, that is as much to say, as Emperor and King. 6 58 30′· 2. 30′· North. The mount Amara, whereas are most carefully kept with continual watch and ward of soldiers, the children and grandchildren of the Emperors of Aethiopia. 7 53. 0′· 7. 0′· South. It is said that this country is inhabited of amazons, which are women that make war. 8 55. 40′· 15. 30′· South. In this place the King of Monomotopa, hath his great and ample jurisdictions. 9 42. 30′· 15. 40′· South. This South part of afric, unknown to the ancient writers, is called by those of Persia and of Arabia, Zanzibar. 10 45. 30′· 1. 0′· South. Hear is digged out of the Ours, great abundance of gold. 11 30. 20′· 20. 0′· North. Libya or inward afric, is at this day called Sarra, or the deserts. 12 19 0′· 15. 30′· North. The people Azanagi, are of colour black grey, and they cover their mouths as a member of shame, and do never uncover them but when they eat.   

The 11. Inscriptions belonging to Mexicana.  
The number of inscrip. Longitude. Latitude North or South. 1 301. 30′· 81. 0′· North. This I'll is thought to be the best and most wholesome of all the North parts. 2 301. 0′· 74. 0′· North. The I'll of Crockland, the inhabitants whereof say that they had their original from Swethland. 3 311. 0′· 61. 30′· North. In the year of our Lord, 1500. one Gaspar Corteriale a Portugal, entered into these Regions, hoping to have found some passage on the North towards the Isle's Molucques, but arriving at the River, which by means of the abundance of snow there falling, is called Rio Nevado, which hath in North Latitude 62. degrees, he did leave to sail any further towards the North, by reason of the great cold there, and turning to the South did fetch in all the sea coasts until he came to Capo Razo, which hath in North Latitude about 48. degrees: and in the year 1504 the Britaines were the first that discovered all the sea costs of new France in America, nigh unto the golf of Saint Laurence, which hath in North Latitude about 50. degrees and in the year 1524. john Verazzan a Florentine, did part from the Port of Diep, the 17. of March, in the behalf of Frances king of France, and sailed towards the South sea coast of new France, whereas he arrived at the 34. degree of North Latitude, and from thence sailed towards the East, viewing all the Sea costs until he came to Cape Britain, which hath in North Latitude about 46. degrees. 3 311. 0′· 61. 30′· North. And in the year 1534. new France was again visited by the Admiral jaques Cartier, and in the year next following, it was conquered to the use of the King of France, also in the year 1577. Martin Furbosher Englishman, arrived at the north strait which is betwixt Groinland & Estotiland, that place having by this Map in the North latitude 65. deg. seeking passage by the North unto Cathay, whereas he found certain Isles, & a mine of gold, wherewith having laden his ships, he returned into England with great hope of profit, but his success was not answerable to his hope. 4 275. 20′· 70. 0′· North. This dangerous beast is called Sucaratha, which being chased of Hunters, doth take her young ones upon her back, to save both herself and them by flight. 5 271. 0′· 64. 30′· North. This is a great lake or sea of fresh water, the limits whereof are unknown as they of Canada do say, and as they have heard by relation of those of Saguenay. 6. 293. 30′· 52. 0′· North. Alongst this River a man may sail very commodiously, towards the country of Saguenay. 7▪ 287. 0′· 47. 40′· North. All those that devil betwixt Terra Florida, & Terra de Laborador be called by one common name Canadois, but there be many divers Nations as those of Hochelada, Honqueda, & Corterealia, they are all very courteous to strangers, they live commonly by fish, & they are clothed with skins of wild beasts, as they be also that devil further towards the North, this country is also called new France, because the Britain's which are French men did first discover it in the year of our Lord 1504 the conquest whereof was achieved by the Admiral jaques Cartier in the year of our lord 1535. to the behoof & use of the French king. In the mountains towards the South, do devil many and divers Nations, which be cruel people, living without any law, & do seek by continual war, to vex and oppress one an other, as the people Auanares, Albardi, Calicuares, Tag●●es, Apalcheni, Mocose, Capaschi, Chilage, and many others, amongst which there be some of such agility and swiftness, as they may contend with horses, who can run fastest, they eat as those of Florida, certain kind of Spiders, Aunts or Pismires, Leazards, Adders & other venomous beasts. The land of Baccalaos, is so called of the fish Baccalaos, which is there taken. Terra Florida is so called of Easter day, because john Ponce of Lion did discover it on Easter day, in the year 1512. the which day is called of the Spaniards Pasqua du flores: this is a fertile country and rich in gold. 8 271. 0′· 39 0′· North. Marcus Nizza testifieth that this Province called the seven Cities, is a very good country with whom Frances Vasker doth not agree, for he saith they be places of no value, and like to Villages, and be under the jurisdiction of Cevola, which at this present is called of the Spaniards Nova Granada. 9 211. 40′· 82. 0′· North. 
The North Ocean sea, entering with 19 mouths betwixt the Isles of the North, doth make four straitss of the sea, and running floods, which continually take their course towards the North, & there are swallowed into the bowels of the earth, like to springs of fountains and floods.  
In mine opinion this Inscription would have been one of the first of Mexicana, and not one of the last, sith it is no small leap, to turn so suddenly from the midst of America to the North pole.  10. 223. 0′· 55. 40′· North. This Country is desert and plain, in which are many wild Horses, and Oxen with high backs like Camels, and wild sheep like unto those which Boetius writeth (in his description of Scotland) to be found in one of the Isles of the Hebrides. 11 240. 20′· 40. 30′· North. New Spain was brought by force of arms under the obedience of the Spaniard, in the year 1518. by their general, Ferdinando Cortes who conquered the same with great loss of his soldiers, but with greater ruin to the inhabitants, who fought for the liberty of their Country, the soil is very fertile, and the Country is rich in gold and silver, for in the floods are found great sands of gold, and in the mountains is drawn out of the mines great quantity of silver, and alongst the sea side they take to their great profit an infinite number of Oysters, wherein are found very fair pearls. In this Province there are many salt lakes, the water whereof by force of the sun is turned into salt: there groweth also great abundance of Cassia fistula, and an other kind of fruit, which the inhabitants call in their tongue Cacao, it is like to an Almond, they have it in great price, for of it they make a certain drink, which they love marvelous well. The sea and floods which do wash this Province, do furnish them with great plenty of fish, and in those floods are many Crocodiles, whose flesh the inhabitants do eat, and this beast will grow to be twenty foot long and above. This Country is full of great mountains and high rocks, there is great diversity of languages, in so much as one understandeth not an other without an interpreter. Mexico is the Metropolitan and Royal town, or rather the Queen of all the principal Towns in the world, it is situated upon the side of a lake or marish, yea the very foundation of it is a very marish, in such sort, as you can neither enter into it, nor come out of it, but by bridges, and it is aswell peoplished with inhabitants and Merchants, as any renowned Merchant town in Europe, the town is very great, for it containeth in circuit 3. Spanish leagues.   

The 6. Inscriptions belonging to Peruana.  
The number of inscrip. Longitude. Latitude. North or South. 1 333. 0′· 3. 30′· South. This great flood Maragnon, is called of some Oreigliana, and also the flood of the Amazons, it was discovered by Vincent john Pinsonio, in the year 1499. and was sailed in a manner from the head spring even unto the sea, by Francisco Oreigiano in the year 1542. the which voyage he performed in eight months, having sailed 1660. leagues. This river keepeth his water still fresh after it is entered certain leagues into the sea, by reason that his course is so swift and violent. 2 330. 30′ 22. 0′· South. 
Peruana is the South part of America, and is divided by the Spaniards into five goodly Provinces, that is to say, Castiliador, that is to say, the golden Castilia, Pompaiana, Peru, Chila, and Bresille: The Province Peru (before the arrival of the Spaniards) did extend a great deal further, when as their Country was yet under the government of their natural King, which then was called Ingas, as Girava and others do writ. At this present it is limited on the North with the Town Quito, and towards the South with the Town of Plata, that is to say, the silver Town, and it was called Peruana as some writ, of the flood and Port of the sea called Peru. It is at this present divided (according to the situation of the Country) into three parts, that is to say, Sie ras, Andes, and the flat or plain country.  
The plain Country is that which lieth alongst near to the Sea-coast.  
Sierras is that part which is full of mountains.  
And Andes is that part of the Mountains which tendeth towards the East. Of all the Provinces in the world this is the richest in gold, and in Emeralds, the Metropolitan Town of this Province, is the City Lyma, otherwise called the Town of ●inges. Castillelledor took his name of the great abundance of gold that is there. The Province Popaiana took his name of the renowned Town Popaian, Chila is a cold country by reason that it is so nigh unto the South pole. The Province Brasilia took his name of the wood called Brasill, which groweth there in great abundance. To these Provinces were meet to be joined these other Provinces, that is to say, Caribana, Charcas, Chica, and the land of the Patagones'.  3 329. 0′· 28. 0′· South. This beast is called of some Haute, but of a certain people of Brasill, it is called Hay, which beast was never seen to eat or drink, as some writ, and therefore some think that she liveth without meat or drink, only by the air. 4 328. 0′· 35. 20′· South. This kind of beast is found in the Province of Parias, which in his fore parts is like to a Fox, and behind he is like to an Ape, saving that he is footed like a man, and he hath the ears of an Owl, and under her ordinary belly, she hath also another belly, which openeth and shutteth, wherein she lodgeth her young ones, until they are able to get their own living, and she never suffereth them to go out but only to suck, Gesner calleth this beast an Ape-foxe, or a Foxe-ape. 5 325. 30′· 36. 30′· South. This is the land of the Patagones', the inhabitants whereof are Giants nine or ten foot high, which do paint their visages with diverse colours, made of herbs. 6 335. 0′· 51. 30′· South. In the year 1582. the king of Spain commanded here to be built certain fortresses, at the entry or mouth of the strait called Mare Magellanicum.   

The 21. inscriptions belonging to Magellanica.  
1 188. 30′· 17. 20. South. This land is new Guinea, so called of the Navigants and Pilots, because the sea coast and situation thereof, is like to that of Guinea in afric, and it was called by Andrew Corsali the Florentine, the land of Piccinacoli, and perhaps it is that which Ptolemy calleth the Isle Labadia, if it may be called an Isle, for it is not yet known whether it be any part of the South firm land or no. 2 209. 0′· 31. 30′· South. It is not unknown to all those that are exercised in Geography, that the degrees of Longitude do diminish and decrease from the Equinoctial to either of the Poles, either North or South, whereby it falleth out, that the Provinces which are next unto the two Poles of the world, do differ greatly from that natural shape which they have by the roundness of the earth, and for that cause we have briefly drawn a description of the whole world in two round figures or Circles at the end of this Map, to the intent that every man might see their natural situation so far as may be showed in Plano, that is to say, in flat form. 3 231. 72. North. These Provinces at this present are little known, yet it is said, that they are full of many kinds of wild beasts. You shall find this inscription in the round figure on the left hand nigh to the North pole, which indeed belongeth to Mexicana, and not to Magellanica. 4 247. 20′· 14. 0′· South. These two infortunate Islands were so called by Magellane himself, because he could found in them neither men, nor any thing else that was meet for man's sustenance. 5 283. 0′ 61. 20′· South. These Provinces were discovered by a Spaniarde, who being separated by tempest of sea, from the fleet or army, ran wandering here and there, through this great Southern sea. 6 350. 40′· 85. 20′· South. In the year 1493. when as the desire to sail into far countries increased more and more amongst the Castilians and Portugals, and that with great contention who should discover most, Pope Alexander ordained, that the Meridian which is 100 leagues distant towards the West, from every one of the islands aswell of Capo Verde, as of those which they call Azores, should be the bounds and limits to either party, for their navigation, determining their rights in such sort, as the Castilians should have the West part of the world, to find out unknown countries, and the Portugese's the East part, but there was such strife and contention betwixt them afterward touching the bounds of Navigation, as this ordinance of Pope Alexander, pleased neither party, and therefore in the year 1524. it was fully determined, that the Meridian which is distant 370. leagues towards the West from the isle Saint Anthonio, being the most Western I'll of all the Isles of Capo Verde, should by the common bound of Navigation to both parties. 7 19 0′· 54. 30′· South. Here under the Latitude 42. degrees, & distant 450. leagues, from the Cape de bona speranza, and also 600. leagues distant from the Cape Saint Augustine, was found the Promontory of the South land, as Martin Ferdinando Denciso hath noted in the Epitome of some of his Geography. 8 22. 0′· 65. 0′· South. In this our Geography, we have in such manner described the circuit of the whole earth, as all the Countries are situated under their proper Meridian's, which could not have been done, without extending them more or less from the West towards the East, notwithstanding being desirous to satisfy those that are practised in the art of Navigation, we have described the North Provinces of Europe in this our Geography, in such sort as their situation wholly agreeth with the particular Mariners Cards. 9 57 20′· 54. 0. South. 
The 9 Inscription is entitled to the lovers of Geographie.  
We have in this Inscription of the whole earth, employed all diligence to describe all the Seas and Provinces, in such sort as every place may have his true Longitude and Latitude, in which matter, we have spared neither labour nor cost, for we have diligently conferred together the Sea Cards, as well of the Castilians as Portugese's, which they use in their Navigations to America and to India, and amongst others we obtained from Portugal a Mariner's Card, describing the whole earth very correctly, and besides that 14. other particular Cards, in the which likewise all the Seas and Provinces of the whole world, with their situation were comprised, all which Cards being compared together, we do here now set forth this new description of all the Provinces and Seas in the whole world, and that as correctly as may be, according to the consideration and observation which hath been used by the most expert Geographers and Pilots, even unto this present hour: for in this Card we do describe all the sea coasts, Promontories, windings in and out, Isles, Ports depths, sands, showlds and rocks, also we have added thereunto in place convenient, the Mariner's compass, and the lines of the winds, which we have set down as correctly as was possible for us to do, for the commodity of Navigation: and for so much as the true Longitude of the places can not be well observed, without extending or enlarging too far those Provinces that are nigh unto the North or South pole of the world, we have therefore briefly comprehended the same in this our description of the world, in two rondles or circles, unto the which we have added another little Geographical Card, comprehending the north Provinces of Europe, to the intent that in them every man may see with his eyes, the natural situation of those Provinces as well as it may be done in plano, as is more amply declared in other inscriptions of this Map. And this little Septentrional Map last mentioned, is placed at the foot of this map, in the very midst thereof comprehending all the North parts that lie betwixt the 52. degree, and 72′· of Latitude.  10 79. 30′· 56. 0′· South. Here is a very strong current of the sea, which ruuneth East & West, betwixt Madagascar and the isle Romoros, in such sort as Navigation there is very troublesome and laborious, as Marcus Paulus Venetus testifieth in his third book & 40. chap. whereof it must needs follow of necessity, that the sea coasts of this country are not far distant from Madagascar, in such sort as the great Oriental sea doth ebb and flow through this strait with great violence into the West Ocean sea, whereunto agreeth the letter missive of a Candiot, who was Ambassador for the Venetians to the king of Portugal. In which letter he writeth that the men of this Country go all naked. 11 91. 0′· 35. 0′· South. To this place arriveth a Portugal ship called S. Paul. 12 212. 36 South. 
The South firm land, is called of some Magellanica, of Ferdinando Magellanus, who first discovered the same.  
This Inscription, together with the four next following, are to be found in the rondle on the right hand.  13 146. 46 South. Marcus Paulus Venetus, and Lewes Vartiman do testify in their books of peregrination, that here be very great and ample deserts. 14 75 52 South. Betwixt the isle of Saint Laurence and the Isles Romeros, do fall a most violent flux and reflux of the Sea, East and West. 15 46 55 South. This is the land of Popeniayes, so called of the Portugese's, because those birds in that Country are of incredible bigness. 16 15 46 South. This Promontory of the South land, is situate 450. Spanish leagues from the Cape de bona speranza, and 600. leagues from the Cape of Saint Augustine. 17 148. 0′· 34. 0′· South. As we have in the first rondle on the left hand set down the description of that part of the world, which extendeth from the Equinoctial to the North pole: so in this other rondle, we have set down a description of that part of the world which extendeth from the Equinoctial to the South pole, in such sort as this rondle containeth all Magellanica, and almost all Peruana together with a great part of afric, and a great number of the most noble and renowned Isles of the world, and herein you may plainly see with your eye, the natural situation of those Provinces that are nigh unto the South pole. 18 148. 30′ 20. 30′· South. The Realm of Maletur, which aboundeth in all manner of spices. 19 148 0′· 15. 40′· South. The Country of Beach is rich in gold, but little frequented by Merchants of other Countries, by reason of the cruelty of the people. 20 161. 0′· 20. 0′· South. java minor, bringeth forth diverse spices which have not yet been seen in Europe, as Marcus Paulus Venetus testifieth in his third book 13. chapter. 21 170. 40′· 3. 0′· North. The Isles Moluccas are much renowned for the great abundance of spices, which are sent from thence into all countries of the world: The chiefest of those Isles are these, ternary, Tidoris, Motir, Machian, and Bachian, unto which some do add Gilolo, Celebes, Burro, Amboino, and Bandar.  
Besides all these Inscriptions, Plancius at the four corners of his Map, setteth down four rondles, two above and other two beneath: and in that above on the left hand, representing the Northern half of the celestial Globe, he describeth all the North stars that are already, & in the other rondle on the right hand representing the southern half of the celestial Globe, he doth not only set down such Southern stars as were known to the ancient Astronomers, but also such Southern stars, as have been found out of latter days by those that have traveled into the East and West Indieses, as the Cross, the Southern Triangle, noah's Dove or Pigeon, and an other in the shape of a man, called Polophilax, and certain others, touching which stars he setteth down nigh unto the foresaid rondle, a certain inscription written in the Latin tongue, which I have here interpreted word for word in our mother tongue as followeth.  
We have here set down the fixed stars in their true places answerable to the year 1592. and not 95. as the Printer hath made it.    

Of the South pole and of the stars that are about the same.  
Lest the South part of this Hemisphere or half Globe, should remain void and empty, I have taken these Southern stars out of the observations of Andreas Corsalius Florentine, and have diligently compared the same with the writings of Americus Vesputius, and of Petrus Medina, and have reduced the said stars into this form or shape. But for so much as I have seen nothing as yet to my satisfaction or contentment touching the Longitude, Latitude, Magnitude or nature of the said stars, I heartily pray all those that have any more certain knowledge of this matter than we, that they will inform us thereof, to the common good of all men: As touching the pole Antarctique, Corsalius writeth that there be two cloudy stars of a mean bigness, which with a circular motion do go about another star, that is distant from the Pole almost 11. degrees, and are sometime above and sometime beneath the said star. Hitherto Plancius. But now to the intent that you may the better understand all the foresaid four rondles, I think it not amiss to describe the same unto you, and to show the use thereof as followeth. You have to note then, that the two upper rondles: that which is on the left hand, signifieth the Northern half of the celestial Globe and the other rondle on the right hand, signifieth the Southern half of the said Globe as hath been said before, and each one of these rondles is traced with certain circles and lines: The outtermost Circle whereof being divided into 360. degrees, and containing the Characters of the 12. signs, signifieth the zodiac, or rather the very Ecliptic itself, the Centre of which Circle, is the pole of the zodiac, which by continual turning about, describeth another lesser Circle hard by it, signifying in the North rondle the circle Arctique, and in the South rondle the circle Antarctique, the Centre of which lesser circle in either rondle, is the pole of the world, both which poles are distant from the pole of the zodiac 23. degrees 28′· which is the greatest declination of the sun. Moreover in either rondle are drawn upon each pole of the world, two other circles, the largest whereof signifieth in both rondles the Equinoctial, and the lesser thereof in the North rondle, signifieth the Tropic of Cancer, and in the South rondle the Tropic of Capricorn: besides these circles each rondle is traced with 12. right lines, signifying those six Meridian's or lines which passing through both the poles of the zodiac, do divide the zodiac into 12. equal parts, every part containing 30. degrees: for so many degrees do belong to the Longitude of every one of the 12. signs, whereby the zodiac hath in Longitude 360. degrees, which Longitude is to be counted from the first point of Aries and so forth, according to the succession of the signs, and by help of these lines, you may know under what sign any fixed star is: I shall not need here to show you, how the said fixed stars are situated, in either of the rondles, nor how they are named, because their images or shapes together with their names, are apparent to your eye. But if you would know the true place, the Longitude, Latitude, Magnitude and the nature of any fixed star herein contained, then do thus: First to know the place and Longitude of any star, lay a ruler or extend a thread, so as it may pass through the Pole of the zodiac, and also through the body of the star, whose place and Longitude you seek, even to the very zodiac, and somewhat beyond, and thereby you shall know in what sign, and in what degree thereof that star is, for that is his place, and you shall know his Longitude by counting from the first point of Aries, unto that degree, for that is his Longitude.  
Now to know the Latitude of any star, you have to note that in each rondle there is a certain right black line extending from the zodiac to the Pole, divided by unequal spaces into 90. degrees, which line is called the scale of the fixed stars Latitude, the use whereof is thus: Set the firm foot of your Compasses in the very pole, and extend the other foot into the midst of the body of that star whose Latitude you seek, and turn that foot standing at that wideness to the scale, and the number of degrees written upon the scale, if you count from the zodiac upward towards the Pole, will show you the Latitude of that star. Again, to know the magnitude of any star, Plancius setteth down in the North rondle the self same mean which Mercator also useth in his celestial Globe, that is to say, by making certain shapes of stars representing the bigness of every star, according to his greatness, that by marking & comparing those shapes together, you might find out, or rather conjecture the greatness of the star which you seek. Lastly he showeth the nature of any star by setting down nigh unto the star, the characters of those Planets, of whose nature that star doth participate, all which things you shall more plainly understand by this example here following.  
Suppose that you would know the place, longitude, latitude, magnitude and nature of the star called Arcturus: here because this is a North star, you must therefore resort to the North rondle, and there seek out the image Boötes, betwixt whose legs is the star called Arcturus which you seek. And by extending a thread which may pass through the pole, and also through the body of that star even to the zodiac, and somewhat beyond, you shall found his place to be in the 19 degree of the sign Libra, and his Longitude counting from the first point of Aries, unto that degree to be 199. degrees, and by observing the rule before given, touching the knowing of the Latitude of any star, you shall with your Compass find the Latitude of this star to be almost 32. degrees Northward, and by his shape you shall know that he is of the first magnitude, and the characters of the two Planets Mars and jupiter, placed hard by him, do show that he is of their nature, that is to say, by participating of Mars he is extremely hot and dry, and by participating of jupiter he is hot & moist, and look what order is to be observed in the North rondle, touching the North stars, the same is likewise to be used in the South rondle containing the Southern stars: Amongst which you may see the Images called the Cross, whereby most Pilots in these days do chiefly direct their course, being once past the Equinoctial towards the South pole, which Cross, though Plancius doth here make to consist of five stars, yet I am sure that Martin Cortes and Peter Medina, and all other late writers do appoint thereunto but four stars, the shape and use whereof, I have set down in my Treatise of Navigation, according to the direction of Peter Medina. And those that have travailed into the Indieses, do all affirm that to the Cross there do belong only four stars and no more, wherefore I marvel much, that Plancius doth set down five, whereunto perhaps he is induced by the relation of some Spaniarde that never saw them. Thus having described unto you the two upper rondles, representing together the celestial Globe, and also showed the use thereof, I will now describe the two neither rondles, whereof that on the left hand representeth the North half of the Terrestrial Globe, & that on the right hand the other half of the same Globe, towards the South.  
You have then to understand that the Centre or middle point in each rondle, signifieth the Pole of the world, that is to say, the North pole in the North rondle, and the South pole in the South rondle, and upon each pole are drawn certain Circles, the outtermost whereof, and furthest distant from the Pole signifieth the Equinoctial, which is divided into 360. degrees, every degree containing 60. miles, which is the whole Longitude of the earth, from which circle at the end of every tenth degree, are drawn certain right lines to the number of 18. which do meet in the very Pole, and do signify half Meridian's, whereof that which passeth through the Isle's Azores, and also the Isles of Capo Verde, is the first Meridian, from whence the longitude of the earth taketh his beginning, and there also endeth: which Meridian in the rondle on the left hand, is divided into 90. parts, proceeding from the Equinoctial to the Pole, signifying the North Latitude of the world, the like division and number of degrees of Latitude, hath also the first Meridian, in the rondle on the right hand, saving that the said Meridian tendeth upward. Moreover you have to understand, that in each of these rondles, are drawn nine Circles, equally distant one from another, called Parallels, which together with the Equinoctial, do make nine spaces, every space containing 10. degr. & besides these circles, there are drawn in each rondle two other circles, the one greater, & the other lesser, the greater in the rondle on the left hand being distant from the Equinoctial 23. degrees 30′· which is the greatest declination of the sun, is called the Tropic of Cancer, and the lesser Circle being of like distance from the pole, is called the Circle Arctique, but the greater circle being of like distance from the Equinoctial, in the rondle on the right hand, is called the tropic of Capricorn, and the lesser circle in the said rondle environing the Pole, is called the Circle Antarctique upon which circles in each rondle you shall find their names written. The chiefest uses of these two rondles are these: first to find out the Longitude of any place, secondly the Latitude, and thirdly the distance betwixt any two places.  
To find out the Longitude of any place, you must do thus: Extend a thread, so as it may pass through the pole, and also through the place whose longitude you seek, even to the very Equinoctial, and somewhat beyond, and holding the thread strait, the numbers of the degrees written upon the Equinoctial will show the longitude of the place.  
And if you will know the latitude of that place, or of any other, do thus, Set the one foot of your compass in the very pole, extending the other to the place whose Latitude you seek, and keeping your compass at that wideness, bring the movable foot to the first Meridian, whereon the degrees of latitude are marked, and there staying it, the number of the degrees, counting from the Equinoctial upward towards the pole, will show the latitude of the place. As for example, suppose that you would know the longitude and latitude of Lisbon, which is a famous town in Portugal, here having first found out that town in Spain, which is nigh unto the West Ocean, extend your thread from the pole through the midst of that Town to the Equinoctial and somewhat beyond, and you shall find that the thread will cut the Equinoctial in the 13. degree, which is the Longitude of Lisbon. Now if you would know the Latitude of the same place, set the one foot of your compass in the Pole, and extend the other foot to Lisbon, and keeping your compass at that wideness, bring the movable foot to the scale of latitude, and so you shall found that Lisbon hath in North latitude 38. degrees and 30′·S Now to know the distance betwixt any two places do thus, Set the one foot of your compass in the one place, and the other foot in the other place, and apply that wideness to the Equinoctial, and look how many degrees of the Equinoctial that wideness comprehendeth, and by allowing 60. Italian miles to every degree, you shall have the distance by a right line, betwixt those two places, for by doing thus, you shall find the distance betwixt Lisbon & Compostella to be 120. miles. Thus I have sufficiently (I hope) expounded every thing contained in Plancius his Map, his general scale made for the same only excepted, whereof I come now to speak. In this scale are set down the miles of Russia, of Italy, of England, of Scotland, the French leagues, the hour leagues, the Spanish leagues, the German and Garscoyne miles, which two are all one, the miles of Swevia in Germany, of Scandia and of Swethland, which last three are likewise all one, the use of which scale is thus. Take with your compass the distance betwixt any 2. places which you desire to know, and apply the same to the scale of such miles as you would know, & so many miles the two places shall be distant one from another as the number of the scale doth show: but if the distance betwixt the two places be longer than the scale, then having first taken the whole length of the scale with your compasses, look how many times that wideness of your compass measuring by a right line is contained in the distance betwixt the two places, and if there be any odd space left, straighten your compass to that odd space, & apply that to the beginning of the scale, and add the number of miles which you there found to the first great number, so shall you have the total sum. And lo here for each an example: first suppose that you would know the distance betwixt Cape S. Marry, & Cape finis terrae, which are two Capes or headlands in the West side of Spain, both having in a manner one self Longitude, & do differ only in latitude by 6. degrees, for the one hath 37. & the other 43. in latitude, which distance if you take with your compass by setting the one foot in the one place, & the other foot in the other place, and applying that wideness to the scale of Italian miles, you shall found the distance to be 390. Italian miles, but if you measure the same distance according to the Geographical manner, which is to allow for every degree of latitude 60. miles, you shall found the distance to be no more but 360. Italian miles. Let your other example be thus: Suppose that you would know the distance betwixt Compostella in Spain and Constantinople, which have all one Latitude and do differ only in Longitude, here because the distance betwixt these two places is longer than the scale, you must take with your compass the whole length of the scale, and then to look how many times that wideness is comprehended (measuring by a right line) betwixt the two said places, & you shall found that wideness to be comprehended in the distance betwixt those two places three times, wherefore if you multiply 840. by 3. it will make in all 2520. Italian miles. But if (according to the Geographical kind of measuring) you do multiply the difference of their Longitudes, which is 43. degrees, by the number of miles, which is also 43. belonging to the Latitude of both places, which Latitude is also very nigh 43. degr. you shall found that the distance betwixt those two towns is no more but 1870. Italian miles, which number of miles is not so great as that of the scale by 650. Italian miles. And therefore I can not think but that there is some error in the scale committed either by the Printer or else by the author through some negligence, and not for lack of skill or knowledge how to make a true scale, being so excellent a Geographer as the Author by this & other his Maps heretofore made showeth himself to be, or else there is some greater mystery therein, than I perhaps do understand, for in seeking to know the distance betwixt two places differing only in Latitude, I find the scale most times to agree with the Geographical kind of measuring, but if the two places do differ either in longitude only, or else both in longitude and also in latitude, than I found the scale to differ very much from the Geographical kind of measuring, wherefore I think it good briefly here to set down certain ready ways of finding out the distance of two places, differing either in latitude only, in longitude only, or in both, which I do show also at large in the second part of my Sphere, Chap. 14.   

How to find out the distance of two places differing only in Latitude.  
IF the two places have both either North or South Latitude, then subtract the lesser Latitude out of the greater, so shall you have the difference: which difference, if you multiply by sixetie, the product shall be the number of miles, and if to the whole degrees of difference there be annexed any minutes, than you must add to the product for every minute one mile. But if one of the two places have North Latitude, and the other South Latitude, than you shall find their difference by addition, and not by subtraction. As for example, suppose that you would know the distance betwixt a town called Pasquali, which is the outermost town in Moroa upon the sea towards the South, having in North latitude 35. degrees 30′· and a certain town in afric called Debsan standing nigh unto the lake Zembre, which hath in South Latitude 12. degrees 30′· here by adding these two latitudes together, you shall found the sum to be 48. degrees, and that is the difference of their Latitudes, which difference if you multiply by 60. the product will be 2880. and that is their distance.   

How to found out the distance of two places differing only in Longitude.  
IF both places have either East Longitude or West Longitude, then subtract the lesser out of the greater, so shall you have the difference, which difference you must multiply by the number of miles belonging to their Latitude, which you shall find on the Northwest side of the Map, or by the Table of miles answerable to one degree of every Latitude set down hereafter in the end of this Treatise: and the product thereof shall be the number of miles whereby the one place is distant from the other. As for example, I find Compostella in Spain and Constantinople, having both almost 43. degrees of North Latitude, to differ only in East longitude, for Compostella hath in East longitude 13. degrees 30′· & Constantinople hath in East longitude 56. degrees, the difference whereof by subtracting the lesser out of the greater you shall find to be 42. degrees 30′· here if you multiply 42. by 43. miles belonging to one degree of the foresaid Latitude 43. you shall find the product to be 1806. then to find out the value of miles for the fraction 30′, by the rule of proportion, you must say thus: if 60. require 43. miles what shall 30. require, and you shall find in the quotient 21. miles, which you must add to the former sum 1806. and it will make in all 1827. miles and ½. which is the true distance betwixt the two foresaid places. But if the one place have east Longitude, and the other West Longitude, than you must find the difference aswell by Addition as by Subtraction. As for example, suppose that you would know the distance betwixt S. Domingo in the isle called Hispaniola, and a certain place in afric called Septem montes, night unto the Ocean sea, both places having 18. degrees of North latitude. And S. Domingo hath in West longitude 310. degrees 30′· and the place called Septem montes hath in East longitude seven degrees. Here you must first subtract 310. degrees 30′· out of 360. degrees, and there will remain 49. degree 30′· whereunto you must add the East longitude of Septem montes, which is seven degrees, and it will make in all 56. degrees 30′· which is the difference of their Longitude. Now if you first multiply 56. degrees by 57 miles belonging to 18. degrees of latitude, you shall found the product to be 3192. miles, and to find out the value of miles for the fraction 30′· you must say thus: If 60. require 57 miles, what shall 30′· require, and working by the common rule of three, you shall have in the quotient 28. miles, and there remaineth 30/60. which is one half mile. Now by adding 28. miles and ½. to 3192. it will make in all 3220. mile's ½. and that is the true distance betwixt S. Domingo and Septem montes, but by Plancius scale you shall find the distance to be 3410. miles, which differeth from the other almost 200. miles.  
 〈◊〉. 〈◊〉.   degrees. degrees.  London. 22. 0′· 51. 32′· The greater 〈◊〉. Jerusalem. 6●. 0. 32. 0. The lesser L●●. The difference of their Longitudes 47. 0.     

How to find out the distance of two places differing both in Longitude and Latitude by help of a semicircle divided into a 180. degrees, which I had from my loving friend Master Wright of Cayes College in Cambridge, of whom I make mention aswell in my treatise of the Sphere as in that of Navigation.  
FIrst draw a Semicircle upon a right Diameter, marked with the letters A. B. C. D. whereof let D. be the Centre like unto this here described, and the greater that such semicircle is, the spaces of the degrees shall be the larger, and thereby the more easy to found out the minutes. Than having drawn your semicircle and divided the same accordingly, suppose that by help thereof you would found out the distance betwixt London and Jerusalem, which two towns do differ both in Longitude and also in Latitude so much as is here set down in the front of the figure according to such Longitude and Latitude as Plancius doth allow to either town in his great Map. Now to found out the true distance of these two towns you must first take the lesser Longitude out of the greater, so shall you have the difference of their Longitudes, which is 47. degrees, then count that difference upon the Semicircle beginning at A. and so proceed to B. and at the end of that difference, make a prick marked with the letter E. unto which prick draw a right line by your ruler from D. the Centre of the Demicircle: That done seek out the lesser Latitude which is 32. degrees 0′· in the foresaid Demicircle, beginning to accounted the same from the prick E. and so proceed towards the letter B. and at the end of the said lesser latitude set down another prick marked with the letter G. from which prick or point draw a perpendicular line, which by help of your squire or compasses, may fall with right Angles upon the former right line drawn from D. to E. and where it falleth, there set down a prick marked with the letter H. That done seek out the greater Latitude which is 51. degrees and 32′·S in the foresaid Demicircle, beginning to accounted the same from A. towards B. and at the end of that Latitude set down another prick, marked with the letter I. from whence draw another perpendicular line that may fall by help of your squire or compasses, with right Angles upon the Diameter A. C. and there make a prick marked with the letter K. That done take with your compass the distance that is betwixt K. and H. which distance you must set down upon the said Diameter A. C. setting the one foot of your Compass upon K. and the other towards the Centre D. & there make a prick marked with the letter L. then take with your Compass the length of the shorter perpendicular line G. H. and apply that wideness upon the longer perpendicular line I. K. setting the one foot of your compass at I. which is the end of the greater Latitude, and extend the other foot towards K. and there make a prick marked with the letter M. That done take the distance betwixt L. and M. with your compass and apply the same to the Demicircle setting the one foot of your compass in A. and the other towards B. and there make a prick marked with the letter N. And the number of degrees contained betwixt A. and N. will show the true distance of the two places which you shall find to be 39 degrees, which being multiplied by 60. maketh in all 2340. miles, and whensoever you have any minutes besides the whole degrees, remember to add unto the sum of degrees for every minute one mile. By Plancius scale you shall found the distance betwixt London and Jerusalem to be 3040. miles which are 700. miles too many. But you have to note by the way, that if the difference of the Longitudes doth exceed the number of 180. then you must subtract that exceeding difference out of 360. and the remainder shallbe the difference of the Longitudes, and then work in all points as is before taught.  
By this rule and the other two rules first declared, you shall easily try the scale of any Map whether it be true or not, so as you first have the true Longitude and Latitude of the two places whose distance you seek to know. And thus I end with Plancius Map, hoping not to offend him with any thing that I have added thereunto for the better instruction of those that have not been exercised in such matters.  
And yet I had almost forgotten one thing, which is this, I have here before as you have read, made mention of drawing certain perpendicular lines in the former figure of the Demicircle by help of your Compasses. Wherhfore I think it necessary here to set down the order thereof.   

How to make with you Compasses, a perpendicular line to fall from any point given upon another right line, making therewith right angles without the help of any squire.  
SEt the firm foot of your Compass in the point given, and extend the other foot a little beyond the line right against the point given, & draw a secret Arch or portion of a Circle that may cut the said live in two points, and divide that part of the Arch which lieth betwixt the two sections into two equal parts, setting a prick in the very midst thereof: then having laid your ruler to that prick, and also to the point given, draw a right line, & that line will fall upon the other line with right Angles, as you may see by this figure.  

The Table of miles answerable to one degree of every several Latitude.  D M S  D M S  D M S  D M S  D M S 1 59 59  19 56 44  37 47 55  55 34 25  73 17 33 2 59 58  20 56 23  38 47 17  56 33 33  74 16 32 3 59 55  21 56 1  39 46 38  57 32 41  75 15 32 4 59 51  22 55 38  40 45 58  58 31 48  76 14 31 5 59 46  23 55 14  41 45 17  59 30 54  77 13 30 6 59 40  24 54 49  42 44 35  60 30 0  78 12 28 7 59 33  25 54 23  43 43 53  61 29 5  79 11 27 8 59 25  26 53 56  44 43 10  62 28 10  80 10 25 9 59 16  27 53 28  45 42 26  63 27 14  81 9 23 10 59 5  28 52 59  46 41 41  64 26 18  82 8 21 11 58 54  29 52 29  47 40 55  65 25 21  83 7 19 12 58 41  30 51 58  48 40 9  66 24 24  84 6 16 13 58 28  31 51 26  49 39 22  67 23 27  85 5 1● 14 58 13  32 50 53  50 38 34  68 22 29  86 4 11 15 57 57  33 50 19  51 37 46  69 21 30  87 3 8 16 57 41  34 49 45  52 36 56  70 20 31  88 2 5 17 57 23  35 49 9  53 36 7  71 19 32  89 1 3 18 57 4  36 48 32  54 35 16  72 18 32  90 0 0  
Though it be the common order of working to know by help of the former table, the distance of two places differing only in Longitude, yet I think it a more sure way to found it out per Tabulas Sinuum, the rule whereof is thus.  
First take the difference of the two Longitudes, by subtracting the lesser out of the greater, and the half of that shall be the Arch which you have to seek in the front of the Tables, then multiply the sine of that Arch by the sine of the compliment of the common Latitude, and divide the product thereof by the total sine, the quotient whereof you must seek out in the Tables amongst the sins, and the Arch of that sine is the one half of the distance, which being doubled shall be the whole distance containing degrees of the great Circle, and every such degree containeth of Italian miles 60. and of Germane miles 15. and by working thus you shall find the distance betwixt Compostella and Constantinople to be 1846. Italian miles, supposing the common latitude to be 43. degrees, and the difference of their longitudes to be 42. degrees 30′· And by working by the common table you shall found the distance of those two places to be 1827. Italian miles as before, because the common Table hath no minutes of miles but only seconds, which are not to be accounted of, & in working by Appian his Table having minutes of miles, you shall found the said distance to be 2184. Italian miles, and by Mercator his Map to be 1980. Italian miles, in whose Map the common Latitude of the said 2. places is 43. & the difference of their longitudes is 44. 0′·S And by the scale set down in Plancius his Map, you shall found the distance to be 2520. Italian miles, in which Map the common Latitude of the two foresaid places is 42. deg. 30′·S and the difference of their Longitudes is also 42. degrees 30′· Truly I must needs confess that it is not so easy to make a scale or trunk for a Map or a Card drawn in plano, as for that which is drawn upon a round body or Globe: and therefore it is no marvel though the scales of Maps drawn in plano, and likewise the trunks set down in the Mariner's Cards do not always show the true distance of places, which I believe is to be done as truly and a great deal more readily by my friend Master Wright his Semicircle before described, then by the rules of Gasparus Peucerus in his book de dimensione terrae, which rules do depend upon the knowledge of the quantity of the Angles and sides of Spherical Triangles, which kind of working is indeed more troublesome and tedious then ready or pleasant. But if Master Wright would make his Demicircle an universal instrument to found out thereby all the three kinds of distances as he promised me to do, there were no way in mine opinion worthy to be compared unto it, neither for the trueness, easiness, nor readiness of working thereby.    

A very brief and most plain description of Master Blagrave his Astrolabe, which he calleth the Mathematical jewel. Together with diverse uses thereof, and most necessary for sea men, written by Master Blundevill.  NON SOLO PANE VIVET HOMO: Luke 4 
Verbum Dei manet in aeternum: IW  
Imprinted at London by john Windet.   

To the Reader.  
OF Astrolabes I have never seen but three sorts, First that of Stofflerus, which for these hundred years past or there abouts, hath been had in most price and estimation, as an instrument containing all the uses, or at the lest the most part of all other Mathematical Instruments, which because it requireth almost for every several Latitude a several Table: Gemma Frisius invented since another kind of Astrolabe, having but one Table to serve for all Latitudes, and therefore he called it the Catholicon, that is to say, an universal Astrolabe, which hath also been most esteemed & used many years. Since whose time, & of very late years, one of our own countrymen a Gentleman of Reading besides London, called M. Blagrave, hath greatly augmented the said Catholicon, and hath thereby as it were newly invented a third kind of Astrolabe, which he calleth the Mathematical jewel, whereby are to be wrought more conclusions then by any other one instrument whatsoever, for which his most excellent invention used therein, he deserveth great commendation. And to the intent that others which have not been exercised in such things, might the more easily attain to the better understanding of the said jewel: I have made a plain description of all the parts belonging to the said jewel, without committing any offence (I hope) to the Author thereof, to the intent that every Gentleman might have the perfect knowledge and use of so worthy an instrument: But broad Astrolabes though they be thereby the truer, yet for that they are subject to the force of the wind, and thereby ever moving and unstable, are nothing meet to take the altitude of any thing, and specially upon the Sea, which thing to avoid, the Spaniards do commonly make their Astrolabes or Rings narrow and weighty, which for the most part are not much above five inches broad, and yet do weigh at the lest four pound, and to that end the lower part is made a great deal thicker than the upper part towards the ring or handle: Notwithstanding most of our English Pilots that be skilful, do make their sea Astrolabes or rings six or seven inches broad, and therewith very massive and heavy, not easy to be moved with every wind, in which the spaces of the degrees be the larger, and thereby the truer, of which kind of Astrolabes or Rings, I shall speak hereafter in my Treatise of Navigation, and also of the Mariners Cross staff. But whensoever you have to take the altitude of the Sun or of any other star be in wandering or fixed, I would wish you to use the Mariners heavy and massive Astrolabe, which in mine opinion for that purpose, is the sittest and most assured instrument of all others and to find out all other conclusions by help of Master Blagrave his jewel, or rather by help of the Celestial Globe, which for Astronomical matters is the perfectest instrument of all, in which are contained all the stars both southern and Northern that have been heretofore known, and those that are lately known or shall be known hereafter may be easily placed therein: which thing can not be so well performed in any Astrolabe were it never so great. But because the Globe is cumbersome and not portable, and therewith costly, and especially if it be of any greatness, (for the greater the better) and also if it chance to be broken in any part of the body thereof, it can never be made again perfectly whole, it is therefore no meet instrument for every Mariner to have, but only for such as be of good ability, in steed whereof to find out many necessary conclusions, Master Blagrave his Mathematical jewel may serve very well, and specially if it had on the back part the like matter and Rete or Net, to serve the South Latitude of the world as it hath in the fore part to serve the North latitude, for than I believe verily, that the chiefest Southern star, as the Cross, the Southern Triangle, Noah's Dove or Pigeon, and another called Polophilax, lately found out by such as have traveled by sea on the South side of the Equinoctial, might easily be placed by Master Blagrave in that Rete, having once learned of the skilful Seamen the true longitude, latitude, and declination of the said Stars, so should his jewel in mine opinion be much more serviceable to the sea men, than now it is, by reason that those few Southern Stars that be contained in his first Net, are nothing so nigh unto the South pole, as the Northern Stars described in his first Net, are unto the North pole, amongst which Northern Stars, it were very necessary for the Sea men, that the seven principal Stars as well of the great Bear as of the little Bear, were all duly placed in his first Net. Truly if Master Blagrave his affairs would suffer him to take pain herein, I believe that his jewel should not be much inferior to the celestial Globe, which indeed it representeth, and in so doing he should greatly profit the sea men, and deserve thereby great good will and commendations at their hands.   

What this word Astrolabe signifieth.  
BEfore I begin to describe unto you the said Instrument and all other parts thereof it shall not be amiss to show you what this word Astrolabe signifieth.  
This word Astrolabe is as much to say as the handle or instrument of the Stars, by help whereof the manifold motions and apparences of the heavens and of the Stars therein contained are known, and it is called of some a planispheare, because it is both flat and round, representing the Globe or Sphere, having both his Poles flat both together, the shape or figure of which Instrument I do not here set down because the Instrument itself is to be had for a small price in divers places of London, which if you lay before you when you mind to read this my description thereof here following, I doubt not but that you shall find every part thereof so plainly explained as you shall need no other teach to instruct you therein, or to help you to understand any of these Conclusions that are to be wrought by that Instrument, and specially those which I have here set down, such as the Table following showeth, and as I thought most meet for sea men to know, and being throughly exercised in them, you shall the more easily understand the manifold and necessary conclusions set down by Master Blagnave himself in his own book, which book I would wish you to buy, and earnestly to study the same. But now I will first describe the said Instrument, and then show the use thereof by so many conclusions as are contained in the table hereafter following.   

The description of Master Blagrave his Astrolabe, otherwise called the Mathematical jewel.  
TThis Astrolabe is divided into two parts, whereof the one is called the forepart, and the other the backepart. The forepart containeth two principal parts, that is, the Mater, which is unmovable, and the Rete, which is movable. Again, the Mater is environed with a great circle, called the Meridian, passing through the 2. Poles of the world marked with the letters A.B. and through the two Solsticiall points of Cancer, and Capricorn marked with the characters belonging to those two Signs, and therefore may very well be called the Collure of the 2. Solstices. And this circle is divided by two cross Diameters iuto 4. quarters every quarter containing 90. degrees, so as the whole circumference of the circle is 360. degrees, which degrees do proceed from 90. to 90.  
Now as touching the 2. cross Diameters, the one passing through the Centre, and also through the two Poles, 
The 2. cross Diameters in the Mater.  is the Axletree of the world, signifying sometime the first Meridian, and sometime the right Horizon, and sometime the line of six hours as well for the Morning as Evening, and then it is the line of East and West, and sometime the line of North and South.  
And the other cross Diameter drawn with read ink, signifieth most commonly the Equinoctial, 
The Equinoctial.  and sometime the line of East and West, and sometime the line of North, and South, and especially when you have to find out the 12. houses of heaven, to which end are set down in the outermost Meridian these four Latin words, first at the North pole marked with the letter A. Oriens, which is as much to say, as the East part: and at the south Pole marked with the letter B. Occidens, which signifieth the West part: and at the end of the overthwart Diameter, drawn with read ink, and most commonly signifying the Equinoctial, marked with the letter C. on the right hand is set down Culmen Coeli, the highest part of heaven, which is the South, at which end the Ringle or handle is fastened. And at the other end of the said Diameter, marked on the left hand with the Letter D. is set down Imum Coeli, the lowest part of heaven, which is the North. And by these two cross Diameters the whole Mater is divided into four quarters, that is, North-east, and Southeast, Northwest and South-west. The North-east quarter lieth betwixt Imum Coeli, and the upper end of the Axletree, whereon is written Oriens. And the Southeast quarter lieth betwixt the same point and Culmen Coeli, and the South-west quarter lieth betwixt Culmen Coeli, and the lower end: of the Axletree, whereon is written Occidens, and the Northwest quarter lieth betwixt the point Occidens and Imum Coeli.  
Besides these 2 cross Diameters the Mater is traced with 180 Meridian's which do pass through both the Poles, 
The Meridian's and hour lines in the Mater.  whereofthe first passing through the centre from Pole to Pole is the Collure of the Equinoxes, because it passeth through the first point of Aries, and the first point of Libra, and is otherwise called the axle-tree, and signifieth sometime the right Horizon, and sometime the first Meridian from which upon the Equinoctial are counted the degrees of longitude both East and West. And all the foresaid Meridian's are most commonly used as hour lines, for every 15th. Meridan, counting from the Limb, doth signify one hour, for 15. degrees do make an hour, and 4. minutes do make a degree. And these hours at the end of every 15th. Meridian are marked in the body of the Mater somewhat above the Tropic of Cancer with Arithmetical Figures, proceeding from 1. to 12. forward towards the right hand, and again are marked somewhat beneath the Tropic of Capricorn with like Figures, proceeding backward towards the left hand. Whereof the upper numbers do signify the forenoon hours, and the lower numbers the afternoon hours, and are placed so as one self hour line doth pass through both numbers: for the line of the eleventh hour in the forenoon serveth also to the first hour in the afternoon, and the forenoon hour of 10. serveth also to the hour of two in the afternoon, and the fore noon of 9 serveth to the hour of 3. in the afternoon: and the forenoon hour of 8. serveth to the hour of 4. in the after noon, and the forenoon hour of 7. serveth to the hour of 5. in the afternoon, and the forenoon hour of 6. serveth also to the 6. hour in the afternoon, and from thence forth towards the left hand, 5. serveth to 7. 4. to 8. 3. to 9 2. to 10. and 1. to 11. as before.  
But when soever you have to find out any hour amongst the hour lines in the Mater, remember that the Axletree which passeth through the Centre and also through both the Poles is always the sixth hour both of the morning and evening, from which you shall the more easily find out any hour that you seek by allowing fifteen Meridian's to every hour, as is before said. And you have to note that all these Meridian's do sometime signify every one a several obliqne Horizon, saving that the first Meridian or Axletree signifieth the right horizon as hath been said before. 
How to find out the Horizon of euey several latitude.  And to found out in the Mater the horizon of every latitude, reckon from the axletree on both hands so many Meridian's as may make up the number of degrees of your latitude, and that Meridian shallbe your Horizon. In counring of which Meridian's you shall find that every fift Meridian is drawn with a greater black line then the rest. Now overthwart these Meridian's are drawn 180. Parallels, 
The Parallels of the Mater.  which proceed from 1. to 90. whose numbers are set down in the dimbe of the matter thus, 10. 20. 30. 40. and so forth both upward and downward, until you come to 90. and these Paralle Is are otherwise called the circles of declination as well of the Sun as of any star. Amongst which there be 2. Parallels painted read, 
The two Tropiques.  where of that towards the North pole signifieth the tropic of Cancer, and the other towards the South pole the tropic of Capricorn. Than there is another obliqne overthwart red line, which passing through the centre from Tropic to Tropic, signifieth the Ecliptic line, 
The Ecliptick-line of the Mater  night unto which on each side are set down the Characters of the twelve Signs, whereof Aries and Libra are placed at the centre, and Cancer at the one end of the Ecliptic on your right hand, whereas it toucheth the said Tropic, and Capricorn at the other end of the said Ecliptic, on your left hand, whereas it toucheth the said Tropic of Capricorn, and the rest of the signs are orderly placed betwixt the said Tropiques, every two Signs one against another that have like declination, as Gemini and Leo, Taurus and Virgo, which are placed betwixt the centre and Cancer, then Pisces and Scorpio, and Aquarius and Sagittarius are placed betwixt the Centre and Capricorn. And every Sign contained in the Ecliptic line is divided with slaunting black stréeks into 15. spaces, every space containing two degrees, which maketh 180. degrees, and being doubled, because the spaces are to be counted doth forward and backward, do make in all 360. degrees, which is the whole circuit or longitude of the Ecliptic, and the degrees of the said Ecliptic are larger towards the Limb than towards the Centre, whereby the spaces betwixt Sign and sign are not equal, but some larger than other some.  
And note that upon the outermost Limb of the Mater are drawn three circles making two reasonable spaces, 
The Limb of the Mater.  both which spaces are divided into 360. degr. both of them beginning at the North pole. And the degrees in the outermost space of the Limb, beginning at the North pole do proceed from 15. to 15. downward towards your left hand, and so round about the instrument until you come to 360. which degrees do show the hours of the Equinoctial. And at the North pole, whereas the division beginneth, is set down the figure of 6. signifying the sixth hour of the forenoon, then counting fifteen degrees downward towards your left hand is set down in the foresaid space the figure of 5. then 4. 3. 2. and 1. And at the end of the Equinoctial on the left hand is set down 12. from which proceeding still downwards are set these numbers of hours, 11. 10. 9 8. 7. and 6. which 6. standeth at the South pole, then from thence proceeding upwards are set down these hours 5. 4. 3. 2. 1. and then 12. which standeth at the end of the Equinoctial on the right hand, from whence proceeding still upwards towards the North pole, are set down these hours, 11. 10. 9 8. & 7. and at the Pole itself is set down the figure of 6. as before hath been said, from whence you first began to count. And in the inner space of the same Limb, the 360. degrees are to be counted by 90. in such sort as the first 90. is placed at the North pole, the second 90. at the one end of the Equinoctial marked with C. The third 90. at the South pole, marked with B. And the fourth 90 at the other end of the Equinoctial, marked with D. Every which quarter is to be counted both upward and downward, according as you see the numbers written with Arithmetical figures, and are placed one above another in the said space, whereof the upper figures are somewhat lesser than the neither figures, both of them proceeding from 1. to 90. the greater figures from the Equinoctial upward to the very Pole, and the lesser Figures from the Pole downward to the Equinoctial, and are thus set down. 10. 20. 30. 40.   

A Description of the Net, called in Latin Rete.  
THis part is first environed round about with a great circle signifying most commonly the Meridian or 90. Azimuth or Vertical circle, and sometime it signifieth the Equinoctial, and especially when the Centre is taken from the Pole, which Circle is divided by two cross Diameters into four quarters, every quarter containing 90. degrees, which degrees are to be numbered in the very Limb of the Rete both upward and downward, and of those two cross Diameters, the one end marked with the letter A. signifieth the Zenith, and the other end marked with the letter B. signifieth the Nadir or point opposite, and the other cross Diameter marked with the letters C. D. signifieth the Horizon, which for distinctions cause is otherwise called the Finitor, because the Meridian's before described in the Mater are to be called and used sometimes as Orisons and this Finitor is a pretty broad Ruler, the very edge whereof is divided with small divisions into 180. degrees, which being doubled by reckoning the same both forward and backward, (beginning at the Centre) do make up three hundred and three score degrees.  
This Finitor signifieth the Horizon of the Globe, 
The Finitor  the very edge whereof being divided by little short stréekes into small portions or degrees, is always to be applied to any several latitude when need is. And the broader part thereof serveth only to contain the numbers that are set therein both beneath and above to know thereby the number of every Azimuth hereafter described, which numbers do proceed from the Centre to the right hand thus, 10. 20. 30. 40. 50. and so forth to 90. Than backward towards the Centre are set down 100 110. 120. 130. 140. and so forth until you come to 180. which is placed at the very Centre. Than from thence towards the left hand are set down these numbers 190. 200. 210. and so forth until you come to 270. and from thence turning again towards the centre are set down 280. 290. and so forth until you come to 360. placed at the Centre.  
And in the said Net are certain circles, 
The Almican teraths.  which are Parallels to the foresaid Finitor, proceeding towards the Zenith, and are in every respect like unto the Parallels described in the Mater, and these Parallels are called Almicanteraths, that is to say, circles of Altitude, which beginning at the Finitor, do proceed to the Zenith, marked with the letter A. from 1. to 90. And though there be out out in the Rete but 30. Almicanteraths, yet for so much as every space contained betwixt every 2. Almicanteraths do contain 3. degrees, they make in all 90. for 3. times 30. maketh 90. which degrees you may see set down in the Limb of the Rete on both hands thus, 10. 20. 30. and so forth, till you come to 90. which standeth at the very Zenith.  
And these Almicanteraths are crossed with other circles called Azimuthes, 
The Azimuthes.  that is to say, Vertical circles, which passing from the Finitor, do meet all in the Zenith, whereof though in this instrument there be set down but 12. being 15. degrees distant one from another, yet you must imagine that there be 180. which the degrees set down in the Finitor, in such manner as is before described do show. And you have to note that if the Sun be in the very beginning of the Azimuthes, which is at the centre, then is he full East, and if he be in the 90. Azimuth, than he is full South, and when he is in the 180. then he is full West, and when he is in the 270. Azimuth, than he is full North.  
And if you would have this account of the Azimuthes to answer the Mariner's compass, 
How to count the Azimuthes according to the Mariner's Compass.  then divide the number of the Azimuthes wherein the Sun is by 11. degrees and ¼ which is 15. minutes, and the quotient will show the rombe or wind of the Mariner's Compass, so as in your account you proceed from the first Azimuth towards your right hand, that is, from East to South, and from South to West, and from West to the North, and so from thence again to the East point whereas you first began. Also you have to note by the way, that these Azimuthes do sometime signify the circles of position, the use whereof you shall found set down hereafter in the 31. position of this Treatise.  
In the mean time I will proceed in describing the zodiac, of the Rete, which in shape is like to an Egg, 
The description of the zodiac of the Rete  called in Latin Figura ovalis, the one half whereof extendeth towards the Zenith, and the other towards the Nadir or point opposite to the said Zenith, in which circle are placed the 12. Signs, whereof the 6. Northern signs, that is, Aries, Taurus, Gemini, Cancer, Leo, and Virgo are placed in the neither half towards the Nadir, and the other six Southern Signs, that is, Libra, Scorpio, Sagittarius, Capricornus, Aquarius, and Pisces, are placed in the upper half of the zodiac towards the Zenith, and every one of these Signs are divided into 30. degrees, which are set down with Arithmetical Figures in the said zodiac thus, 10. 20. 30.  
Moreover, in the said Net are placed certain fixed Stars to 
Of Stars 71 contained in the Rete.  the number of 71. whose names here do follow. Caput Ophiuci, that is, the head of Serpentarius, Aquila, the Eagle. Caput Engon, the head of Hercules. Cuspis Sagittarii, the shaft of Sagittarius. Palma Ophiuci, the hand of Serpentarius. Cauda Delphini, the tail of the Dolphin. Romboides Delphini, which is a star in the Dolphin's back. Cor Scorpionis, the hart of the Scorpion. Frons Borealis, Media, and Australis Scorpionis, the Northern, Southern, and middle front of Scorpio. Lucida Lyrae, the bright Star of Lyra, or Vultur cadens. Lanx Chaele Borealis & Australis, the North and South star of the Balance. Corona Gnosiae, the crown of Ariadna. Praecedens & sequens caudae Capricorni, that is, the former and follower in the tail of Capricorn. Caput Draconis, the head of the Dragon Hastile Bootis, the Boar-spear of Arcturus. Cauda Cigni, the tail of the Swan. Fomahand, a star in the mouth of the Southern Fish. Arcturus, a star betwixt the legs of Bubulcus or Boötes. Humerus Bootis, the shoulder of Bubulcus. Dexter Humerus Cephei, the right shoulder of Cepheus. Crus Pegasi, the leg of the winged Horse. Spica Virgins, the Wheat ear in the hand of Virgo. Tres stellae in cauda Vrsae maioris, three stars in the tail of the great Bear. Previndemiatrix, that is Virgo. Cauda Ceti, the tail of the Whale. Andromedae Scapulum, the shoulder blade of Andromeda. Cingulum Andiomedae, the girdle of Andromeda. Humerus Vrse maioris, the shoulder of the great Bear. Corvi rostrum, the beak of the Crow. Corviala dextra, the right wing of the Crow. Cauda Leonis, the tail of the Lyon. Ceruix Leonis, the neck of the Lyon. Cor Leonis, the hart of the Lion, otherwise called Regulus. Lucida Hydrae, the bright Star of Hydra. Capita Geminorum, the heads of the two Twins named Apollo and Hercules. Cancer, the Crab. Canis minor, the little Dog. Canis maior, the great Dog. Canopus, a fair star in the left oar of the Ship Argus. Humerus dexter Aurigae, the right shoulder of Auriga. Hircus, the Goat hanging at the back of Auriga. Hedi, the two little Goats sucking behind at her paps. Humerus dexter Orionis, the right shoulder of Orion. Tres stellae in cingulo Orionis, three stars in the girdle of Orion. Humerus sinister Orionis, the left shoulder of Orion. Pes sinister Orionis, the left foot of Orion. Oculus Tauri, the Bulls eye. Pleyades, the seven little stars in the bulls neck. Extremum Eridani, the last end of the flood Eridanus. Caput Medusae, the head of Medusa. Dextrum latus Persei, the right shoulder of Perseus. Cornu Arietis, the former Star of the Ram's horn. Venture Ceti the belly of the Whale. juba Ceti, the mane of the Whale.  
And note that all these Stars in the Net, 
How to know which stars be North or South.  whose longest tips or points do point from the centre outward, towards the Limb, are Northern Stars having North declination, and those whose longest tips do point inward from the limb towards the Centre, are Southern Stars, having Southern declination. All which Stars are set down in a Table in the beginning of his third book, which doth not only show their names, but also their longitudes, latitudes, and declinations, their natures, their right ascensions and magnitudes or greatness. And whereas both in the Mater, and also in the Rete one self circle is made to have divers significations, the cause thereof shall plainly appear unto you by the use of the Astrolabe, in seeking to find out thereby the propositions of the foresaid book.  
And besides the parts before described, 
A description of the Label.  there is yet another part belonging to the forepart of the Instrument, called the Label, the one end whereof is fastened to the Centre of the Astrolabe, so as it may turn round about, and this Label is divided into 90. degrees twice set down therein with Arithmetical Figures to be reckoned as well from the Centre to the point of the Label, as from the point thereof to the Centre serving to diverse uses, yea the 90 degrees of the Label are sometimes to be repeated 4 times to make up the number of 360. degrees, as shall hereafter plainly appear by the 12. proposition, showing how to found out the right ascension of any degree or portion of the Ecliptic line.  
And remember that the right line drawn from the Centre of the Label alongst the inward edge thereof is called the Fiducial line, 
The fiducial line of the label.  and is divided into 90. small parts called degrees, which line or inward edge is always to be used in any proposition, and not the outward edge or back part of the Label.  
Thus much touching the fore part of M. Blagrave his jewel with every particular part, with which forepart I wish you to be throughly acquainted before you deal with the propositions here following, or with any other proposition contained in M. Blagrave his book.   

A brief description of the back part of the said jewel.  
IN the Limb of the back part is described the Theoric of the Sun, to know thereby in what Sign and degree the Sun is every day throughout the year, by laying the Diopter thereto, M. Blagrave calleth it a Ruler, which Diopter is made with two Pinules or square Tablets, each one pierced with two holes one greater than another, the lesser to take the height of the Sun by his beam passing through the said lesser holes, and the greater holes do serve to take the altitude of the Sun, being something darkened in the day time, so as he casteth no beam, or else the altitude of any Star in the night season by looking with your one eye, the other being shut, through the two greater holes of the Pinules or tabrets. And the midst of this Diopter is fastened with a pin to the centre of the Astrolabe, so as the said Diopter may turn round about, and the middle line of the said Diopter is called the fiducial line, because it rightly directeth the sight of the eye to the foresaid holes, the degrees of the altitude of the Sun or of any Star are set down in the outermost space of the Limb, divided by 2. cross Diameters into four quarters, every quarter containing 90. degrees, the number of which degrees are set down in the outermost space of the Limb of the said back part with arithmetical figures right over the heads of the said degrees of altitude. And you have to note, that the perpendicular Diameter signifieth the Meridian, that is, the line of South and North, that is to say, the South point at the ring or handle, and the North at the point opposite, and the other overthwart Diameter signifieth the right Horizon, that is to say, the line of East and West, the East being placed on the left hand, and the West on the right hand. And under the Theoric of the Sun you may draw as many circles of such reasonable distance one from another, as in the spaces thereof may be set down, the years of our Lord, the Dominical letter for the leap year, and the Dominical letter for the common years, the Prime or golden number, the Epact, and on what day of March or of April Easter day every year falleth, in such order as M. Blagrave hath himself described in a Table made of purpose in the second book of his jewel the 11. chapter.  
Thus having described every particular thing as well in the forepart as in the back part of the said instrument, I will now show you how to use the same, and how to find out thereby all the conclusions contained in the Table here next following.   

The Table containing 32. necessary conclusions to be wrought by this Astrolabe.  
FIrst, how to find out the place of the Sun (that is to say) in what sign and degree thereof the Sun is every day throughout the year, being not leap year, and also the opposite point of that degree. Proposition. 1.  
How to know the place of the Sun in the leap year, and how to find the leap year. Prop. 2.  
How to take the altitude of the Sun or of any star. Prop. 3.  
How to take the Meridian altitude, that is to say, the highest or greatest altitude of the Sun or of any star. Prop. 4.  
How to know the altitude of the Sun at any hour without seeing the Sun. Prop. 5.  
How to know the Meridian altitude of the Sun or of any star in the Net without seeing them. Prop. 6.  
How to know the declination of the Sun, or of any star contained in the Net. Prop. 7.  
How to found out the latitude of any Region divers ways. Prop. 8.  
How to know the hour of the day by the Sun, and also in what part or coast of heaven he is at that instant. Prop. 9  
How to find the rising and setting of the Sun every day in every latitude, and thereby the length of the day, and also in what coast or part of the Horizon he riseth and setteth. Prop. 10.  
How to know every day at what hour the Moon riseth and setteth, and how long she continueth above the Horizon and also when she is full South. Prop. 11.  
How to found out the right ascension of the Sun, or of any degree or portion of the Ecliptic. Prop. 12.  
Another more ready way to find out the right ascension of any degree or portion of the Ecliptic by the Rete. Prop. 13.  
How to find out the ascentionall difference of the Sun or of any degree or point of the Ecliptic. Prop. 14.  
How to found out the obliqne ascension of the Sun, or of any point of the Ecliptic. Prop. 15.  
How to found out the right ascension of any Ark or portion of the Ecliptic, and therewith to know what time it spendeth in rising in a right Sphere. Prop. 16.  
How to find out the obliqne ascension of any Ark of the Ecliptic in any latitude, and what time it spendeth in his rising. Prop. 17.  
How to find out the obliqne descension of any point of the Ecliptic in any latitude. Prop. 18.  
How to found out the obliqne descension of any Ark given of the Ecliptic, and therewith to know the time which it spendeth in his setting. Prop. 19  
How to know the height of any star at any hour without seeing the star, and thereby to find out in the firmament all the stars that be described in the Net, and are to be seen with the eye▪ Prop. 20.  
How to find out the ascentionall difference of any star, Prop. 21.  
How to know the obliqne ascension of any star. Prop. 22.  
How to know what stars do never rise nor set in any latitude. Prop. 23.  
How to know at what hour of the day or night any star riseth or setteth. Prop. 24.  
How to know how long any star continueth above the horizon in every latitude, Prop. 25.  
How to found out the stars hour, and thereby to know the hour of the night. Prop. 26.  
How to find out the distance betwixt any two stars contained in the Net. prop. 27.  
Another way to know the distance of any two stars, theri longitudes and latitudes being first known, and also by that means to find out the distance betwixt any two places upon the earth. prop. 28.  
How to found out the degree of Medium Coeli at any hour of the day, that is to say, the degree of the zodiac that is in the Meridian at any hour that you seek, and also the degree called Imum Coeli. prop. 29.  
How to found out the horoscope or ascendent at any time of the day or night, and thereby to have the four principal angles of heaven. Prop. 30.  
How to found the circles of position, and to know how much the pole is elevated above every such circle in any latitude, without the knowledge whereof you cannot found out the 12. houses by this Astrolabe. prop. 31.  
How to found out all the 12. houses of heaven, and thereby to erect a figure at any hour of the day or night. prop. 32.   

The uses of the Astrolabe, and first how to find out the place of the Sun (that is to say) in what sign and degree thereof the Sun is every day throughout the year, being not leap year, and also the opposite point of that degree. The first. Proposition.  
LAy the Diopter which is on the backside of the Astrolabe upon the day of the month and that end of the Diopter with his fiducial line will show you in what sign and degree thereof the Sun is that day, and the other end of the Diopter will show you the opposite point to that degree, as for example: I would know the place of the Sun the 17. of july, here by laying the Diopter upon that day, I found the Sun to be in the fourth degree of Leo, and the opposite point thereof to be the fourth degree of Aquarius.   

How to know the place of the Sun in the Leap year and how to find the leap year. Proposition. 2.  
WHen it is leap year, you must always add one degree more every day during that year unto the place of the Sun found by the first proposition, which leap year you shall know by dividing the year of the Lord by 4. for if there be no remander left, than that year is leap year, so shall you find the year of our Lord 1596. to be leap year.   

How to take the altitude of the Sun, or of any star. Proposition. 3.  
TO take the altitude in any time of the day when the Sun shineth, you must turn your face and also the left Tablet or Pinule of the Diopter towards the Sun, holding the Astrolabe by the ring with your right forefinger or middle finger being the somewhat bowed, in such sort as the Astrolabe may hung plumb, and then with your left hand lift the Diopter up and down until the Sun with his beam due justly stréeke through both the holes of each Pinule of the Diopter, so as you may see the shadow of the two holes of the upper Pinule to play upon the two holes of the neither Pinule, then mark upon what degree of altitude the thinnest edge or fiducial line of the Diopter falleth in the outermost skirt or border of the back of the Astrolabe, for that is the suns altitude for that present. But if the Sun be covered with a cloud, so as it shineth not clear mough to cast any shadow, and yet so as it may be seen with the eye, then hung the Astrolabe by the ring upon your right thomb, and turning your face towards the Sun▪ lift up your hand with the Astrolabe so high, as by moving the Diopter with your left hand up and down, you may with your right eye (the other being shut) see the Sun through the greater holes of both the Pinules of the Diopter, and mark upon what degree of altitude the upper end of the Diopter falleth, and that is the altitude of the sun at the time, and in this manner you must also take in the night season, the altitude of any star.   

How to take the Meridian altitude, that is to say, the highest or greatest altitude of the Sun or of any star. The 4. Proposition.  
Go into some open place whereas the sun shineth somewhat before noontide, and there hanging the Astrolabe upon your right fore finger or middle finger, take the altitude of the sun in such manner as is before taught, at diverse times with some pause betwixt every time to know thereby whether such altitude increaseth or decreaseth, for if it increaseth, than the sun is not yet at the Meridian, but if it decreaseth, than it is passed the Meridian, and therefore you must watch diligently to take him when he is at the highest. And you must do the like to take in the night season, the Meridian altitude of any known star, saving that then you must hang your Astrolabe upon your thumb before your right eye, and to do as is taught in the last Proposition.   

How to know the altitude of the sun, or of any star at any hour of the day without seeing the sun or star. The 5. Proposition.  
SEt the Finitor at your Latitude, and seek out amongst the hour lines or Meridian's of the Mater in what point the Parallel or Circle of declination of the sun crosseth the hour line which you seek, and the Almicanterath and Azimuth passing through that point do show both the altitude and also the coast or part of heaven, wherein the sun or star is at that instant. This is a very necessary Proposition, for by knowing the height of the sun at every hour of the day in whatsoever sign the sun is you may make Tables for particular dials, as Cylinders, hour Quadrants, and such like to serve any latitude, yea rather this Instrument as M. Blagrave rightly sayeth is a Table of itself ready made to serve such purposes. Let the example of this Proposition be thus, Suppose that at 8. of the clock in the morning the 21. of April 1592. the sun being then in the tenth degree of Taurus, and his declination 15. degrees Northward, you would know the altitude of the sun at that hour, which by working as the rule teacheth, you shall find to be almost 30. degrees, and that he is about 13. degrees distant from the East towards the South.   

How to know the Meridian altitude of the sun, or of any star every day throughout the year, without seeing either sun or star. The 6. Proposition.  
FIrst seek to know the declination of the sun or star either by the seventh Proposition next following, or else by some Table, & whether it be North or South. and knowing his declination, bring the Finitor to your Latitude, and staying it there seek out in the limb of the matter on your right hand the said declination, and there mark what Almicanterath toucheth that point, for that Almicanterath being counted upon the limb of the Rete from the Finitor, doth by and by show the Meridian altitude of the sun or star for that day. As for example, I would know the Meridian altitude of the sun the first of july 1592. at which time his Parallel or declination is 22. degrees and certain minutes Northward. Here having laid the Finitor to the Latitude 52. I find that the 60. Almicanterath toucheth the Parallel, and that is the Meridian altitude of the sun that day. Again suppose that I would know the Meridian altitude of the star Oculus Tauri, in the Latitude 52. Here having brought the Finitor to the said latitude, I find that the 53. Almicanterath toucheth his Parallel or declination, which is 15. degrees 49′· Northward, so as I found his Meridian altitude in that latitude to be 53. degrees. But you have to note that if the sun or star have south declination, than you must count his parallel from the Equinoctial downward towards the south Pole, so shall you found the Meridian altitude of the great dog called Canis maior, whose Southern declination is 16. degrees to be 22. degr. & 30′· And by knowing the Meridian altitude of any star, you may also know how far he is distant from the Meridian or South line, if you subtract from his Meridian altitude his altitude taken at any other time of the same night, for the remainder will show his distance from the Meridian, & if the star at the time of taking his altitude, be in the East part of the firmament, than he is so much short of the Meridian, and if he be in the West, than he is so much past the Meridian. As for example, knowing the Meridian altitude of Oculus Tauri to be 53 degrees, and his other altitude newly taken to be 30. deg. I found his distance from the Meridian to be 23. deg. aswell by subtraction, as by counting upon the limb of the Rete the degrees contained betwixt the two foresaid Almicanteraths. And because the said star was in the East part of the firmament when I took his altitude, I conclude that he wanted 25. degrees of arriving to the Meridian, which maketh one hour and a half, and a little more.   

How to found the declination of the sun, or of any star described in the net. The 7. Proposition.  
THis may be done two manner of ways, first knowing the place of the sun, seek his place in the Ecliptic line of the matter, and look what Parallel of the matter passeth through that degree, and the number of that Parallel will show the declination (that is to say) how far the sun is distant from the Equinoctial, counting from the Equinoctial upon the outermost Meridian or inner limb of the matter, and according as the sign wherein the sun is be it Northern or Southern, so is the declination of the sun, and must be counted either upward or downward accordingly. The second way is thus, bring the Fiducial live of the label to the degree of the sun in the zodiac of the net, and the number of degrees counted upon the label betwixt the limb of the net, and the place of the sun which the label toucheth upon the outward edge of the said zodiac will show his declination. By either of these two ways you shall find the declination of the Sun, being in the tenth degree of Taurus to be 15. degrees Northward. And by this last way you may know the declination of any star contained in the net thus. Having found in the net the star whose declination you seek, lay the Fiducial line of the label to the longest tip of that star, then count upon the label how many degrees are contained betwixt the limb of the Rete, and that point whereas the label toucheth the longest tip of the said star, and that shall be his declination either Northern or Southern according as the said tip point either outward or inward: for if outward from the Centre, than it is Northern, if inward towards the Centre, than it is Southern, as hath been said before in the description of the Net, and by doing thus you shall found that the star called Canis maior, that is the greater dog, hath in South declination 15. degrees 55′· Again you shall find the first star of the Ram's horn called Cornu Arietis to have in North declination 18. degrees.   

How to found out the Latitude of any Region. The 8. Proposition.  
FIrst you must know the place of the sun, and also his declination, and having taken his Meridian altitude, reckon the same amongst the Almicanterathes from the Finitor upwards, and turn about the Rete from the Pole arctique towards the Equinoctial on your right hand, until it toucheth the Parallel of the sun, for then look on your left hand and you shall find the Finitor to stand at that Latitude which you seek. As for example, the 12. of April 1591. the sun being in the first degree 20. minutes of Taurus, and his declination being then 12. degrees Northward, I find his Meridian altitude to be 50. degrees, which I count upon the limb of the Rete proceeding from the Finitor upwards towards the Zenith, and then I turn the Rete until I have brought that Almicanterath to the Parallel of the Sun, which is 12. degrees, counting the same from the Equinoctial on the right hand of the matter towards the North Pole, and there staying the Rete, I find that on the left hand the Finitor lieth upon the 52. degree of Latitude counting from the North pole down towards the Equinoctial. The common way of finding the Latitude is thus, if it be in the day time, then take the Meridian altitude of the Sun, and if the Sun be in any of the six Northern signs, then subtract the declination of the sun out of his Meridian altitude, and the remainder shall be the altitude of the Equinoctial above your Horizon, which being taken out of 90. the remainder will show the altitude of the Pole, but if the sun be in any of the six Southern signs, than you must add his declination to his Meridian altitude, and the sum thereof shall be the altitude of the Equinoctial, which being taken out of 90. the Remamder will show the Latitude or elevation of the Pole. But to know the latitude of any place in the night season, you must take the Meridian altitude of some known star which both riseth and setteth, then after that you have taken his Meridian altitude with your Astrolabe, you must learn to know his declination, and whether it be Northern or Southern, for if the star have North declination, than you must subtract his declination from his Meridian altitude, and the remainder shall be the Altitude of the Equinoctial, which being taken out of 90. shall be the latitude or elevation of the Pole, but if the declination of the star be Southernly, than you must add his declination to his Meridian altitude, and that sum shall be the altitude of the Equinoctial, which being taken out of 90. the remainder shall be the elevation of the Pole. And there be divers other ways of finding out the latitude of any place, which I have partly set down in my Treatise of the two Globes about the latter end thereof, and partly in my Treatise of Navigation, whereas I speak of the North star and of his Guards.   

How to know the hour of the day by the Sun, and also in what part of heaven he is at that instant. The 9 Proposition.  
TAke the altitude of the Sun, and knowing the latitude of the place where you are, bring the Finitor of the Rete to that Latitude, and having stayed it there, look in what point the Almicanterath or altitude of the Sun crosseth the Sun's Parallel or Circle of declination in the matter, and the hour line passing through that point will show the true hour, and at that instant you may also know in what Azimuth (that is to say) in what part of heaven, or as the Mariners term it in what ram or wind, as East, West, North, or South etc. the Sun is at that instant, for that Azimuth which passeth through the foresaid point, is the Azimuth of the Sun. As for example, the 21. of April 1592. the sun being in the tenth degree of Taurus, and his declination being then 15. degrees Northward, I found by the Astrolabe, Quadrant, or cross-staff, the altitude of the Sun in the forenoon to be 30. degrees. Wherhfore I having laid the Finitor to my Latitude which is 52. and stayed it there, I mark in what point the Almicanterath cutteth the suns foresaid Parallel in the matter, and I found that the hour line of eight in the forenoon cutteth that point, which showeth that it was then eight of the clock in the morning, and that the sun was about 14. degrees distant from the East towards the South.   

How to found the rising and setting of the sun every day in every Latitude, and thereby the length of the day, and in what coast or part of the Horizon be riseth and setteth. The 10. Proposition.  
BRing the Finitor to your Latitude, and staying it there, look in the matter where the Parallel of the Sun doth cut the Finitor, and the hour line which crosseth that point, will show the hour of his rising and setting, and the number of hours betwixt his rising and setting is the length of the day, and the number of the Azimuthes betwixt that point of the Finitor, and the Centre or first Azimuth will show you in what part or coast of the Horizon he both riseth and setteth. As for example, seeking by this rule to know at what hour the sun riseth the 19 of june 1592. he being then in the sixth degree 39 minutes of Cancer, and his declination then 23. degrees 19′· I found that he riseth 12. minutes before four, and goeth down 12. minutes after 8. and thereby I found the length of the day counting from sun rise to sun set to be 16. hours and 24. minutes, and that he riseth from the East towards North 40. degrees, which according to the Mariners reckoning is North-east and by East two quarters and somewhat more towards the North.   

How to know every day at what hour the Moon riseth and setteth, and how long she continueth above the Horizon, and also when she is full South. The 11. Proposition.  
FIrst you must learn by some Almanac or Ephemerideses in what sign and degree the Moon is, and whether it be a Northern sign or a Southern sign, for if she be in a Northern sign, then bring her place to the Horizon of your latitude in the North-east part of the Astrolabe, but if she be in a Southern sign bring her place to the said Horizon in the Southeast part of the Astrolabe, & there having stayed the Rete bring the label to the place of the Sun for that day, & the label will point to the hour of the Moons rising in the limb of the matter: but because the Almanac or Ephemerideses do not set down the true place of the Moon but only at noon, you must therefore consider whether it be in the forenoon or in the afternoon that you seek, for if it be in the afternoon you had need to know how many hours are run from noon, and then for every hour to add half a degree to the place of the Moon which you found at noontide, but if it be in the forenoon, than you must subtract from her place at noon, for every hour half a degree, so shall you go very nigh to finds her true place in the zodiac for that hour, though you know not her latitude which is but 5. deg. at the most, & therefore can 'cause no great error in this matter. Now to know when she setteth you must do thus, if the Moon be in any Northern sign, them you must bring her place to the foresaid Horizon in the Northwest part of the Astrolabe, & by laying the label to the place of the sun, it will point to the hour of her setting, but if she be in any Southern sign, you must bring her place to the Horizon in the Southwesth part of the Astrolabe, and the label being laid to the place of the Sun will point to the hour of her setting. Now if you would know how long time she is above the Horizon, and also at what hour she is full South, then count the hours betwixt her rising & setting, and that shall be the time of her continuance above the Horizon, and the very midst of that is the true hour that she is full South: As for example, the fourth day of September 1592. in the latitude 52. at nine of the clock at night the Sun being in the 21. degrees 47′· of Virgo or there abouts, & the Moon being in the 2. degree 30′· of Capricorn, I am desirous to know when the Moon did rise that day, and by working according to the rule before set down, I found that she did rise about three of the clock in the afternoon, and that she went down at ten of the clock at night and half an hour past, and that she was full South or at the Meridian a little before seven of the clock in the afternoon. And as by this rule you may find out the time of the rising and setting of the Moon, so may you find the time of the rising and setting of the other five Planets, that is Saturn, jupiter, Mars, Venus, and Mercury, any day throughout the year, so as you know their places in the zodiac, which the Ephemerideses of Stadius doth show, not only at noontide, but also at any other hour of the day by help of certain Tables made of purpose, the use of which Tables I have set down in the latter end of my Treatise of the two globes.   

How to found out the right ascension of the sun, or of any degree or portion of the Ecliptic. The 12. Proposition.  
TAke the Rete and the label clean from the Astrolabe, and seek out in the Ecliptic line of the matter the sign and degree whose right ascension you would know, and mark what Meridian cutteth that point: that done, place the label upon the pin which standeth in the very Centre of the jewel, and make the fiducial line thereof to lie right upon, and alongst the Equinoctial line either towards your right hand or towards your left, according as the sign & degree whose ascension you seek is placed in the matter. Than mark where the foresaid Meridian cutteth through the label, and also through the Equinoctial line, and the number of degrees contained in the label, betwixt the Centre and that point of the Equinoctial is the right ascension of that degree of the Ecliptic which you seek, which number of degrees you must count upon the label in this manner. For if that sign and degree be contained betwixt the first point of Aries and the first point of Cancer, than you must begin to count upon the label at the Centre, and so proceed forward towards Cancer, the right ascension of whose first point is 90. degrees, but if the sign and degree which you seek, be betwixt the beginning of Cancer and the beginning of Libra, which is at the very Centre right against Aries, than you must count upon the label backward from 90. to 180. by adding to every tenth space of the label 10. degrees, so as the first number proceeding towards your left hand shall be 100 and next to that 110. and so forth until you come to 130. which is the right ascension of the first point of Libra, and from Libra you must count towards Capricorn 190. then 200. and so forth till you come to 270. which is the right ascension of the first point of Capricorn: and from thence you must count 280. 290. then 300. and so forth towards the Centre until you come to 360. which is the right ascension of the last point of Pisces, so as though there be set down in the label but 90 degrees both forward and backward, yet 90. being four times repeated, do make in all 360. which is the whole Longitude of the Equinoctial. As for example, you would know perhaps the right ascension of the tenth degree of Sagittarius, here by seeking in the matter you shall found that the Meridian passing through that degree will cut both the label being laid towards your left hand, and also the Equinoctial in the 248. degrees 21. minutes, which is the right ascension of the tenth degree of Sagittarius, in counting whereof remember to begin from 180. that is from the Centre, and this ascension agreeth with the Table of right ascensions set down in Stadius his Ephemerideses the 44. page of his book. Again in seeking to know the right ascension of the 10. degree of Taurus, if you begin to count upon the label from the Centre, which is the first point of Aries, you shall find that the Meridian which passeth through the 10. degrees of Taurus will cut the label, and the Equinoctial line in the 37. degree 35′·   

Another morereadie way to find out the right ascension of any degree or portion of the Ecliptic by the Rete. The 13. Proposition.  
LAy the Finitor even with the Axletree, signifying here the right Horizon, so as the first point of Aries may meet with the East point of the said Horizon, and lay the label right upon and alongst the Equinoctial line, either towards Cancer or towards Capricorn, according as the sign and degree which you seek is placed in the Ecliptic of the Mater, and mark therewith what Meridian passeth through that degree, and follow the same until it cutteth the Equinoctial, and also the label lying thereon, and by counting upon the label the number of degrees to that point in such order as is before taught, you shall have your desire. As for example, if you seek to know the right ascension of the tenth degree of Leo, here having placed the Finitor as before is taught, seek out in the Mater what Meridian passing through that degree cutteth the said Equinoctial, and in what point, and lay the label to the said Equinoctial according as the sign requireth, be it towards Cancer or Capricorn, and by counting the degrees upon the label as is before taught, that is from 90. proceeding from the limb backward towards the Centre you shall find the right ascension of the 10. degree of Leo to be 132. degrees, which differeth from the foresaid Table but 27′· a thing of small moment considering the narrow spaces of the Meridian's in the Mater.   

How to find out the ascentionall difference of the sun, or of any degree or point of the Ecliptic. The 14. Proposition.  
BRing the Finitor to your Latitude, and there staying it look amongst the Meridian's in the Mater in what point the Parallel of the Sun cutteth the Finitor, and the number of the Meridian's contained betwixt the axle-tree, and that point shall be the ascentionall difference. As for example, in the Latitude 52. I would know the ascentionall difference of the Sun being in the fourth degree of Cancer, at which time his Parallel of declination is 23. degrees Northward, here by working according to the rule I find the ascentionall difference to be 33. degrees 30′·   

How to found out the obliqne ascension of the sun, or of any point of the Ecliptic. The 15. Proposition.  
Having found the right ascension and also the ascentionall difference by the former Propositions, consider whether the declination of the sun or of any other point of the Ecliptic be North or South, for if it be North, then subtract the ascentionall difference out of the right ascension, and the remainder shall be the obliqne ascension. But if the declination be South, then add the ascentionall difference to the right ascension, and the sum thereof shall be the obliqne ascension. As in the former example knowing the declination of the sun to be 23. degrees Northward, I subtract the ascentionall difference which was 33. degrees 30. minutes, out of the right ascension of the sun, which was 94. degrees, and there remaineth 60. degrees 30′· which is the obliqne ascension of the sun being in the fourth degree of Cancer, and his declination being 23. degrees Northward as is before supposed.   

How to found out the right ascension of any ark or portion of the Ecliptic, and therewith to know what time it spendeth in rising in a right Sphere. The 16. Proposition.  
BRing the end of the given ark found out in the zodiac of the Rete, unto the left end of the Equinoctial marked with D. and there staying it, lay the Fiducial line of the label upon the beginning of the said Ark, and the number of degrees in the innermost limb of the matter contained betwixt the left end of the Equinoctial, and the Fiducial line of the label shall be the right ascension of that Ark, and the hour set down in the outermost limb whereunto the label pointeth, shall be the time which the said ark spendeth in his rising, so shall you find the ark of the whole sign Taurus to be 30. degrees, and to spend in his rising two hours.   

How to find out the obliqne ascension of any ark of the Ecliptic in any Latitude, and what time it spendeth in his rising. The 17. Proposition.  
FInd▪ the obliqne ascension both of the beginning and ending of the ark by the 15. Proposition, then subtract the said ascension of the beginning of the ark out of the ascension of the ending of the said ark, always remembering that if the obliqne ascension of the beginning be greater than the other, then to add to the lesser 360. and out of that sum make your subtraction, for the remainder shall be the obliqne ascension of the whole ark, then count the number of that ascension in the innermost limb of the matter beginning at the left end of the Equinoctial marked with D▪ and so proceed upward towards your right hand, and where the said ascension endeth, there lay the label, & the hour whereunto the label pointeth showeth the time which the said ark spendeth in his rising in that Latitude. As for example, I would know the obliqne ascension of the whole ark of the sign of Taurus, and what time that ark spendeth in his rising: first here having found the obliqne ascension of the first point of Taurus to be about 13. degrees in the Latitude 52. and the obliqne ascension of the last point of that sign to be about 30. here by taking 13. out of 30. there remaineth 17. degrees, which is the obliqne ascension of the whole ark of the sign Taurus, which 17. degrees, I count in the limb of the matter from the end of the Equinoctial marked with D. upward, and there laying the label, I find that it pointeth to one hour and two degrees, which maketh 8′· so as I conclude thereby that the whole ark of the sign Taurus, spendeth in his obliqne ascension one hour and eight minutes.   

How to find out the obliqne descension of any point of the Ecliptic in any Latitude. The 18. Proposition.  
Having found out the obliqne ascension of the point opposite to the point given, add thereunto 180. degrees, and the sum thereof shall be the obliqne descension of the point given, always remembering if the sum of addition do exceed 360 to subtract out of that sum 360. and the remainder shall be the descension of the point given. As for example, you would know the obliqne descension of the first point of Taurus, whose point opposite is the first point of Scorpio, and his obliqne ascension according to Reynholdus his Tables is 222. degrees 36. minutes, whereunto if you add 130. you shall make the total sum to be 402. degrees 36. minutes, from which sum if you subtract 360. degrees according to the rule before given, there will remain 42. degree 36. minutes, which is the obliqne descension of the first point of Taurus.   

How to find out the obliqne descension of any ark given of the Ecliptic, and therewith to know the time which it spendeth in his setting. The 19 Proposition.  
Having found the obliqne descension of the beginning, and also of the ending of the given ark by the last Proposition subtract the descension of the beginning out of the descension of the ending, and the remainder shall be the obliqne descension of the given ark, always remembering if the subtraction can not be made to add thereunto 360. that done, divide that by 15. and the quotient will show the number of hours which the given ark spendeth in his setting: and remember that if in making that division there be any remainder left, to multiply that remainder by four and so you shall have the minutes. By working thus you shall find the obliqne descension of the whole Ark of Gemini to be 36. degrees 49. minutes, and in the Latitude 52. to spend in his going down two hours and 24. minutes.   

How to know the height of any star at any hour, without seeing the star, and thereby to found out in the firmament all the stars that be described in the net, and are to be seen with the eye. The 20. Proposition.  
LAy the label to the hour supposed upon the limb of the matter, and then bring the place of the sun for that day to the Fiducial line of the label, and there having stayed the Rete, bring the Fiducial line of the label to the tip of the star whose altitude you seek, and the label will show you upon the limb of the Mater how many degrees that star is distant from the South. Again by counting upon the label, the degrees contained betwixt the point of the label, and the tip of the said star you shall have his declination, which is North or South according as the star is Northern or Southern. Than bearing in mind, aswell the stars distance from the South, as also his declination, work thus, Bring the Finitor to your Latitude, and upon the stars Parallel or Circle of declination in the Mater count from the limb of the Mater, the distance of the star before found, and mark what Almicanterath crosseth that point, for there is the altitude of the star at that instant, and the Azimuth which cutteth that point, showeth in what part or coast of heaven the star is at that present. And remember to seek the Parallel of the star on that side of the Mater either on your right or left hand, so as it may fall amongst the Almicanterathes, for otherwise you shall not find that you seek. As for example the 26. of October 1591. at nine of the clock at night, you would know the height of the first star of the Ram's horn called Cornu Arietis the Sun being then in the 12. degrees and 12. minutes of Scorpio. Here after that you have laid the label to the hour supposed, and brought the degree of the Sun for that day to the Fiducial line of the label, and stayed the Rete there, bring the label to the tip of the star in the Rete, and the label will show in the limb of the Mater how many degrees the star is distant from the South, which you shall find to be 28. degrees and 30. minutes, and the declination of the said star being counted upon the said label, to be 18. degrees, then keeping those two numbers in mind, bring the Finitor to your Latitude supposing the same to be 52. degrees, and there staying the Rete, seek for the Parallel of the star which is 18. degrees on your right hand, and upon that Parallel count by the Meridian's from the limb inward, the stars distance from the South, which was 28. degrees 30. minutes, and you shall find that the 49. Almicanterath cutteth that point, so as you may conclude that the altitude of the star called Cornu Arietis, was at that hour 49. degrees, and that the 47. Azimuth doth also pass through that point which showeth that the star is 47. degrees from the East towards the South, which according to the Mariner's account is Southeast, and somewhat more to the Southward. Now to find out the said star in the firmament, if it be a star light, you have no more to do but to lay the Diopter of your Astrolabe at that altitude, and to turn your face towards that coast (that is to say) Southeast and somewhat more to the southward, and the next bright star which answereth in that coast to that altitude is the star which you seek.   

How to find out the ascentionall difference of any star. The 21. Proposition.  
BRing the Finitor to your Latitude, and staying it there, look in what point the Parallel of the stars declination cutteth the Finitor, and the number of the Meridian's contained betwixt that point and the Centre, is the ascentionall difference, so shall you found the ascentional difference of the bulls eye called Oculus Tauri, whose declination is 15. degrees 48′· to be 21. degrees.   

How to know the obliqne ascension of any star. The 22. Proposition.  
IF the declination of the star be Northern, subtract the ascentionall difference out of the right ascension, and the remainder shall be the obliqne ascension of the star, but if his declination be Southern you must add the ascentionall difference to the right ascension, and that shall be the obliqne ascension of the star. As for example, because Oculus Tauri is a North star, subtract his ascentional difference which is 21. degrees, out of his right ascension which as the foresaid Table showeth is 62. degrees 30′· and there will remain 41. degrees 30′· which is the obliqne ascension of the said star. But if it were a South star as Spica Virgins, whose declination is almost 9 degrees southward, and her ascentionall difference is 11. degrees 30′· then you must add the ascentionall difference to her right ascension, which is 195. degrees 51′· so shall you find her obliqne ascension to be 207. degrees 21′·   

How to know what stars do never rise nor set in any Latitude. The 23. Proposition.  
IF the star be a North star having a greater declination be it never so little, then is the altitude of the Equinoctial answerable to your latitude: As for example, suppose your latitude to be 52. which if you take out of 90. then the remainder is 38. degrees, which is the altitude of the Equinoctial answerable to that Latitude, otherwise called the compliment, than that star never setteth in that Latitude: Again if it be a South star having greater declination never so little than the compliment of your Latitude, than that star never riseth above your Horizon. As for example, in the latitude 52. the star called Hircus, that is the Goat, being a North star never setteth in that latitude because his North declination is 45. degrees, which is greater than the compliment of your Latitude by 7. degr. Also the star called Lyra, whose declination is 38. degr. like unto the compliment of your Latitude doth never set, but only toucheth your Horizon aswell at his rising as setting, so contrariwise the star called Canopus, being a South star having 51. degr. 38′· of South declination, never riseth above your Horizon in the foresaid latitude 52. And by this rule you may judge in like manner of all the rest of the fixed stars both North and South.   

How to know at what hour of the day or night any star riseth or setteth. The 24. Proposition.  
FIrst mark by the outward or inward shooting of the longest tip of the star, whether it be North or South: for if it be North, then count amongst the Meridian's in the Mater so many Meridian's as your Latitude amounteth to, beginning at the Axletree, and so proceeding towards the North-east, which is betwixt the North pole and the Equinoctial on your left hand, but if the star be South, then count your Latitude proceeding from the said Axletree towards the Southeast, which is betwixt the ringle and the North pole on the right hand, and bring the tip of the star to that Meridian which now signifieth your Horizon, and there staying the Rete, bring the label to the place or degree of the sun in the zodiac of the Rete, in which the sun is that day you seek, and the hour in the limb whereto the label pointeth, is the hour at which the star riseth that day or night. Now to know when the same star setteth, you have no more to do but to work with the star in the Northwest part in such order as you observed before in the North-east part of the Astrolabe. As for example, I would know at what hour the bulls eye called Oculus Tauri, doth rise the last day of june the Sun being then in the 17. degree and 40. minutes of Cancer. Here because this star is a North star, I bring the longest tip thereof to the 52. Meridian which is our Latitude counting from the Axletree towards the North-east part of the Astrolabe which is on my left hand, for that Meridian is always the Horizon serving the Latitude 52. and there staying the Rete, I bring the Fiducial line of the label to the place of the sun which at that day is the 17. degree 40′· of Cancer as I said before, and I found that the label pointeth to one of the clock 30′· after midnight: wherefore I conclude that Oculus Tauri riseth that day at that present hour. Now to know at what hour that star goeth down the same day, I bring his longest tip to the said Horizon towards the Northwest, and staying the Rete there, I lay the label to the 17. degree 40′· of Cancer as before, so that the label pointeth to four hours and 20. minutes in the afternoon, at which time he goeth down, so as he continueth at that time above the Horizon in the Latitude 52. 14. hours 48. minutes. And to know the abode of any star above the Horizon, the next Proposition doth also show.   

How to know how long any star continueth above the Horizon in every Latitude. The 25. Proposition.  
BRing the Finitor to your Latitude, and look in what point the Parallel or declination of the star cutteth the Finitor, and the number of the Meridian's in the Mater contained betwixt the limb▪ and that point do show the half time of his abode above the Horizon, which being doubled is the whole time of his abode above the Horizon. And in numbering the said Meridian's whereof 15. do make an hour, remember to begin to count from the right side of the Mater, proceeding towards your left hand, and remember also that the middle Meridian or Axletree, signifieth always the sixth hour, so shall you not err in your account. As for example, having brought the Finitor to the Latitude 52. look in what point the Parallel of the foresaid star Oculus Tauri cutteth the Finitor, & by numbering the Meridian's proceeding from the limb on your right hand towards the left, you shall find the Parallel of Oculus Tauri, being 15. degrees 49′· to cut the eleventh Meridian, which being doubled and then the sum thereof divided by 15. maketh 14. hours and 48′·S as before.   

How to find out the stars hour, and thereby to know the hour of the night. The 26. Proposition.  
FIrst having taken the stars altitude, set the Finitor to your Latitude, and look at what houre-line in the Mater that stars Almicanterath and the Parallel of his declination do meet, & having sought out the same hour in the limb right against that point, bring the label thereunto, for that is the stars hour, and there staying with your fingers end the very point of the label, bring the longest tip of the star to the Fiducial line of the label, and staying there the Rete, turn the label to the degree of the zodiac of the said Rete, wherein the sun is that day, and the label will point to the true hour of the night set down in the outward limb of the Mater. As for example, I suppose that the seventh of October 1591. the sun being then in the 23. degree 15′· of Libra, I took the altitude of the star Hircus, that is the Goat, which I found to be 20. degrees, here having set the Finitor to my Latitude 52. I look in the Mater at what hour line that Almicanterath, and the Parallel of the same star which is 45. degrees Northward do meet or cross one an other, and I found that they meet just upon the fourth hour line of the forenoon, wherefore I seek right against that point the same hour in the outermost limb of the Mater, for that is the stars hour, and having placed the label at that hour, I stay it there with my finger until I have brought the longest tip of the star Hircus unto the Fiducial line of the label, and there staying the Rete, I turn the label to the degree of the sun which is 23. degrees 15. minutes of Libra, and I see that the label pointeth to the seventh hour of the night set down in the limb of the Mater and half an hour past. Master Blagrave saith that the sooner you take the altitude of the star whereby you seek to know the hour of the night, you shall have the hour more truly.   

How to find out the distance betwixt any two stars contained in the Net. The 27. Proposition.  
FIrst consider whether the declination of both stars be either South or North, or that the one be South and the other North, for if both their declinations be South or North, than you must not leave to turn the Rete to and fro until you have brought the longest tips of both stars to one self Meridian in the Mater, that done count upon the said Meridian how many degrees are contained betwixt the tips of the said stars, for that is the distance betwixt them. But if the one star have North declination and the other South then turn the Rete to and fro until both stars do lie upon two such several Meridian's as each of them is equally distant on each hand from the Axletree, then count how many degrees or Meridian's are contained betwixt the Axletree and either of those stars (which you will) and that is their distance, so shall you find the distance betwixt Oculus Tauri and Canis minor being both North stars to be 46. degrees 20′· and the distance betwixt Oculus Tauri and Canis maior, whereof the first is a North star, and the other a South star to be just 46. degrees.   

Another way to know the distance of any two stars, their Longitudes and Latitudes being first known, and also by that means to find out the distance betwixt any two places upon the earth. The 28. Proposition.  
FIrst seek to know the difference of their Longitudes by subtracting the lesser Longitude out of the greater, then count that difference from the outermost Meridian of the Mater towards the Centre, and mark well that Meridian at which your account endeth, than number upon the limb of the Mater from the Equinoctial the greater Latitude either Northward or Southward according as the Latitude is, and to that point bring the Zenith of the Rete, then upon the self same Meridian before marked count from the Equinoctial the lesser Latitude, and look what Azimuth passeth through that point, for the degrees which are contained betwixt that point and the Zenith shallbe the distance: and thus doing you shall found the distance betwixt Oculus Tauri & Canis maior to be 46. degrees and 15′·S and the distance betwixt London and Venice to be 12. degrees and 20′·S which 12. degrees being multiplied by 60. maketh 720. miles, whereto if you add for the 20′· 20. miles, it will make in all 740. miles.   

How to found out the degree of Medium coeli at any hour, of the day (that is to say) the degree of the zodiac that is in the Meridian at any hour that you seek, and also the degree called Imum caeli. The 29. Proposition.  
FIrst seek out the place of the sun for that day in the zodiac of the Rete, and having laid the label to the hour supposed upon the limb of the Mater, bring the place of the sun to the Fiducial line of the label, and there staying the Rete, look what degree of the said zodiac cutteth the noon line or Meridian at noontide, which you shall easily found by laying the label to the hour of 12. at noon, for the Fiducial line of the label crossing the zodiac, will show the degree of mid heaven at that hour. As for example, the 26. of june 1592. I would know the degree of Medium coeli, at eight of the clock in the morning, the sun being that day in the 14. of Cancer. Here by laying the label to the said hour in the limb of the Mater, I bring the place of the sun to the Fiducial line of the label, and there having stayed the Rete, I bring the label to the twelfth hour at noon, and I found that the label cutteth the zodiac of the Rete in the 18. of Taurus, which at that hour is the degree of Medium coeli, whose point opposite is the 18. degree of Scorpio, and that is at that hour Imum coeli▪ which the label being laid to the 12. hour of midnight will show.   

How to find out the Horoscope or ascendent at any time of the day or night, and thereby to have the four principal angles of heaven. The 30. Proposition.  
Having laid the label to the hour given upon the limb of the Mater, stay it there with your finger until you have brought the place of the sun for that day unto the Fiducial line of the label, and there staying the Rete, look what degree of the zodiac thereof toucheth or crosseth the Horizon answerable to your Latitude, for that is the ascendent at that present hour. As for example, I would know the ascendent at eight of the clock at night the twelfth of October 1580. the sun being then in the 28. degree 14′· of Libra. Here having laid the label upon the said hour, I bring the 23. degree 10′· of Libra to the Fiducial line of the label, and there staying the Rete, I found by help of the label that my Horizon which is the 51. 40′· Meridian, counting on both hands from the Axletree, that the first degree of Cancer, doth cross my Horizon in the North-east quarter, wherefore I affirm that to be the ascendent or first house, whose point opposite being the first of Capricorn, is the descendent or seventh house, then by bringing the Fiducial line of the label to the South end of the Equinoctial at which the ringle hangeth, I find by help of the label that the 24. degree 40′· of Aquarius cutteth the Equinoctial, which is the tenth house or Culmen coeli, whose point opposite being the 24. degree 40′· of Leo, is the fourth house, otherwise called Imum coeli, and thus you have all the four principal houses of heaven for that hour, as you may see in this figure here following.  
A figure of the 4. Angles of heaven, made the 12. of October 1580. the sun being then in the 28. degree 10′· of Libra, for the latitude 51. 40′·S  
But the difficulty of finding out the true ascendent consisteth in knowing whether it is to be sought in the North-east quarter, or in the Southeast quarter of the jewel. The North-east quarter is that which lieth betwixt Imum coeli, and the North pole or East end of the Axletree, because in this case the Axletree signifieth the line of East and West, and the Equinoctial signifieth the line of South and North, at the South end whereof is fastened the ringle or handle. But in the former example you may plainly see that the North part of the zodiac of the Rete doth cut the Horizon, as well in the North-east quarter as in the Southeast quarter with two several degrees and signs, for in the North-east quarter the zodiac cutteth the Horizon with the first of Cancer which is the ascendent, and in the Southeast quarter it cutteth the Horizon with the eight of Taurus, which is not the ascendent: for you have to understand that every degree of the zodiac doth both rise and set either towards the North or towards the South, the first point of Aries and of Libra only excepted, both which do rise right East, and go down right West, even as the Equinoctial doth, whereof M. Blagrave doth gather a rule how to found out by his jewel when the ascendent is to be sought either in the north-east quarter of the Mater, or in the southeast quarter, which is thus: The ascension of any of the six Northern signs is to be sought for in the North-east part of the jewel, & the ascension of any of the six Southern signs is to be sought for in the southeast part of the jewel. For although that the North part of the zodiac of the Rete containing the 6. Northern signs cutteth the Horizon answerable to your Latitude, aswell in the North-east as in the Southeast part of the jewel, yet you must seek the ascendent in the North-east part, and not in the Southeast part of the jewel, because that every degree of any of the 6. Northern signs riseth Northernly: so contrariwise if the South part of the zodiac containing the six Southern signs do cut the Horizon, aswell in the North-east part as in the Southeast part of the jewel, yet you must seek the ascendent in the Southeast part, and not in the North-east part of the jewel. As for example, the second day of August 1592. the sun being in the 20. of Leo, I would know the ascendent at four of the clock in the afternoon: here having laid the label to that hour, & brought the place of the sun to the Fiducial line thereof, I found that the 13. degree 30′· of Aquarius doth cut the Horizon serving to your latitude 52. in the North-east part of the jewel, and that the 19 of Sagittarius cutteth the said Horizon in the Southeast part of the jewel, which must be the ascendent, because that every degree of any of the southern signs riseth Southernly, and not Northernly.   

How to found the Circles of position, and to know how much the pole is elevated above every such Circle in any Latitude, without the knowledge whereof you can not find out the twelve houses by this Astrolabe. The 31. Proposition.  
FIrst bring the Zenith of the Rete to your Latitude, so shall the Azimuthes become circles of position, then upon the Parallel of declination of the point given, according as the declination thereof is north or south: count from the limb on your right hand, the number of the hours given amongst the Meridian's in the Mater, & the Azimuth which the Meridian of that hour cutteth shallbe the Circle of position, which had, you shall found the elevation of the pole above the Circle of position thus: Count your Latitude amongst the Almicanteraths from the Zenith upon the Circle of position found, & to that point whereas your Latitude endeth, bring the Fiducial line of the label, & reckon upon the label how many degrees are contained betwixt the limb of the Rete & that point, for the number of those degrees is the elevation of the Pole above the Circle of position. As for example used by M. Blagrave himself, the 12. of October 1580. the sun being then in the 29. degree 10′· of Libra, and his South declination 11. deg. 10′· I would know at 8. of the clock at night in what Circle of position the sun at that time was, here I bring the Zenith to the Latitude of Reading, which is 51. deg. 40′· then for 8. of the clock I count 8. hours amongst the Meridian's in the Mater, from the South part of the jewel upon the Parallel of the sun being then 11. deg. 30′·S and by attributing to every hour 15. Meridian's, I found that the Meridian whereas the 8. hour endeth doth cross the 28. & ½. Azimuth, counting the Azimuths from the Zenith line, and that is the Circle of position wherein the sun was then under the earth in the North-east quarter. Now to know how much the Pole is elevated above that circle of position I do count from the Zenith the foresaid Latitude 51. degrees, 40′· upon the said Circle of position, and to the point where that Latitude endeth I bring the label, & counting thereon from the limb of the Rete the degrees contained betwixt the said limb, and the foresaid point, I found the number of them to be 43. degrees 30′· wherefore I conclude that the Pole is elevated above the said Circle of position 43. degrees 30′·   

How to found out all the 12. houses of heaven, and thereby to erect a figure at any hour of the day or night. The 32. Proposition.  
YOu shall understand the order of this better by this one example given by M. Blagrave himself, then by manifold rules which be also so plainly expressed and observed in this example, as you need none other instruction to erect the like figure in any Latitude, and at any hour of the day or night. Suppose then that you would erect a figure for one that was borne the twelfth of October 1580. at eight of the clock at night in the Latitude 51. 40′·S the sun being then in the 28. degree 10′· of Libra, and his declination being at that time 11. degrees 10′· Southward: Here first set down in your figure already drawn the four Angles before found by the 30. Proposition answerable to the said day & hour, then to found out the rest of the houses do thus, bring the Zenith of the Rete to the latitude presupposed, which is 51. degrees 40′· & staying it there, mark what number of Azimuthes or Circles of position doth cut every 30. degree of the Equinoctial, counting the degrees of the Equinoctial by help of the Meridian's from the limb towards the Centre, but you must count the number of the Azimuthes or Circles of position from the Zenith line towards the limb, of which Circles of position there be no more but six at the most (that is to say) three betwixt the limb and the Centre on the one side, and as many on the other side of the Centre, the elevation or rather depression of every one whereof you must seek to know by the last Proposition. But M. Blagrave saith that you need to know the elevation or rather depression, but only of two Circles of position, that is of that which showeth the eleventh and third house, and of that which showeth the twelfth & second house, and so saith Stadius also in the beginning of his Ephemerideses, treating of the 12. houses, for having them you have all the rest. The order of working according to M. Blagrave his rule is thus: having brought the Zenith to the foresaid Latitude, look what Circle of position cutteth the Equinoctial in the first 30. degrees next to the limb, and you shall find that the 47. Azimuth or Circle of position (counting from the Zenith line) cutteth that point of the Equinoctial, and therewith serveth to the eleventh house, and also to the third house, and the elevation of this Circle is 32. deg. then from thence tell upon the Equinoctial 30. degrees more, which do make in all 60. degrees, and through that point you shall find the 19 Azimuth or Circle of position to pass, whose elevation is 47. degrees and this Circle serveth to the twelfth house & to the second house. Now keeping well in mind those two last elevations, that is, 47. and 32. work thus: Lay your label to the eight hour of the morning, which is 30. degrees distant from the right Horizon or Axletree, and bring to the Fiducial line thereof the Culmen coeli or 10. house first set down in your figure which is the 24. of Aquarius, and staying it there, look what degree of the Ecliptic cutteth the 32. Horizon which serveth to the 11. house counting from the right Horizon towards your left hand. And by following that Horizon up towards the North pole, you shall find that the 25. degree 30′· of Pisces cutteth the same Horizon very nigh unto the Pole, but by my instrument I find it to be the 27. degree of Pisces, which perhaps is not truly made, and therefore set down in the eleventh house of your figure 25. degrees 30′· of Pisces. Than bring the foresaid Culmen coeli together with the label to the tenth hour of the morning, which is 30 degrees further towards the South, and staying it there, look what degree of the Ecliptic cutteth the 47. Horizon, serving to the twelfth house, and you shall found that the 20. degree of Taurus cutteth that Horizon, wherefore set down the 20. of Taurus in the twelfth house of your figure, then bring the Culmen coeli together with the label to the 12. hour at noon, and you shall see the ascendent which is the first of Cancer to cut the obliqne Horizon, which is 51. degrees 40′· and from thence bring the label and culmen coeli to two of the clock in the after noon, and staying it there you shall see the 21. degree of Cancer to cut the 47. Horizon which showeth the second house and also the twelfth house as before, wherefore set down 21. of Cancer in the second house of your figure, that done, remove the label together with Culmen coeli to four of the clock in the afternoon, and there staying it, you shall see the seventh of Leo to cut the 32. Horizon which showeth both the third house and the eleventh house as before, wherefore set the seventh of Leo in the third house of your figure, so have you eight of the houses, that is to say, the first, the second, the third, the fourth, the seventh, the tenth called Culmen coeli, the eleventh and the twelfth, so as there want only four, that is, the fifth, the sixth, the eight and the ninth house. The fifth is opposite to the eleventh, the sixth to the twelfth, the eight to the second, and the ninth to the third, whose opposite signs each one having like number of degrees are to be placed in the four houses of your figure that be wanting. As in the fifth houses the 25. degree 30′· of Virgo, in the sixth house the 20. degree of Scorpio in the eight house the 21. degree 30′· of Capricorn, and in the ninth house the 7. degree 10′· of Aquarius, so have you all the twelve houses. And by this means M. Blagrave saith that you may make the places of the twelve houses to serve for ever in any Latitude, so as you do distinguish from the rest with some colour those two Orisons whereof the one doth show the eleventh and the third house, and the other Horizon showeth the twelfth and second house, yea and by this means you may (as he saith) make as true Tables to find out the twelve houses in every Latitude as those that be calculated of purpose.  
The figure of the foresaid twelve houses.  
The figure of the heavens the twelfth day of October 1580. at eight of the clock in the afternoon, for the Latitude 51. 40′·S   

A new and necessary Treatise of Navigation containing all the chiefest principles of that Arte. Lately collected out of the best Modern writers thereof by M. Blundivile, and by him reduced into such a plain and orderly form of teaching as every man of a mean capacity may easily understand the same.  
 They that go down to the Sea in ships, and occupy their business in great waters: These men see the works of the Lord and his wonders in the deep. Psalm. 107.  

Of Navigation, what it is, and with what order the principles thereof are here taught by Master Blundevill, according to the rules of the best modern writers of that Arte.  
NAVIGATION is an Art which teacheth by true and infallible rules, how to govern and direct a ship from one Port to another safely, rightly, and in shortest time: I say here safely, so far as it lieth in man's power to perform. And in saying rightly, I mean not by a right line, but by the shortest and most commodious way that may be found: And by saying in shortest time, I mean thereby according as the ship is good of sail, and according as both wind and tide shall serve.  
The means to attain to this Art as to all other Arts are two parts, whereof every Art consisteth, that is Method and Practice, which is as much to say here as instruction and experience: For of the chief points of this Art, some are to be learned by instruction, and some only by experience: for what instruction I pray you will serve to make a good coaster, that is to say, to know any Cape or cliff when he seeth it, unless he hath first seen it & hath before taken good marks thereof. Also to know the currents in all places, the depth and qualities of waters, sands, flats, or shouldst, and such like, the perfect knowledge whereof consisteth chiefly in experience. Of which experience I mind not here to treat, because I know that a Dutchman called Wagoner or Aurigarius hath lately written of these things notably well, & especially for these our seas and the North-east seas, whose book I doubt not but that some of our learned sea men will set forth in our mother tongue, with some augmentation of their own experience for the East and West Indieses, for the profit of their country and to their own praise & commendation. In the mean time I will proceed with instruction, the chief points whereof are these here following: first to know the uses of such instruments as every skilful sea man aught to have with him that mindeth to sail any long voyage, which are these here following: First a perfect Calendar or Ephemerideses, the Mariner's ring or Astrolabe, the cross-staff called of the Spaniards Balla stella, the two Globes both Celestial and Terrestrial: Of all which things I have in a manner already written several Treatises. Item an universal Horologe, to know thereby the hour of the day in every Latitude, and a Nocturnlabe to know thereby the hour of the night. And it were necessary for him that saileth long voyages to carry with him a topographical instrument to describe thereby those strange Coasts & Countries whereby he saileth, the use of which instrument is plainly taught by William Borne in his book called the Treasure of travailers, and by Master Digges in his book called the Pantometria, wherefore I shall not need to speak any further thereof: Also the Mariner's compass, whereof I do not only make a plain description and show the uses thereof, but also how to find out the variation of the same, according to the precepts of the best modern writers thereof. And lastly the Mariner's Card, whereof whilst I speak, I do not only show the making thereof, and how to draw the Parallels in better sort than they are used to be drawn in the common Cards, but I set down therewith the chiefest uses of the Card, amongst which is taught how to know by help of the Card and certain Tables made of purpose, what way your ship hath made in sailing by any rombe or wind, yea in sailing right East & West, which heretofore (as Cogniet saith) hath been thought a thing unpossible to be known, but only by conjecture: And being out of your way how to know in what place you are, & to mark the same in your Card that you may the more readily direct your ship again to the place whereunto you would go. But all these instruments serve to little purpose, unless you know also the North star with his guards, & diverse other stars situated aswell towards the North as towards the South pole, together with their Longitudes, Latitudes, declinations, and greatness, to know thereby the Latitude of any place, & the hour of the night: also the course of the sun and his declination, by help whereof you may know the Latitude of any place, the times and seasons of the year, the hour of the day, and the length of the day and night in every latitude. And finally you must know the course of the Moon, whereon dependeth the knowledge of the tides in all places: Of all which things I mind here to treat in such order as the Chapters hereafter following do show.   

A general Calendar or Almanac for ever, containing these 9 Chapters next following.  
HOw to found out the golden number every year. Chap. 1.  
How to find out the Epact in every year. Chap. 2.  
How to know the Epact by the Mariners rule upon your thumb. Chap. 3.  
How to know the age of the Moon in every month throughout the year. Chap. 4.  
How to know the change, full, and 4 quarters of the Moon, every month throughout the year. Chap. 5.  
How to know in what sign and degree the moon is every day throughout the year. Chap. 6.  
How to found out the movable feasts every year, only by knowing the day of conjunction in the month of February. Chap. 7.  
How to find out the circle of the Sun, called in Latin Cyclus Solaris, and thereby the Dominical Letter in every year. Chap. 8.  
How to found out the number of indiction. Chap. 9  
A brief description together with the use of the diurnal Table or Almanac of johannes Stadius. Chap. 10.  
How to found out the place of any Planet by the Ephemerideses.  
Chap. 11.  
Of the Mariner's Ring or Astrolabe, and of his cross staff Chap. 12.  
A brief description of M. Hood his cross staff, and of all the parts thereof. Chap. 13.  
How to set the parts of M. Hoods staff together to serve such Astronomical uses as do chief belong to the Mariner. Chap. 14.  
The shape or figure of the foresaid staff, having all his parts set together to serve for Astronomical uses. Chap. 15.  
How to take the altitude of the Sun at any hour that he is to be seen with the eye by M. Hoods staff. Chap. 16.  
How to take the altitude of any Star with M. Hoods staff. Chap. 17.  
How to take the distance betwixt two stars with M. Hoods staff. Chap. 18  
Of the Mariner's Astrolabe. Chap. 19  
A brief description of the Mariner's Astrolabe, and the use thereof. Chap. 20.  
A brief description of the Mariners cross staff. Chap. 21  
The uses of the Mariners cross staff. Chap. 22.  
Of the Wind, what it is, and of the diverse kinds and names thereof. Chap. 23.  
A brief description of the Mariner's Compass, and use thereof. Chap. 24.  
Of the loadstone, and of the variation of the Compass in Northeasting and Northwesting. Chap. 25.  
How to find out the variation of the Compass in every latitude. Chap. 26.  
Of the Mariner's Card, and of the making thereof. Chap. 27▪  
The shape and figure of the first lineaments of the Mariner's Card drawn after the old manner, and how to set down the places of the land or sea therein. Chap. 28  
A Table to draw thereby the Parallels in the Mariner's card, together with the use thereof in truer sort than they have been drawn heretofore. Chap. 29.  
The draft of the Meridian's and Parallels of the Mariner's Card or nautical Planispheare according to the former table. Chap. 30.  
The four chiefest uses of the Mariner's card. Chap. 31  
How to know the way of your ship, and how many leagues are to be accounted for one degree of latitude in every rombe whereby you sail. Chap. 32.  
How to accounted the leagues in sailing directly East or west without changing latitude or altitude of the Pole. chap. 33.  
A Table to help you to know what way your ship hath made in sailing right East or West without changing your latitude together with a brief description & use thereof. chap. 34.  
An example of counting the way of your ship in sailing right West. chap. 35.  
another example of counting the way of your ship in sailing right East. chap. 36.  
To know how much you go out of your way in sailing by one wrong rombe or by more. chap. 37.  
Of the North Star, otherwise called the Lodestarre, and of his guards, and how to know the same. Chap. 38.  
The uses of the North star and of his guards. chap. 39  
To know by help of a little Table made according to the Mariners rule touching the 8. principal rombes, showing how much and when the Lodestarre is either above or beneath the Pole, that you may know thereby the true altitude of the Pole in taking the height of the Lodestarre with your Astrolabe or cross staff. chap. 40.  
How to make an Instrument which will show at any hour of the night how much the Lodestar is either above or beneath the Pole in every other rombe as well as in the 8. principal rombes, and also the true hour of the night. chap. 41.  
How to know by the foresaid twofold instrument as well the mounting and descending of the North star as the true hour of the night both at one instant and also the elevation of the Pole, Chap. 42.  
What stars are to be observed by those that sail beyond the Equinoctial under the South pole. chap. 43  
Of the Sun, and of his motion, and of the chiefest appearances belonging to him. chap. 44  
A Table showing the declination of the Sun every day throughout the year, and the use thereof. Chap. 45.  
Of the four seasons of the year, that is, Spring time, Summer, fall of the lease, called otherwise Autumn, and winter. Chap. 46.  
How to know when the Sun riseth and setteth in every latitude, and thereby the length of the day and night, and also in what rombe or wind he riseth and setteth, and how much he declineth every day from the Equinoctial either Northward or Southward. Also how to know the elevation of the Pole, otherwise called the latitude of any place, by knowing the Meridian altitude of the Sun, & his declination. Chap. 47.  
Of the shadow of the Sun, and how to know thereby the hour of the day in any latitude by help of an universal dial. Chap. 48.  
Of the Moon and of all her divers motions. Chap. 49.  
How to know in what sign the point Auge of the Moon is in any year. Chap. 50.  
When the Moon is said to be in Conjunction with the Sun, or to be at the full, and what her greatest latitude is aswell from the Ecliptic line, as from the Equinoctial. Chap. 51.  
How to know in what part of the zodiac the head of the Dragon is every year. Chap. 52.  
How to know the tides in any place by the Moon. Chap. 53.  
How to know by help of an instrument the tides at any place. Chap. 54.  
How a general Rutter showing the tides in all places should be made. Chap. 55▪   

A general Calendar or Almanac for ever.  
WHat this word Kalends signifieth, & from whence it is derived is before set down in the first book of my Treatise of the Sphere the 45. chapter. But because there be general rules to know the conjunction of the Moon with the Sun, her full, and all her four quarters, and also the movable feasts and Dominical letter, and such like things easily to be learned without the help of any particular Calendar, I thought good first to set down this general Ralender, containing 9 propositions, as followeth.   

How to found out the Golden number every year. Chapter. 1.  
THe Golden number is the number of 19 proceeding from 1. to 19 and so to begin again at 1. And it is so called because it was sent in golden letters from Alexandria in Egypt to Rome: For in 19 years the Moon doth make all her sundry motions & changes, and returneth again to the place where she first began. And to find the foresaid number the way is thus. Add 1. to the year of the Lord whereof you inquire, and divide the same by 19 and the remainder shall be the Golden number for that year, as for example, being desirous this present year 1590. to know the Golden number, I add 1 to the said year, and so make it 1591. which being divided by 19 there remaineth 14. which is the Golden number of this present year.  
But when there is no remainder, than 19 is the Golden number, and remember that the Golden number beginneth always at the first of january, and the Epact the first of March.   

How to find out the Epact in every year. Chapter. 2.  
THe Epact is a number not exceeding 30. because the Moon betwixt change and change, never passeth 30. days, and thereby the common Lunar years consisting of twelve Moons is lesser than the Solar year by 11. days, for to every Moon are attributed no more but 29. days and a half, which make in all 354. days, so as the common Solar year consisting of 365. days exceedeth the Lunar year 11. days, from whence the Epact taketh his original, which Epact is found thus. Multiply the golden number of the year by 11. the product whereof if it be under 30. then it is the Epact. But if the product be above 30. then divide the product by 30. and the remainder shall be the Epact. As for example, to know the Epact in the year 1590. the golden number being 14. as before: here having multiplied the same by 11. and divided the product thereof by 30. I found the remainder to be 4. which is the Epact of the said year. Also by knowing one former Epact you shall have it ever after, by adding thereunto 11. and if the number do exceed 30. than you must divide the same by 30. and the remainder shall be the Epact, as by adding 11. to 4. I know the Epact shall be the next year, which is 1591. the number of 15. and by adding 11. to 15. the Epact in the year of our Lord 1592. shall be 26. and so forth.   

How to know the Epact by the Mariners rule upon your thumb. Chapter. 3.  
FIrst, you must suppose the inside of your left thumb to be divided into three spaces, and the nethermost space to contain 10. the middle space 20. and the highest space towards your thumbs end, to contain 30. and knowing first the golden number, begin to tell the same at the neither space saying there 1. at the middle space 2. at the third space 3. then begin again at the lowest space, and there say 4, and so continued your account still after that manner, until you have the full sum of the Golden number, and mark upon what space the full sum falleth, for the Golden number being added to the number of that space, doth show the Epact, so as the total sum doth not exceed 30. for then you must subtract 30. and the remainder shall be the Epact: as for example, in the year 1591. the Golden number is 15. which being counted upon your thumb in such order as is before taught, it will fall upon the highest space, which is 30. to which if you add the Golden number, which is 15. it will make in all 45. from which sum if you subtract 30. there will remain 15. which shall be the Epact for that year, so as the Epact and Golden number in that year are like numbers. For every three years they are always like, as when the Golden number is either 3. 6. 9 12. 15. or 18. the Epact hath also like number.   

How to know the age of the Moon in every month throughout the year. Chapter. 4.  
Add to the Epact the number of months from the beginning of March, together with the month wherein you seek, and also the number of the days passed of that month, wherein you seek, and the sum of such addition will show you the age of the Moon, as for example, I would know the age of the Moon the sixth day of December in the year 1590. Here knowing the Epact of that year to be 4. I add thereunto the number of the months from the beginning of March, which are 10. months, and also the number of days of December, which are six, and the sum thereof is the age of the Moon, if the sum be less than 30. But if the sum of such addition do exceed 30. than you must subtract 30. and the remainder shall be the age of the Moon, so as the month wherein you seek have 31. days, for if it hath less than 31. days, than you must subtract but 29. and the remainder shall be the age of the Moon, as for example, suppose you seek the age of the Moon the 22. of November in the year 1589. in which year the Epact was 23. here by adding to the Epact first 9 Months, and then twenty two days, it maketh in all 54. out of which by subtracting 29. because November hath but 30. days, the remainder is 25. which was the age of the Moon in that month, day, and year.   

How to know the change, the full, and four quarters of the Moon every month throughout the year. Chapter. 5.  
MArtin Cortes in his book called the Art of Navigation teacheth a rule to found the day of the change, in every month by knowing the age of the moon that day you seek, and then by reckoning from that day backward the number of the days that went next before, or else by taking the age of the Moon out of the days of the month that went next before, as for example, the last of October 1592. I find by the fourth proposition the age of the Moon to be 5. which being taken out of 31. (for so many days October hath) there remaineth 26, which day by this means should be the day of the change, but the very day indeed was the 25. of the said month, for the former rule to know the age of the Moon is not so true itself, but that sometime it will fall a day either over or under. But Gemma Frisius teacheth to found out the day of the change in every month thus. Add to the Epact the number of the months from the beginning of March whereof that month wherein you seek the change, must be counted as one, and then subtract the product or sum of that addition from 30. and the remainder will show the day of the change, which rule I find to be true so long as the sum of the addition doth not exceed 30. but when the sum of the addition is more than 30. which will commonly chance when the Epact is a great number, as 26. or 29. he teacheth no rule for that, wherefore I think it best to take such sum out of 59 and the remainder shall be the day of change, or at the most but one day over or under, and having the day of the change, you shall have the day of the full by adding 15. days more to the day of change: And having the change and the full, you shall easily have all her four quarters by adding or subtracting 7. days.   

How to know in what sign and degree the Moon is every day throughout the year. Chapter. 6.  
SOme do set down rules to know in what sign and degree the Moon is every day, but such as are not true. And to say the truth the Moon hath so many divers motions as it cannot be done but by special Tables calculated of purpose. And by what rule soever you work, you must first know the place of the Sun, which the Ephemerideses most truly showeth, and in looking for that, you shall also found hard by it the place of the Moon, that is to say, in what sign and degree she is every day, and so orderly the place of any other Planet, and therefore I leave to speak any further thereof.   

How to found out the movable Feasts every year, only by knowing the day of conjunction in the month of February. Chapter. 7.  
THe way to find out the Conjunction in every month is before taught in the fift proposition. And having the day of the conjunction in February, you may assure yourself that the next tuesday following is always Shrove-tuesday, for though the conjunction itself do fall upon a Tuesday, yet the next Tuesday after that, shall be Shrove-tuesday, and the next Sunday after that, is called Quadragesima, which is the first Sunday in Lent, and six weeks next af-that is Easter day, whereunto if you add five weeks more, that is to say, 35. days, you shall have Rogation Sunday, and 4. days next after that is ascension day, and ten days next after ascension day is Pentecoste or Whitsunday, and seven days next after that is Trinity Sunday, and four days next after that is the Festival day called Corpus Christi. And you have to note that the Sunday called Adventus Domini, which we call the first Sunday in Aduent, is always the fourth Sunday before Christmas day, and the Sunday called Septuagesima is the third Sunday before Quadragesima, otherwise called the first Sunday in Lent, and betwixt Septuagesima and Quadragesima there are other two Sundays, whereof that which is next before Quadragesima is called Quinquagesima, and the next Sunday before that is called Sexagesima.   

How to find out the circle of the Sun, called in Latin by a Greekish name Cyclus Solaris, and thereby the Dominical Letter in every year. Chapter. 8.  
THis Circle was invented more to find thereby the Dominical Letter than to show any great changes of the suns motions therein: And because there be seven days in the week, commonly signified by seven letters, A. B. C. D. E. F. G. This Circle therefore is made to contain twenty eight years, for four times seven do make twenty eight, by help whereof is known the true order of the letters, whereof A. signifieth always the first day of every year, and for every leap year are appointed two Dominical letters, whereof the one continueth from the beginning of that year until Saint Mathias even, and the other from thence to that years end, as you shall more plainly perceive by that which followeth. And first I will show you how to find out the just number of the circle of the Sun every year, which is done thus.  
Add to the year of the Lord given or supposed 9 and divide that product by 28. and the remainder shall be the number of the foresaid Circle: As for example, I would know the number of the said Circle in the year 1590. whereunto by adding 9 I make the sum to be 1599 which being divided by 28. there remaineth 3. Now to find out the Dominical letter by the number of the said Circle, you must resort to the figure following, consisting of three Circles, making two spaces, in the upper space whereof are set down the aforesaid seven letters, and in the neither space the numbers of the suns Circle, in such sort as every number hath his proper Dominical letter standing right over his head, and for every leap year there are set down two Dominical Letters. All which letters are to be counted backward, & not forward as they are placed in the Almanacs. As for example, having fowd the number of the suns circle for the year 1590. to be 3. in the lower space, you shall find the letter D. standing over his head, which is the Dominical letter for that year. And note that when 28. is the number of the suns circle A. is always the Dominical Letter, from which number you must begin to count again at 1. and so proceed backward to 28.   

How to found out the number of Indiction. Chapter. 9  
THis number consisteth of 15. years, and is commonly set down in all the Charters of the Bishops of Rome, and in the instruments and writings of their pronotaries, and therefore is called Indictio Romana, wherein they seem to follow the ancient Romans, which used the like indiction of years, but to other purposes, whereof I make mention in my Sphere: The way to found out this number every year is thus. Add to the year of the Lord given 3. and divide the product thereof by 15. and the remainder shall be the number of the said indiction. But you have to understand that this Indiction is to be counted from September, and not from March as is the Epact.  
But the surest, and therewith the most general Calendar to serve in all places is the Ephemerideses or daily Almanac and especially that of johannes Stadius, which will serve for 14. years yet to come: And before those years be all expired, I doubt not but that the like will be set forth by some one or other of our learned Astronomers, amongst whom for many years past the Germans have been most famous. I chose the Ephemerideses of Stadius because he is more portable, and of less price than that great Ephemerideses of Leovitius. The chiefest uses whereof, and most meet for Mariners, though I have already set down in the latter end of my Treatise of the Globes, to the intent you might by the help thereof found out in the Globe the true place of the Moon and of every one of the other 5. Planets that hath both longitude and latitude, yet I think it good once again here in this place, most properly requiring the same, briefly to show the use of the said Ephemerideses, and specially of the diurnal, Table which beginneth at the year of our Lord 1583. endeth with the year 1606.   

A brief description together with the use of the diurnal Table or Almanac of johannes Stadius. Chapter. 10.  
THis Table beginneth at the 202. page of his book called the Ephemerideses or daily Almanac. And every page of the said Table that is on the left hand is divided into 9 collums. In the first collum whereof on the left hand are set down the days of the month. First, the Gregorian days according to the Roman account, and next to that the days of the month according to our English account, and then in the front of every other collum are set down the characters, first of the Sun, and then of the other six Planets, that is to say, of the Moon, Saturn, jupiter, Mars, Venus, Mercury, and last of all, the head of the Dragon figured thus ♌. And right under these seven Planets, and also under the head of the Dragon are set down the signs and degrees whererein every of these is every day of the month throughout the year at noontide. And in the foot of the said Table is set down the latitude of every one of the five planets, proceeding by the days of the month divided into three parts. And in the margin of every left page are set down the chiefest feasts and Saints days that fall in every month throughout the year.  
And moreover, in the Table on the right hand right against the left Table are set down first the days of the month, and then what conjunction or any other aspect the Moon hath with any of the other six Planets, that is, with the Sun, with Saturn, with jupiter, with Mars, with Venus, and with Mercury, which Planets are set down in the front of the said Table, and under them the Characters of such aspects as the Moon hath that day with any of the other Planets. The characters of which aspects are these here following.  
☌ ☍ △ □ ⚹  
Whereof the first signifieth a Conjunction, the second an opposition, the third a trine aspect, the fourth a quadrat aspect, and the fift a sextile aspect.  
Two Planets are said to be in a conjunction when they are both in one self sign.  
And to be in an opposition when they are in two several signs distant one from another, that is to say, 6 signs distant one from another.  
And to be a trine aspect when they are distant one from another by four signs:  
And to be in a quadrat aspect when they are distant one from another by three signs.  
And to be in a sextile aspect when they are distant but 2 signs one from another.   

How to found out the place of any planet by the Ephemerideses. Chapter. 11.  
NOw to found out the place of any Planet, or of the head of the Dragon by this diurnal Table, you must first seek out the day of the month in the first Collum of the left Table, and right against that on your right hand in the said left Table, you shall found in the common angle right under the Planet or Dragon's head, which so ever of them you seek, the sign and degree wherein the said Planet or Dragon's head is the said day at noontyde.  
And to found out the aspects which the Moon hath with any of the Planets the same day, you must resort to the other Table on the right hand, observing like order as before.  

The Example.  
As for example, the 21 of April 1592. which is the first of May according to the Roman account, I found by the Table on the left hand, the Sun to be in the 10. degree, 15. minutes of Taurus, the Moon to be in the third degree 47 minutes of Capricorn, Saturn to be in the 8. degree 10 minutes of Cancer, jupiter to be in the 18. degree 28 minutes of Sagittarius, Mars to be in the 12 deg. 6 minutes of Gemini, Venus to be in the 2 degree 0 minutes of Aries, and Mercury to be in the 6. degree 20 minutes of Taurus, and the head of the Dragon to be in the 29 degree 45 minutes of Gemini. And right against this in the Table on the right hand, you shall found the Moon to be in a trine aspect with the Sun, to be in an opposition with Saturn, to be in a trine aspect with Mercury.  
Thus much touching general Calendars or Almanacs. Now as for particular Calendars I take that which Robert Norman hath set down in his new Attractive to be the fittest for our country men, which containeth many necessary things, as the contents thereof together with the Table next following the same doth show, which Calendar I think it superfluous to be set down again here, and the rather for that I wish every Mariner not to be without that book, and especially containing besides the Calendar many other good precepts touching navigation, to which book is also joined a very learned and Mathematical discourse touching the variation of the Mariner's Compass made by Master William Borough controller of her majesties Navy, who in mine opinion is one of the skifullest men in the Art of navigation that is in this Realm.    

Of the Mariner's Ring or Astrolabe, and of his cross staff. Chapter. 12  
NOw according to the order set down in the beginning of this Treatise, I must needs speak somewhat of the common Marmers' Ring or Astrolabe, and of his cross staff which serve not to so many purposes as that of Stofflerus or of M. Blagrave before described, but only to take the altitude of the Sun, or of any Star or Planet, neither are so many conclusions to be wrought by their common cross staff as by that of Gemma Frisius, or by that which Master Hood hath lately invented. Sigh our Marmers for the most part do use their common cross staff to none other end but either to take the altitude of the Sun or of any fixed star or planet, or else to take the distance betwixt two stars, the making of which staff is plainly set down by Martin Cortes in his art of Navigation, and also the making of their Astrolabe.  
But Cogniet and Wagoner do set down a new kind of Cross staff, having 3. transoms or crosses every one longer & shorter than another by the one half, affirming that so many conclusions may be wrought thereby as by that of Gemma Frisius, but in mine opinion they are not to be compared in all respects to that of Gemma Frisius which though by reason that the yard thereof is of so great a length as it is not maniable in a ship, yet upon the land it is most serviceable, nor yet to that of Master Hoods invention, which is most maniable, and therewith very light of carriage, by whose staff I think verily that as many things may be wrought as by any other kind of cross staff whatsoever, and with less trouble, yea, and in matters of Astronomy more truly by reason that the yard and transome be of one self length, and thereby the degrees be larger throughout the whole quadrant than they be in the common cross staves, for in them the degrees from 50. upwards to 90. are very small and have narrower spaces than M. Hoods staff hath. Again, whereas in using the Mariners cross staff in such latitude, as the suns beams be of great force, they are feign to have glasses made of purpose to save their sight, and in some places all too little. But in using M. Hoods staff they shall not need to behold the Sun itself at all, but only to mark upon what degree of the yard the shadow of the Vane stréeketh. Moreover, when the Sun or star is 50. or 60. degrees high, they are feign to use their Astrolabe and not their staff, which Astrolabe in mine opinion, as I have said before, is the best instrument of all others to take the altitude of the Sun in the day, or of any star in the night, and because I have here commended unto you M. Hoods staff, I will first set down a plain description thereof together with those few Astronomical uses which do chiefly belong to the Mariner, without committing any offence I hope to the author thereof, and then I will describe unto you the Mariner's ring and his common staff together with the uses of the same.  
But as I was about to describe unto you M. Hoods staff, a friend of mine coming in the mean time desired me that I would first set down the making and use of the cross staff with three transomes, which Wagoner and Michael Cogniet do so much commend, and as I hear, is used of many sea men in these days, whose request I could not well deny, and therefore lo here followeth both the making and use thereof.  

Of the cross staff having three Transames or Cursours, commonly used in these days, the use and making whereof doth hereafter follow according to the description of Wagoner and Michael Cogniet.  
FIrst prepare a right staff of firm wood that is square every way, bearing in thickness three quarters of an inch, and in length three or four foot, for which you must also prepare three Transames or Cursours every one shorter than another by the one half, for the longest would contain in length twelve inches, the second six inches, and the shortest or lest three inches. And every one of these Transames or Cursours must be cut with a square hole in the very midst, so as they may be made to run just upon the staff to and fro, these things being prepared, you must divide three sides of the staff into certain degrees to serve the three several cursours as followeth.  
First you must upon a smooth square Table somewhat longer every way then the staff, and for want of one such Table you may join two Tables that may stand even together, draw by help of a true long squire a right Triangle marked with the letters A. B. C. and let A. be the right Angle, and the Centre as you see in the figure following, and let both the sides of the said Triangle be of like length to your staff, then putting the one foot of your compass in the Centre A. and the other foot in B. or C. draw a quarter of a circle from C. to B. and having divided that quarter into two equal parts, make a prick in the midst marked with the letter D. and laying your ruler to that prick and to the Centre A. draw a right line which shall be A. D. then divide the half Quadrant D. B. into nine equal parts making 90. equal degrees proceeding from B. to D. so as the first degree may be at B. and the 90 at D. that done, take with your compass the just half of the longest Transame, and keeping your compass at that wideness, set the one foot in the Centre A. and the other foot in the line A. C. so far as that wideness will extend, and there make a prick marked with the letter E. then from that point draw a right line that may be a Parallel to the line A. B. which line shall be E. F. then take with your compass the half of the middle Transame, and keeping your compass at that wideness, set the one foot in A. and the other foot in the line A. C. so far as that wideness doth extend and there make a prick marked with the letter G. and from that prick draw a right line that may be a Parallel to A. B which line shall be G. H. Thirdly take the one half of the lest Transame with your compass and transfer that wideness to the line A. C. setting the one foot of your compass in A. and make a prick with the other foot in the line A. C. as before, marked with the letter I. and from that prick draw another right line that may be a Parallel to the line A. B. which shall be the line I K. Now to graduate the first side of your staff to serve the longest Transame, you must lay the ruler to the centre A. and draw right lines from thence to every degree of the circumference contained betwixt 90. and 30. and those lines shall divide the line E. F. into so many unequal spaces as do belong to the first side of the staff, for you must lay the first side of the staff to that line to be marked according to the division of that line, the neither section whereof towards the lower end of the staff must be marked with 90. and the upper section with 30. so is the first side of your staff truly divided to serve the longest Transame, now to serve the middle Transame you must divide the line G. H. by drawing right lines from the Centre A. to every degree of the circumference contained betwixt 30. and 10. which lines will divide the line G. H. into unequal spaces, of which spaces the lowest must be marked with 30. and the highest with ten, according to which divisions you must mark the second side of your staff by laying the side close to the line G. H. so shall that side be marked to serve the middle Transame, then lay your ruler again to the Centre A. and draw right lines to every degree of the circumference contained betwixt 10. and the first or second degree next to B. and those lines shall divide the line I K. into unequal spaces, the lowest whereof is to be marked with 10. and the highest with 2. or 5. according as your instrument will bear, and according to those sections you must divide the third side of your staff by laying the same close to that said line I. K. and remember to make your lines of division so finely as is possible, so shall your staff be the more truly graduated. And noce that in graduating your staff, you shall not need to draw right lines from the Centre to every degree of the circumference contained betwixt the 90. degree and the 60. degree, for when the sun or any star is higher than 60. degrees, you must use your Astrolabe and not your staff which will not serve you to look so right up to take so great an altitude either of sun or star.  
A figure of the foresaid Triangle.  
 
The shape of the staff with his three Transames, together with the use thereof.  
Whensoever you would take the Altitude of the Sun or of any star, you have first to consider whether the sun or star be 30. degrees or more high: for than you must place the longest Transame upon your Staff. And set the lower end of your staff marked with ninety to your eye, which is always to be done how high or low so ever the sun or star be, and you must move the Transame either forward or backward until you may see by the upper end of the Transame the body or midst of the sun or star, and with the neither end of the Transame the Horizon, and then look in what degree the Transame cutteth the staff, for that is the Altitude of the Sun and star at that present, but if the sun be not 30. degrees high, than you must put on the middle Transame, and if he be less than ten degrees high, you must put on the shortest Transame, and then do as before. Thus much touching the Cross staff with 3. Transames, now I will describe unto you Master Hood his staff, and show you the use thereof.    

A brief description of Master Hood his cross staff, and of all the parts thereof. Chap. 13.  
The figure or shape of every part of the said staff.  
The transome.  
The yard.  
THis instrument as you see consisteth of four parts.  
The cross sockett  
The Vane.  
First of 2. square rulers each one bearing in thickness three quarters of an inch or there abouts every way square, and in length I would wish each ruler not to contain above one yard, for then in some uses they would wax top heavy.  
Of which two rulers the one marked with the letters A. B. is called the Transame, which is divided on the one side into 45. degrees beginning at one and so forth to 45. And every degree is divided again into six lesser parts making 60. minutes, for six times ten maketh 60. which serveth for Astronomical uses.  
And on the opposite side to that, the said Transame is divided into 1000 equal parts beginning at 25. and so increaseth by 25. until you come to 1000 Every which 25. parts is divided into five lesser parts, and every one of those again into five parts, which maketh in all 25. for five times five is 25 and this serveth for Geometrical uses.  
The other square ruler called the yard & marked with the letters C. D. is divided into degrees like to those of the Transame proceeding from 45. to 90. which together with those degrees that be in the first side of the Transame do make up the number of a just Quadrant to serve Astronomical uses. And the side opposite to that is divided into 1000 degrees proceeding from 25. to 25. like in all respects unto the opposite side of the Transame to serve Geometrical uses.  
The third part belonging to this instrument is a double socket of brass marked with this letter E. joined together with right Angles, and standing cross one to another. And each socket hath a screw to keep the same fast to his staff at any degree that you list to set the same: And also a long notch to the intent it may be laid close to any degree of the ruler whereto it belongeth.  
The fourth part belonging to this instrument is a vane of brass marked with the letter F. the upper edge whereof is pierced with a little round hole for the beam of the sun to pass through the same. And this vane is made with a socket, and with a screw to hold it fast to that ruler whereon it is set. And for certain Geometrical uses it were necessary (as some think) that there were two such vanes.  
Thus having described all the parts of the foresaid instrument, I will now show you how to set those parts together: which to serve Astronomical uses is thus done.   

How to set the parts of Master hoods staff together to serve such Astronomical uses as do chiefly belong to the Mariner. Chap. 14.  
FIrst put the neither end of the Transame marked with the 45. degree into the double socket, so as the Transame may stand right up, and that the notch of the socket may meet even with the said 45. degree, and turn the screw that the socket may stand fast at that degree, that done put the yard into the cross socket, so as the notch of the said cross socket may lie just upon the 45. degree of the yard, and there make it fast by turning the screw: then put on the vane, which to serve Astronomical uses must be set at the highest end of the Transame, in such sort as the upper edge of the vane pierced with the hole may stand even with the first point or stréeke of the said Transame, and when the Transame standeth on the left hand of the yard, than the vane must be placed on the right side of the Transame: But if the Transame do stand on the right hand of the yard, than the vane must be placed upon the left side of the Transame: And the yard and Transame would be so set together as the degrees of the Transame proceeding downward from one to 45. may look towards you, that is to say, may stand right before your face: And the degrees of the yard proceeding from 45. to 90. would lie upward, so as the end of the yard marked with 90. may point to your breast or be set to your eye as occasion shall require, for such Astronomical uses as here do follow.   

The shape or figure of the foresaid staff, having all his parts set together to serve for Astronomical uses. Chap. 15.  
  

How to take the altitude of the funne at any hour that he is to be seen with the eye by M. Hoods staff. Chap. 16.  
THe Staff being set in such order as is before taught, go into some open place whereas you may see the sun, and turning the end of the yard marked with ninety towards your breast, hold the yard so level as you can, that it may be a just Parallel to the Horizon, and turn both your face, and also the vane of the transame towards the sun. Than you have to consider whether the sun at that present be either just 45. degrees high, or more, or else less than 45. degrees, which you shall easily know thus: For if he be just 45. degrees high, than the shadow of the upper edge of the vane will stréeke just upon the 90. degree of the yard lying at your breast: But if he be more than 45. degrees high, than the shadow of the vane will stréeke short of the 90. degree: And if it be less it will cast no shadow at all upon the yard, but stréeke clean beyond it over your shoulder. Now knowing by this means whether the altitude of the sun be more or less than 45. degrees, you shall take his true altitude thus:  
If his altitude be less than 45. degrees, then make the Transame to sink down through his socket until the upper edge of the vane standing upon the Transame do cast his shadow just upon the nintieth degree of the yard. Than look at what degree the notch of the Socket wherein the Transome standeth, doth cut the said Transame, for that is the true altitude of the sun at that instant.  
But if the altitude of the sun be more than 45. degrees, then draw the double socket upon the yard nigher towards your breast, until you see the shadow of the vane to fall just upon the ninetieth degree of the yard, that done, look upon what degree of the yard the notch of the double socket cutteth, and that is the true altitude of the sun at that instant.   

How to take the altitude of any star with Master hoods staff. Chap. 17.  
THere is no difference betwixt the order of the taking the altitude of the sun and of any star, but only that in taking the altitude of any star, you must set the end of the yard marked with the 90. degree to your eye, staying the same upon the upper bone of your cheek, and that in such sort as you may see the star by a right line passing from the end of the yard, lying at your eye to the upper edge of the vane at one instant, being always sure that the yard lie level, making a just Parallel to the Horizon, which Horizon is more easy to be seen upon the sea, then upon the land, for on the sea there be neither hills nor trees to hinder the sight thereof.   

How to take the distance betwixt two stars with Master hoods staff. Chap. 18.  
THe instrument being set in such order as was first taught: Set the end of the yard marked with 90. at your eye: and let the other end of the yard directly point to one of those stars whose distance you seek, and turn the edge of the vane to the other star be it on the right hand or on the left, according as you think good yourself: Than you must consider whether the distance of the two stars be just 45. degrees or more or else less, for if it be just 45. degrees, than the outward edge of the vane, and the end of the yard marked with 90. will be answerable in your sight to that distance without any more ado. But if the distance be less than 45. degrees, which you shall easily perceive by your eye (for then the star that is on your right or left hand, will appear within the vane and fall short thereof) than you must thrust down the Transame in his socket until you may see the two stars by a right line passing from the end of the yard marked with 90. and lying at your eye, right forth to the outward edge of the vane, that done, look in what degree the double socket cutteth the Transame, and that shall be the true distance of the two stars. But if the distance of the two stars be more than 45. degrees, which you may know also by your eye, because the star which is on your left or right hand, will appear clean without the vane: then draw the double socket together with the Transame towards your eye, until you may see the two stars by a right line as before, and the degrees which the notch of the double socket cutteth on the yard, shall be the true distance of the two stars.  
Thus much touching the Astronomical uses of M. hoods staff meet for the Mariner. Now as for setting of the said staff to serve Geometrical uses, and to find out the manifold conclusions to be done thereby, I wholly refer you to M. hoods own book, thinking it good now to speak somewhat of the Mariner's Astrolabe or Ring, and of his usual cross-staff.   

Of the Mariner's Astrolabe. Chap. 19  
Michael Cogniet setteth down the making of a new kind of Astrolabe, which he calleth an universal Astrolabe, only because the declination of the sun every day is added in manner of a Table to the one end of the Diopter of the said Astrolabe, which is of no great importance sith the suns declination doth altar in the space of 30. years and less, which being expired, the declination is to be new calculated: In the mean time that Table which Norman hath set down in his Attractive, and also that which is more lately calculated set down by myself in the first part of my Sphere, and likewise in this Treatise whereas I treat of the motion of the sun and of his declination, may serve these 20. years and more without any greater error: Wherefore leaving to speak any further of Cogniet his universal Astrolabe, I will describe here the common Astrolabe and cross-staff used by most Mariners, the shape or figure of both which here followeth.  
  

A brief description of the Mariner's Astrolabe, and the uses thereof. Chap. 20.  
THe limb of the Mariner's Astrolabe is traced as you see with three Circles, making two spaces to contain therein the degrees and numbers of altitude: and these Circles are divided by two cross lines called Diameters, and cutting one another in the very Centre of the Astrolabe into 4. equal parts, of which quarter's the uppermost on the left hand towards the Ringle is only marked with degrees and numbers, as you may see in the figure: In which figure the perpendicular Diameter signifieth the Zenith, or the line of South and North, at the upper end whereof is fastened the Ringle or handle. And the other overthwart Diameter signifieth the Horizon the one end whereof on the left hand signifieth the East point, and the other end on the right hand the West point. And to this Astrolabe (as to all other) doth belong a ruler or Diopter, which as you see hath at each end a square tablet pierced with two holes, the one greater and the other lesser, the greater to look through with your eye, to take the altitude of any star or of the sun being so darkened by some cloud as though it casteth no shadow, yet it may be seen with the eye: And the lesser hole is for the suns beam to pass through when he shineth clear, and some Astrolabes are divided in like manner of the Zenith on both sides, and have two Diopters, whereof the one is pierced with a great hole, and the other with a smaller hole to serve to such purposes as is above said.  
The use of this Astrolabe is only to take the altitude of the sun at any time of the day, or of any fixed star or Planet in the night, in such sort as is before taught in my description of Master Blagrave his Astrolabe, in the third and fourth Proposition thereof.   

A brief description of the Mariner's cross-staff. Chap. 21.  
THis staff consisteth of two four square rulers of wood, whereof the one is called the yard, and the other the Transame or Cross: And the yard containing in length most commonly three quarters of a yard is the longer piece, and is divided into 90. unequal degrees: for from 10. to 90. the degrees do grow lesser and lesser, and the Transame containing for the most part but one third part of the yard, is cut with a square hole in the midst, so as it may run up and down upon the yard, & some Transames have two running vanes to be set at what wideness you list, to take thereby the distance of any two stars.   

The uses of the Mariner's cross-staff. Chap. 22.  
THe use of the Mariner's staff chief consisteth in two points, that is, to know thereby the altitude of the sun in the day time, or of any star in the night season, and the other is to know thereby the distance betwixt any two stars.  
The first is thus done, having put the staff or yard through the square hole of the Transame, set the end of the yard which is marked with 90. to your eye, laying that end upon the upper point of your cheek bone keeping your legs close together, and having directed the other end of the yard towards the sun or star whose altitude you seek▪ move the Transame to and fro or up & down, until you may see with the one eye (winking with the other) the one end of the Transame to meet▪ just with the Centre or midst of the sun or star, and the other end to touch the Horizon both at one instant, and that degree of the yard which the square edge of the Transame cutteth, will show how high the sun or star is at that present: And in taking the altitude of the sun or of any star, remember always to hold the yard so level as it may be a just Parallel to the Horizon, for so shall you take the altitude the more truly. Now as touching the second point, which is to know the true distance betwixt any two stars you must do thus.  
Set the end of the yard marked with ninety to your eye as before, and move the Transame to and fro until you may see the one end of the Transame to answer the one star, and the other end thereof to answer the other star, for that degree of the yard which the Transame cutteth will show the distance betwixt the two stars: And if the two stars be so nigh together, as they will not conveniently answer the two ends of the Transame, than you may use the two movable vanes, which (as I said before) you may set at what wideness you list, but yet so as they may both stand equally distant from the Centre of the Transame or from both ends thereof.  
Thus having showed the use of the Mariner's Ring and cross staff, I will now proceed to the other instruments, whereof the two Globes are part, which Globes I have already described and showed the chiefest uses thereof in a Treatise by itself, and as for the universal Horologe to know thereby the equal hour of the day in every Latitude, and also the Nocturnlabe to know the hour of the night, I shall speak of the one when I come to speak of the sun, & of the other whereas I treat of the North star and of his guards, for there I will set down the figure and shape of each instrument, and show how to use the same: But the two chiefest instruments belonging to a Mariner, are his compass and his Card, whereof I come now to speak.  
But sith the Fly of the Compass representeth the 32. winds, I think good first briefly to define what the wind is, and to show how many winds the ancient Mariners did use, I say here briefly because I have already spoken sufficiently thereof in the latter end of my Sphere.   

Of the wind, what it is, and of the divers kinds and names thereof. Chap. 23.  
THe wind according to Aristotle, is an exhalation hot and dry, engendered in the bowels of the earth, and being gotten out is carried sidelong upon the face of the earth. And of winds there be four principal which do take their names from the four quarters of the earth from whence they blow, that is North, South, East and West: And though the Greeks and Latins divide every quarter into three parts, and thereby make in all but twelve winds, whose names both Greek and Latin are set down in the latter end of my Sphere, yet our latter Sea men to be the more assured of their Rowtes and Courses, do divide every quarter of the Horizon into eight winds, so as in all they make 32. winds, giving them such names as are set down in this figure here following representing the Fly of the Mariner's compass.  
  

A brief description of the Mariner's compass, and the use thereof. Chap. 24.  
THe Mariner's Compass may be very well divided into two essential parts, that is, the Fly, and the wyars touched with the Lodestarre called in Latin Magnes. And first you have to understand, that the fly is a round white Card traced with 32. lines all passing through the Centre of the Circle, which lines do signify the 32. winds in such sort as the figure before set down doth show, of which lines that which is marked with the Flower-deluce signifieth the north, whose point opposite is the South, and that which is marked with a Cross signifieth the East, whose point opposite is the West, and the outtermost Circle of the said Fly signifieth the Horizon, which circle is also divided into 360. degrees like unto those of the Equinoctial, so as every space bétwixt point and point cóntaineth 11. degrees and 15. minutes, which is the fourth part of a degree. Moreover, this Circle is divided in 24. hours by allowing to every point three quarters of an hour, which is 45. minutes, for an hour containeth 60.m m and half an hour containeth 30.m m and the quarter 15 m. Moreover, the common Mariners do divide every point of the Compass into four quarters to make the more exact account of their Routs or Rombes. But now as touching the other essential part of the Mariner's Compass which is the wyars, you shall understand that they are of iron or steel, and made in this form, and being touched at the one end with the loadstone, they are fastened to the back side of the Fly, either right under the line marked with the Flower-de-luce, or else somewhat distant from the same, either towards the East or else towards the West, for such cause as shall be declared in the next Chapter, for those wyars being thus touched, do make that part of the Fly that is marked with the Flower-de-luce, to stand always towards the North, and to that end the Fly having in the Centre a latin socket, is put into a turned box, in the midst whereof is a sharp pointed Latin pin, upon which the Fly turneth about, and that turned box is covered with glass, partly to keep the Fly clean, but chiefly that the wind should not move it to and fro. And this turned box is hanged by two round narrow plates of Latin in another square box of thin wainscot board, so as it may always hung level, howsoever the ship swayeth or inclineth on either side, and though of the Mariner's Compass there be divers uses, as you shall perceive hereafter, yet the chiefest end and use thereof is to show the North part of the world, whereby the shipmaster knowing what course he hath to hold and how the port or place to which he goeth, beareth from the place from which he departeth, may by looking always to his Compass, know how to direct his ship accordingly.   

Of the loadstone, and of the variation of the Compass in Northeasting and Northwesting. Chapter. 25  
THe loadstone or Adamant, called in Latin Magnes, & in Italian Calamita, hath two marvelous great and secret properteys or virtues, the one to draw steel or Iron unto it, and the other to show the North and South part of the world, of which stones some be of more force than others according to the place from whence they come. For those are counted best which are found in the East Indies upon the coast of China and Bengala, which is no shell but a whole stone of sanguine colour like to Iron, and is firm, massy, and heavy, and will draw or lift up the just weight of itself in Iron or steel. And as Norman saith in his new Attractive, such stones are commonly sold for their weight in Silver.  
Next to this, Norman commendeth the red stone of Arabia, he commendeth also the stone of Almain, which is in colour like to Iron, but spongeous, and thereby lighter than the other. And as for the black and white stone of Elba, which is an Island, not far from Piombino, he saith that the virtue thereof is but of small force, and of no long continuance. But the worst are those that come from Norway, whose colour is mixed with grey, I have heard that there be of them here in England, but how good I know not, it is marvel that Norman maketh no mention thereof, whose Treatise called the new Attractive, together with Master Borough his addition I would wish all that be studious in this Art to read most diligently, for truly in mine opinion the secrets of this Stone and variation of the Compass, was never better deciphered, nor by more experiences tried than it hath been by these two men last named. The book is most necessary for navigation and of an easy price, meet for every poor man's purse, though perhaps in some points not meet for every man's understanding. Now whensoever the wires are to be touched with this stone, they must be made very clean and voided of all rust, to the intent that the Iron may the more firmly receive the virtue of the stone. And it is well known by good experience, that by virtue of the loadstone the North point of the Compass declineth always from the true North, either to the East, or West more or less according to the latitude of the place wherein you are unless you be right under the Meridian of the Azores. And therefore, most men in these parts of the world do use to set the North point of the wires not right under the Flower de Luce signifying the North part of the world, but rather somewhat inclining toward the East half a point or thereabout to avoid the Northeasting and Northwesting of the Compass, the cause of which declination by divers learned men hath been diversly taught, but not rightly as Michael Cogniet saith, until Mercator found out the true cause, who first learned by the experience of one Francis of Deep, an excellent Pilot, that in sailing under the Meridian which passeth through the Isles called the Azores, the needle doth decline neither East nor West, whereupon Mercator by calculating the variation of the Compass at Ratisbone, found that the Pole of the loadstone aught to be put in that Meridian which passeth through the foresaid Isles so as it may be distant from the Pole of the world sixteen degrees and ½ or rather as Master Borough saith, sixteen degrees and twenty two minutes, and by that calculation Cogniet hath found the variation of the Compass at the town of Antwerp to be 9 degrees. And Master Borough by the help of the new instrument of variation first made by Robert Norman, and afterward perfected by himself, hath found the variation for London to be 11. degrees and fifteen minutes, which is a whole point from North to East. But whensoever you departed from the foresaid Meridian of the Azores, be it never so little, either towards the East or West, the needle will vary and decline accordingly. And his greatest declination is when you come to a full quarter of that Parallel wherein you sail, for from thence it declineth less and less until you come again under the foresaid Meridian, which thing Cogniet doth plainly demonstrate by this Figure here made of purpose.  
In which Figure the letter A. which is in the very centre of the Circle representeth the north Pole, and the right line, marked with the letters C. B. signifieth the Meridian which passeth through the Isle's Azores, in which line is a point or prick marked with the letter D. signifying the Pole of the loadstone, distant from the North Pole sixteen degrees and a half. Now, if your ship be in the point B. then your needle declineth on neither side, but pointeth right to the North Pole, and also to the Pole of the loadstone. But if you sail Eastward and arrive to the point E. then the right line of the North is E. A. but your needle declineth on the right hand towards his own Pole D. which is to the East, so much as the angle A. E. D. doth show. Likewise if you sail from B. Westward to the point F. the right line of the North shall be F. A. but your needle will decline to his own Pole D. towards the West, so much as the angle A. F. D. doth show. And to be short, the needle doth never show the right North, but only in C. or B. to either of which points the nigher that you approach the less your needle declineth, and the more that you go from any of these two points the more your needle declineth either East or West, but the greatest declination thereof is in H. or G. for than you are a just quarter from the foresaid Meridian. Also by this Figure he plainly showeth that of two sundry towns having one self Meridian, that which is nearest to the Pole of the world hath greatest declination. As for example, suppose the letter I. in the foresaid Figure to signify that town which is nigher to the Pole, and the letter E. to signify that town which is more distant from the Pole, both towns being under one self Meridian E. A. Here you see that the Needle being in E. doth show the Pole of the loadstone by the line E. D. and being in I. it showeth the Pole of the loadstone by the line I D. Now according to the doctrine of Euclyde, the angle A. I D. is greater than the angle A. E. D. whereby it followeth that the needle declineth more in I. than in E. But whereas Mercator affirmeth that there should be a mine or great rock of Adamant, whereunto all other lesser rocks or needles touched with the loadstone do incline as to their chief fountain, that opinion seemeth to me very strange, for truly I rather believe with Robert Norman, that the properties of the stone, as well in drawing steel, as in showing the north Pole, are secret virtues given of God to that stone for man's necessary use and behoof, of which secret virtues no man is able to show the true cause.   

How to found out the variation of the compass in every latitude. Chapter. 26.  
WIlliam Boorne in his Regiment of the Sea teacheth divers ways how to find out the variation of the Compass, as well by the Sun in the day time, as by the North star in the night season. First, thus mark at what point of the Compass the Sun riseth and setteth, for if he riseth at the East point of the Compass, and goeth down at the West Northwest point, than the Compass is varied one whole point, that is to say, the North point of the Compass standeth North and by East. But because the air is seldom clear at the rising and setting of the Sun, you may do thus, take with your Astrolabe or cross staff the altitude of the Sun in the forenoon, the sooner the better, at some just point of the compass, & take again his altitude in the afternoon, when he is in like degree of altitude, and mark therewith at what point of the Compass the Sun hath such altitude, and by the difference thereof you shall know the variation of the Compass. As for example, you find by your Astrolabe or cross staff, that the Sun is 20. degrees high at the Southeast point of the Compass, and observing the same again in the afternoon you find his height to be 20. degrees at the West Southwest point of the Compass, whereby you see that the compass is varied one whole point, that is to say, that the North point standeth North and by East, and the South point is South and by West. He teacheth it also another way, which is thus.  
Take with your cross staff or Astrolabe the height of the Sun at noontyde, which is called the Meridian altitude of the Sun, and thereby you shall have the true Meridian of the place where you are, with which Meridian if the South point of your Compass doth agree, than your Compass hath no variation at all, but if the south point thereof do serve or incline on either side from the said Meridian, mark how much it differeth, and that difference will show you the variation. Now to know the variation of the Compass by the North star, do thus, set your Compass with the North star, and if you find them to agree, than there is no variation, but note that this is to be done when the two guards or pointers of Charles wain are right over or right under the North star, for if these two stars be West from the North star, than the North star is a third part of a point unto the Eastward of the North Pole, and if the said two stars be right East from the North star, than the North star is a third part of a point unto the Westward of the north Pole.  
The compass (as Bourne saith) doth vary most in sailing long voyages East and West, and though it varieth two or three points, yet you may know what course to hold without alteration of the Wires any manner of way. As for example, suppose the North-east point to stand right North, and your course is to go right West, here in this case you may use the South-west point in stead of the West point, whereby you may perceive that it maketh no great matter which point standeth due North, so as you take that point of the Compass for the North which directly pointeth to the North. But Robert Norman and Master Borough by the help of their new invented Instrument of variation, do show how to find out the variation of the Compass much more exactly than ever it hath been heretofore taught, which instrument together with the book, I would wish all sea men to have, and thereby to learn the perfect use of the instrument, which use that book teacheth both plainly and learnedly, in which book is also proved by divers demonstrations that there is no such attractive point as some have dreamt, but rather a respective point whereunto the needle of your Compass will always turn in what part of the world soever you sail. But of the place where that respective point should be, divers learned Pilottes have had divers opinions, for some have imagined it to be in the heavens, and some above the heavens, if it were in the heavens, than the needle would daily turn about, and altar according to the motion of that heaven, wherein the said point is, which is nothing so. And to be above the heavens, it is contrary to the old rule of Philosophy, which saith that Extra Coelum non est locus, and therefore Norman and Master Borough have great reason to say that it is in the body of the earth beneath the Horizon, for they have tried by divers lively experiments that the North part of the needle of his own accord and nature would always decline downward if it be not otherwise counterpoised or letted, & by their demonstrations supposing the Meridian of the Azores to be the first or common Meridian, and also knowing the altitude of the Pole at London to be 51. degrees, 32. minutes, they found that according to that latitude the Pole of the loadstone, being in the upper face of the earth, and right under the foresaid common Meridian is 25. degrees 44. minutes distant from the Pole of the world. And the point respective to be distant in a right line from under the Horizon of London 71. degrees, 50 minutes and the variation of the Compass to be as hath been said before, 11. degrees and 15. minutes, but to found out the true place of the respective point in every latitude, they say that no certain rules can as yet be set down, by reason that the compass doth vary more suddenly in one place than in another, for in some place it will vary more in sailing 200. leagues than in another place in sailing 400. leagues. Again, it will sometimes be retrograde, for M. Borough in sailing betwixt the North Cape and Vaigates, towards the North-east, and looking by his computation that the needle should have increased his variation towards the East, he found that it was suddenly turned backward toward the West, notwithstanding both Norman and M. Borough do affirm that though the Compass hath in several Orisons several variations, yet in any one Horizon the needle always respecteth one only point without alteration, and that in his declining it keepeth the like order and that certainly in every place. And although the needle of the Compass by reason of the weightiness of the Fly cannot decline downward according to his own property, but only showeth the point respective always upon the Horizon, which indeed, as they say, is most necessary for navigation, yet by such means and conclusions as are set down in the foresaid book, a diligent Pilot having with him a perfect stone may by exact observations found out the increasing or decreasing of the declnation of the needle, which declination you shall found, as they think to be more or less according as the point respective is more or less distant from the place whereas the trial is made, which being diligently observed in sundry places, with the certain variation of the needle from the Meridian, the true place of the point respective may be found out as they think.   

Of the Mariner's Card, and of the making thereof. Chapter. 27.  
THe Mariner's Card, which some call a nautical Planisphear, is none other thing, but a description made in plano upon paper or parchment of the places that be in the sea, or on the land next adjoining to the sea, as points, Capes, Bays, Portes, Floods, Islands, Rocks, Sands, and such like, And such Cards are either universal or particular, the universal Cards are those wherein are described the most parts of the world, such as is the universal Card or Map of Mercator, or of Plancius. The particular are those wherein some special parts of the sea and land are described. And both these kinds of Cards are traced with certain lines, whereof some are called Meridian's, some Parallels, and some the Lines of the Mariner's compass, showing the 32. winds before described in the beginning of this Treatise. The order of making which cards in times past was wont to be thus.  
First draw with a pair of compasses a secret circle that may be put out, so great as you shall think meet for your card, which circle shall signify the Horizon, then divide that circle into four equal quarters, by drawing two Diameters crossing one another, in the centre of the foresaid circle with right angles, whereof the perpendicular line is the line of North and South, and the other crossing the same is the line of East and West, at the four ends of which cross Diameters you must set down the four principal winds, that is, East, West, North and South, marking the North part with a flower deluce in the top, and the East part with a cross, as you may see in the figure following. Than divide every quarter of the said circle with your compasses into two equal parts, setting down pricks in the midst of every quarter, through which pricks, and also through the centre of the circle, draw two other cross lines, which must extend somewhat beyond the circumference of the Horizon, which two cross lines together with the first two cross lines shall divide the circle into 8. parts, and thereby you shall have the eight principal Winds. That done, divide every eight part of the said Horizon into two equal parts by drawing other two cross lines through the centre, and extending somewhat beyond the circumference of the Horizon, as before▪ whereby the whole circle shall be divided into 16. parts, which shall suffice without making any more divisions, which would 'cause a confusion of lines, & at the end of every one of these 16. lines you must draw a little circle, whose centre must stand upon the circumference of the Horizon, every one whereof must be also divided into 16. parts by the help of 16. lines, diversly drawn from the centre of one little circle to another, in such order as the figure here placed more plainly showeth to the eye, than I can express the same by mouth. And these little circles do signify 16 little Mariners compasses, the lines whereof signifying the winds, do show how one place beareth from another, and by what wind the ship hath to sail. But besides those little circles there is wont to be drawn also another circle somewhat greater than the rest upon the very centre of the Horizon, which Circle by reason of the 16. lines that were first drawn passing through the same, is divided into 16. parts, and the Mariners do call this circle the mother compass.   mariners 'card  

The shape and figure of the first lineaments of the Mariner's Card drawn after the old manner, and how to set down the places of the land or sea therein. Chapter. 28.  
NOw the true setting down of the places in the Mariner's card, as Points, Capes, Bays, Floods, islands, and such like is to be done by knowing what latitude and longitude every place hath, which is to be learned either by modern tables made of purpose: for the ancient Tables make no mention of any longitudes or latitudes of such places as are in the new found land, which land to them was never known, or else by such Mariners cards already made as do show the true longitudes and latitudes of those places which you would describe in your Card, of which longitudes and latitudes, and especially the longitudes of the places in the West Indies, few or none are as yet truly set down,  
Moreover, to know the distance of places, that is, how many leagues or miles one place is distant from another there is wont to be set down in the Mariner's Card a scale, otherwise called by the Mariners a Trunk, the making whereof is plainly taught by Martin Cortes in his art of Navigation, in the second chapter of his third book, and also how to graduate the Cards, to show what latitude every place hath, and there also he teacheth how to translate one Card into another, and how to reduce a greater Card into a lesser, and contrariwise. To which book I refer you, and the rather for that it is in English, translated many years since out of Spanish by M. Richard Eden.  
But for so much as the sea and the earth do make together one whole round body, the lines of the 32. rombes in the Card being drawn right, and made to signify great circles, can never show the true course that the ship hath to hold, which Michael Cogniet proveth by a Figure demonstrative, and thereof gathereth three conclusions. First, that a man may sail right north and South round about the world, if the Sea in all that course be navigable, and so return again to the port from whence he first departed. secondly, that making the Equinoctial his Parallel he may sail East and West round about the world, and so return to the port from whence he departed, Also if he sail East and West in any other Parallel that is distant from the Equinoctial, he may return by the same Parallel to the port from whence he first departed, and yet not about the whole world, for that cannot be done, but only when the Equinoctial is his Parallel, thirdly, that whosoever saileth by any other rombe than by one of the four principal, he by often changing his Meridian and Horizon must needs sail by a line spiral, which is neither perfectly right, nor perfectly round, and thereby he may well approach the Pole, and also go round about it, but yet with unequal distance, so as he shall be nigher beyond it than on this side, by means whereof he cannot return to the place from whence he came, as you may plainly perceive by this Figure demonstrative here placed.  
In which Figure the letter A. doth signify the North Pole, and the letters B. C. the Meridian passing through the Pole A. then suppose your ship to be in Q. whereas the Pole is elevated 30. degrees. and Q. to be your Zenith, and the right line G. Q. D to be your right line of East and West cutting the foresaid Meridian with right angles, and let D. be the East point, and E. the West point.  
Now you may sail from Q. towards the North with a South wind, and from A. you may sail again southward with a North wind until you come to the South Pole, and from thence you may sail again Northward with a South wind until you come again to Q from whence you first departed, and so you shall have gone round about the world under one self Meridian B. A. C. But in sailing East or West, you shall continually change your Meridian, and thereby change your East point, for in sailing from Q. towards D. you come immediately to the Meridian A. F. whose right East point is G. and in sailing further Eastward, you come to the Meridian A. H. whose right East point is I. and so forth from one Meridian to another, notwithstanding in keeping still in one self parallel marked with K. L. M. you may sail round about the Pole A. which is the centre, and so come again to the Port Q. from whence you first departed, but not about the whole world: for that you cannot do unless the Equinoctial were your Parallel, as hath been said before, but if you sail by any other rombe than by one of the four principal, that is, East, West, North, or South, your course shall neither be by right line, nor yet by true circle, but by a spiral line, which is partly right, and partly round, so as you cannot with like course return to the place from which you departed, by reason that you change so often both Meridian and Horizon, as by the spiral line Q. N. O. drawn in this Figure you may easily perceive, which approacheth nigher to the Pole beyond it than on this side. Moreover the sea men by making the Meridian's and parallels in their cards all of equal distance, they make some countries far greater than they should be by the one half. Also by that means he that saileth East and West round about the Pole in the Parallel whose latitude is 60. degrees, should make as long a voyage as he that saileth East and West alongst the Equinoctial, the voyage whereof is twice as long as the other, for the redress and remedy of which faults, Cogniet hopeth to find out some more perfect rule of making cards when opportunity of time shall serve: in the mean time to reform the said faults, Mercator hath in his universal card or Map made the spaces of the Parallels of latitude to be wider every one than other from the Equinoctial towards either of the Poles, by what rule I know not, unless it be by such a Table, as my friend M. Wright of Caius college in Cambridge at my request sent me (I thank him) not long since for that purpose, which Table with his consent, I have here plainly set down together with the use thereof as followeth.  
The Table followeth on the other side of the leaf.   

A Table to draw there by the parallels in the Mariner's Card together with the use thereof in truer sort than they have been drawn heretofore, and the use thereof. Chapter. 29.  
Degrees of the Meridian, beginning from the Equinoctial circle. Equal parts of the Meridian in the Mariner's Card▪ of which parts every degree of the Equinoctial containeth 60 miles. 1 60 2 120 3 180 4 240 5 300 6 361 7 421 8 482 9 542 10 603 11 664 12 725 13 787 14 849 15 911 16 973 17 1035 18 1098 19 1162 20 1225 21 1289 22 1354 23 1419 24 1484 25 1551 26 1617. 27 1684 28 1752 29 1820 30 1889 31 1959 32 2029 33 2100 34 2173 35 2245 36 2319 37 2394 38 2470 39 3546 40 2624 41 2703 42 2783 43 2865 44 2948 45 3032 46 3118 47 3205 48 3294 49 3385 50 3477 51 3572 52 3668 53 3767 54 3868 55 3972 56 4078 57 4187 58 4299 59 4414 60 4532 61 4655 62 4781 63 4911 64 5046 65 5189 66 5331 67 5482 68 5639 69 5804 70 5976 71 6156 72 6346 73 6547 74 6759 75 6985 76 6226 77 7484 78 7764 79 8067 80 8399  
THe use of this Table for making the sea Card is thus. Overthwart the midst of the plain superficies wherein you would draw the lineaments of the Card, describe a right line marked with the letters A. B. C. whereof B. is the very midst or centre, and this line representeth the Equinoctial, the end whereof on the right hand, marked with the letter C. signifieth the East, and the other end on the left hand marked with the letter A. signifieth the West, which Equinoctial line must be divided into 36. degrees, then cross the same squirewise with perpendicular lines, passing through every tenth or fift degree, as you see in the example following. Than take with your compasses the length of half the Equinoctial, that is, 180. degrees, and set one foot of your Compasses in the mutual intersection of the Equinoctial, and of that perpendicular or Meridian which passeth through the East end of the Equinoctial line, marked with the letter C. and with the other foot make a prick in the same perpendicular or Meridian, and mark that prick with the letter D. that done, divide the space contained betwixt this prick and the Equinoctial first into three equal parts, and every one of those into other three equal parts, so have you 9 parts. And again, every one of those into three, so have you twenty seven parts, and divide every one of those parts into four parts, so shall you have a hundred and eight parts, and if there be space enough, divide again every one of those into ten, so shall you have a 1080. parts, and if it be possible, divide again every one of those parts into other ten, so shall you have in all 10/800. parts, but this can hardly be done, unless the Card be very large, wherein every degree of the Equinoctial is near an inch long, which happeneth very seldom, and therefore 1080. parts shall suffice.  
And for the easier numbering of these parts, set to them Arithmetical figures with black lead, which may afterward be put out when your work is done, beginning at the Equinoctial and so proceed from thence both Northward and southward, then look what number standeth right against every degree in the former Table, which degrees do extend from 1. to fourscore degrees, and omitting always the first figure on the right hand of that number which you find, for so you must always do when your division containeth no more but 1080. parts, count that number which remaineth upon the line of division, and there make one prick, and make another prick of the same distance from the Equinoctial upon the outermost Meridian on the left hand, through which two pricks draw a right line, and that shall be your first Parallel of latitude, and so proceed with all the rest first Northward, and then Southward, if you mind to make an universal Card. But the example following containeth no more but the one half of an universal Card proceeding from the Equinoctial line Northward unto the 80. degrees of latitude: for further Northward than 80. degrees is no land as yet discovered or known, nor yet all that, so far as ever I could learn.   

The draft of the Meridian's and parallels of the Mariner's Card or nautical Planispheare according to the former Table.  
IN this last figure or Table is first drawn (as you see) the Equinoctial line marked with the letters A. B. C. and that line is divided into 360. degrees, and therein also are drawn perpendicular lines, as well through the beginning and ending of the said Equinoctial line, as also through every tenth degree thereof, which be the Meridian's, and are every where equidistant each one from other, then take half the length of the Equinoctial which is A. B. or B. C. with your Compasses and setting one foot in the end of the Equinoctial marked with C. make with the other foot a prick at D. in the Meridian or perpendicular line marked with the letters C. D. E. then divide the space contained betwixt C. & D. into 1080. parts in such sort as before hath been showed, and set the figures unto them as here you see, to the intent that you may the more readily number the parts. Than look in the first Table what number answereth to every 10. degree of the Equinoctial, & casting away the first figure of that number on the right hand, found out the parts answerable to the number remaining in the line C. D. and at those parts set pricks in both the outermost Meridian's through which pricks you shall draw the Parallels. As for example in the first Table you see that the number right against 10. degrees is 60. (the first figure 3. towards the right hand being rejected) therefore look 60. in the line C. D. and by that part draw the first Parallel distant 10. degrees from the Equinoctial. And after this manner all the rest of the Parallels are to be drawn.  
Many do use to paint the vacant places in their Cards with over many flags, and the compasses thereof with diverse and superfluous colours which William Borne misliketh, wishing that in stead thereof they would show by letters or other Characters what moon doth make a full sea, in such places as are necessary to be known, and also to draw the true shape and fashion of every Cape or headlande that is needful about the coast, and at what point of the compass the land riseth of this or that fashion, for being near the land it will seem to be of one fashion, & being far off to be of an other fashion, and to mistake any place on the sea is very dangerous to the Mariner. But above all things let him that saileth by Card & Compass be sure that the needle of his compass have the like declination that his needle had which made the Card. For Cogniet reporteth that certain Mariners being in the west Indies, and seeing the North star to be North-east, marveled thereat very much not knowing the cause of that error, which indeed was for that the needle of their compass was made to decline North-east, whereas it should rather have declined Northwest, for the West Indieses do stand Westernly from the Azores. It is necessary also to set down in the Card such places as are dangerous, as sands, flats, or shouldst, rocks, and such like things as are not always to be seen with the eye, to the intent that the Mariner being advertised thereof may shun the same. All which things before mentioned, Wagoner hath in all places contained in his book very well observed. Thus having spoken sufficiently of making the Mariner's Card, Now I think it good to show the most necessary and chiefest uses thereof.   

The chiefest uses of the Mariner's Card. Chap. 31.  
THe chiefest uses of the sea Card are these four here following:  
The first is to know thereby how that place whereunto you would sail, beareth from the place or Port from whence you set off, or departed. And that is to be known by the lines of the Mariners Compasses painted in the Card in this manner following: Take a pair of Compasses and having opened them, set the one foot thereof in the very place from whence you departed, and the other foot in the next line of that Compass which is nearest unto your place of departing, I mean such line as doth most rightly direct you to the place to which you would go, and your Compass being opened at the fit wideness to serve that line, draw them from the place of your departing unto the place whereto you would go, suffering that foot of the Compass which standeth upon the line of the wind whilst you draw it forward, not to swerver one jot from that line, and that line will either rightly direct you to the place assigned or fall short thereof, or else overrech the same, if it fall short, then take another line nearer to the place from which you departed, and if it overreach, take some line that is further off from the place of your departing, and having found a line that pointeth directly to the place, consider what wind or rombe it is, for by that wind the place assigned beareth from you, and the rombe or wind opposite to that is the wind whereby you have to sail.  
The second use is to know by the Card how far the place whereto you go is distant from the place of your departing, which is done by help of the scale or trunk set down in the Card thus: Take the just distance betwixt the two places with your compasses by setting the one foot in the one place, and the other foot in the other place, and apply that wideness of the compasses to the scale or trunk, and the trunk will show how many leagues the one place is distant from the other, and if the distance betwixt the places be longer than the trunk, then take first the length of the trunk with your Compasses, and look how many times that is contained in the space betwixt the two places, and if there do remain any odd measure, then having taken that odd measure with your compasses by setting them at such wideness as is answerable to that odd measure, apply that wideness to the first part of the trunk, so shall you know the just measure of the whole. And this rule serveth to take the true distance of any other two places whatsoever set down in the Card.  
The third use is to know by the Card what Latitude or altitude of the Pole any place set down in the Card hath, which is done by help of the line of degrees of Latitude, otherwise called the Graduation of the Card in this manner following. Set the one foot of your compasses in the very place whereof you would know the Latitude, and the other foot in the line of East & West, which is next unto that place, and keeping that foot still upon that line, draw your compasses forward until you come to the line of degrees, and mark what degree of the said line the foot of the compass which was first set in the place doth cross or touch, for that is the degree of Latitude for that place, numbering from the lowest degree of Graduation upward, so shall you find the Latitude of Lisbon in Portugal by Mercator his universal Card, and by the Card set down in Martin Cortes book, and also by Medina his Card drawn in his book of Navigation to be 38. degrees 30. minutes and somewhat more. But by the Tables of Ptolemy you shall find it to be 40. degrees 24. minutes, and by Appian his Tables to be 39 degrees and 38. minutes.  
The fourth use chanceth when you are driven out of your right course by storms or tempest, which storms how to foresee and to prognosticate, is plainly taught by Martin Cortes in the 19 Chap. of his second book. Also you may be driven by force of contrary winds, by surging of the sea, or by overthwart tides, currants, & such like impediments, so as you can not lay your course right to the place assigned, for remedy whereof you must seek in what place you are, & to note the same in your Card which as the Mariners term it, is to make a prick in the Card, which to do truly in time of need many things are to be known & well observed and kept in memory. First to know what latitude the place from whence you first departed hath, then to keep in mind what way your ship did make good at every shift of wind, that is to say, how many leagues, and in how long time you sailed by every several wind: and then not knowing well where you are, nor how far you are distant from the place whereto you would go, learn to know by help of your Astrolabe or cross staff, in such sort as is before taught, the altitude of the Pole in that place where you are, which if you find to be all one with the Latitude of the place of your departure, than you may assure yourself that you have sailed by the line of East and West without altering your Latitude, but if you find the Latitude of the place where you are, to be more or less than the Latitude of the place from whence you departed, then resort to your Card, and take two pair of Compasses opened at such wideness as the one foot of the one Compass may stand in the place from which you departed, and the other foot of the same Compass to stand in the line of the rombe whereby you sailed: and let the one foot of the other Compass stand in that degree of Latitude which you last found, and the other foot of the same Compass in the next line of East & West, and holding the Compasses so ordered in each hand one pair: draw them both so as they may meet together, taking good heed in drawing them, that the foot of that compass which was placed in the line of the wind, may at no time serve from that line, nor the one foot of the other Compass to serve from the line of East and West, wherein it was first placed, and whereas the two feet of those Compasses do meet, that is to say, that foot of the one compass which was drawn from the place of your departing do meet with that foot of the other compass which came from the degree of Latitude last found, where these two feet (I say) do meet, there make a prick or mark in your Card, for that is the place where you ship is at that instant: And from thence you must take your right course again to the place whereunto you would go. But because it is necessary aswell at this time as at all other times, to know what way your ship hath made, and that the same is not in mine opinion, so plainly nor so commodiously taught by any one that I have read, as by Michael Cogniet, I mind therefore in the two chapters next following to set down his way not only how to found out the way of your ship when you sail South and North under one self Meridian, or in any other place where you are to change in your gate the latitude or altitude of the Pole, but also how to find out the way of your ship in sailing right East and West, without changing the altitude of the Pole, which way as he saith, was never heretofore known to any Pilot but to himself first author and inventor thereof.   

How to know the way of your ship, and how many leagues are to be counted for one degree of Latitude in every Rombe whereby you sail. Chap. 32.  
Than suppose C. D. to be another Parallel equally distant from A. B. by one degree of altitude. Now if you sail right East or West, than you shall always remain in the Parallel A. B. equally distant from the Pole. But if you sail from A. right North so far as the altitude of the Pole is augmented one degree, than your ship shall be in C. and if you sail by the first rombe towards the East so far as the Pole doth altar in altitude one degree, than your ship shall be in E. and thereby your way must needs, be longer, and so consequently the more rombes you decline from North to East, the longer is your way, and the more leagues must be accounted to one degree of altitude, as the lines drawn from A. to E. F. G. H. I. K. and D. do show. And what same ever is said here of the quarter from North to East, the same is to be understood in all the other three quarters, that is, from North to West, from South to East, and from South to West. But if you will know how many leagues do belong to every degree according to the rombe whereby you sail, then consider well this Table here following.  

The first Table.  Rombes.  Leagues. 1  17 ⅚ 2  18 14/15 3  21 ½0 4  24 ¾ 5  31 ½ 6  45 ¾ 7  89 ⅔  
For in sailing from North or South towards East or West so far as you come to change one degree of altitude of the Pole, the said degree doth require for the first rombe, etc.  
Now then to know how much way you have made in sailing, you must first know aswell the Latitude of the place from which you departed as of that place whereunto you be arrived: then by the foresaid Table seek to know how many leagues do belong to a degree of that rombe whereby you have sailed, for in multiplying the number of the leagues by the degrees of the difference of the two Latitudes, the product thereof will show you how many leagues you have sailed, notwithstanding sith the way may be made longer or shorter by changing or shifting of the wind, it is needful that the Pilot have consideration thereof, who by skilful conjecture must sometime either add to, or take fro according as need shall require. Moreover by the foresaid figure marked with letters, you may also easily understand how much you change in Longitude, that is to say, how much you are distant from the Meridian of that place from whence you departed, be it either towards the East or West, for he that saileth from the point A. as is aforesaid right North or South, he remaineth always under one self Meridian: but he that saileth by the first rombe towards the East or West so far as he changeth one degree of Altitude, and arriveth to the point E. is now distant from his first Meridian so much as is the space betwixt C. and E. which we find by computation to be three leagues and a half for one degree of Latitude, which amounteth to twelve minutes of a degree, and so of the rest of the rombes, as appeareth by this Table here following.  

The second Table.  Rombes' leagues Degrees and minutes or Longitude. first 3 ½ 0 12 second 7 ¼ 0 25 third 11 ⅔ 0 40 fourth 17 ½ 1 0 fift 26 ⅕ 1 30 sixth 42 ¼ 2 25 seventh 880 5 2  
For in sailing from North or south towards East or West so far as you change one degree of altitude of the Pole you change also your Meridian, and thereby your Longitude, the quantity whereof answerable to every rombe, is set down on the right side of this Table.  
And to make this more plain by example suppose that you sail from Lisbon, which is a famous port in Portugal, by the wind Southwest and by west, which is the fift rombe from south to West, so far as you find the altitude of the Pole to be 18. degrees less than at Lisbon. Now if you would know how many leagues you have sailed, and also how much the Meridian of that place is more westward than the Meridian of Lisbon, then do thus. Look in the first Table, and you shall find that to one degree of the fift Rombe do belong 31. leagues and a half, which leagues being multiplied by 18. do make in all 582. leagues and ¾. which you have sailed. then look in the second Table, and you shall find for the fifth rombe one degree and 30. minutes of Longitude, which being multiplied by 18. and a half, do make 27. degrees and ¾. of a degree, for by so much is Lisbon more Eastward, than the place where you are. And whereas the first Table is made according to the proportion of right lines such as are commonly drawn in Mariners Cards, Cogniet maketh another Table according to the proportion of circular lines, which for that it differeth very little or nothing from the first Table, I omit here to set it down. But now because the first Table doth chiefly serve those that fail either East or West in any Parallel betwixt the Equinoctial and the 60. degree of Latitude or Altitude of the Pole: And that from thence forth by reason that they sail by more obliqne and spiral Circles do make the longer voyage, Cogniet thought good to add a third Table showing how many leagues be answerable to one degree of altitude to those that sail either East or West in any Parallel that is betwixt the Pole and the 60. degree of altitude, which Table differeth not much from the others in the four first rombs, but in the three last, that is in the 5. 6. and seventh rombe it differeth greatly, and most in the seventh as you may easily perceive by comparing this and the first Table together.  

The third Table.   Deg M  Leagues. The first rombe hath 1 1 Which according to the proportion of 17. leagues & a half, for one degree do make for every rombe so many leagues as this Table showeth. 17 19/24 the second rombe 1 5 18 23/24 the third rombe 1 12 21 0 the fourth rombe 1 26 25 1/12 the fift rombe 1 50 32 3/12 the sixth rombe 2 42 47 ¾ the seventh rombe 5 44 100 3/3   

How to accounted the leagues in sailing directly East or West without changing the latitude or altitude of the pole. Chap. 33.  
IN sailing ritht East or West, you continued still in one self Parallel without making any charge of Latitude of the Pole. Most men therefore think it impossible for you to make any true account of the leagues but only by conjecture, for remedy whereof Cogniet hath invented a rule most certain, as he saith, the foundation whereof is thus: First you must suppose the Meridian of that place from whence you departed to be a firm and fixed point, and to be the very beginning of the Parallel wherein you sail, than you must know what hour it is aswell at the place from whence you first departed, as also at the place whereunto you are arrived, and having the difference of the hours, you must know how many leagues every hour yieldeth, according to the Parallel of that altitude of the Pole under which you sail, so shall you easily know how many leagues you have sailed. This foundation being first laid, Cogniet setteth down his general rule in this manner following: When soever you have to sail (saith he) right East or West, you must first provide yourself of these two things, the one is of an Astronomical ring that may justly show the hour at any time: In steed whereof I think it better to have an universal Dial, such a one as is described by William Borne in the one and twentieth Chapter of his Regiment: the other thing is to have a true hour glass that will run continually four and twenty hours, and therefore had need to be three times more long and large then common hour glasses be, whereof the glass makers can quickly provide you, and because the ship leaneth sometime on the one side, and sometime on the other, it shall be needful to hang the said hourglass with rings of brass or latten, to the intent that it may always hang level like as the Mariner's Compasses are wont to be hanged in their boxes. Now being thus provided of these two instruments, you must at your departing prepare your hour glass, that is to say, you must set it a running just at noon when the sun is right South, not forgetting to turn it in your voyage once a day in the very instant that it is ready to run out. Than having sailed certain days, and being arrived at the place where you would know what way you have made, look to your hourglass, and tarry until it be full run out, at which instant seek to know what hour it is by your Astronomical ring, or rather by the universal Dial, for if you sail East, you shall found it to be past noon, but if you sail West, than it will want of noon, and keep those hours in mind that you may know how much it is more or less than noon, for those hours by showing the difference of the hours that are betwixt the Meridian of the place from whence you departed, and the Meridian of the place whereto you are arrived, will certify you with the help of this Table here following, what quantity of way your ship hath made.   

A Table to help you to know what way your ship hath made in sailing right East or West without changing your Latitude, together with a brief description and use thereof.  
De. of Latit. leagues. 0 262 ½ 1 262 5/12 2 262 ¼ 3 262 ⅛ 4 262 5 261 ½ 6 261 7 260 ½ 8 259 ⅞ 9 259 10 258 ½ 11 257 ½ 12 256 ⅔ 13 255 ¾ 14 254 ⅔ 15 253 ½ 16 252 ¼ 17 251 ⅛ 18 249 ⅔ 19 248 ¼ 20 246 ¾ 21 245 22 243 ¼ 23 241 ½ 24 239 ¾ 25 238 26 236 27 234 28 231 ⅞ 29 229 ½ 30 227 ¼ 31 225 32 222 ½ 33 220 ¼ 34 217 ½ 35 215 36 212 ⅓ 37 209 ¾ 38 206 ⅞ 39 203 ⅚ 40 201 41 198 42 195 43 192 44 188 ¾ 45 185 ½ 46 182 ¼ 47 179 48 175 ½ 49 172 50 168 ½ 51 165 52 161 ½ 53 157 54 154 ¼ 55 150 ½ 56 146 ⅔ 57 143 58 139 59 135 60 131 61 127 62 123 63 119 64 115 65 111 66 106 ¾ 67 102 ½ 68 98 ¼ 69 94 ⅓ 70 89 ⅚ 71 85 ½ 72 81 73 76 ⅔ 74 72 ⅓ 75 68 76 63 ½ 77 59 78 54 ½ 79 50 80 45 ½ 81 41 82 36 ½ 83 32 84 27 ½ 85 22 ¾ 86 18 ⅜ 87 13 ¾ 88 9 ¼ 89 4 ⅔ 90   
This Table as you see consisteth of four collums, in every one whereof are set down on the left hand the degrees of altitude of the Pole, and next to that on the right hand the leagues answerable to every degree, the use whereof is thus: First seek out in the said Table the degree of altitude belonging to that Parallel under which you sailed, and next to that on the right hand you shall find the number of leagues incident to that degree: which number of leagues, if you multiply by the number of hours before found, the product thereof will show you how many leagues you have sailed. And if there be any minutes annexed to the hours, then multiply also the number of the foresaid leagues found in the Table, by those minutes, and divide the product thereof by 60. and the quotient shall be leagues, which you must add to the former leagues, and the sum thereof will show how many leagues you have in all sailed.   

An example of counting the way of your ship in sailing right west. Chap. 35.  
SVppose that you have to sail from the Cape Saint Vincent in Spain right West, and therefore you prepare your hour glass as it may begin to run just at noon, and having sailed 8. or 9 days, (not forgetting once every day to turn your hourglass) you do arrive at one of the Isles of the Azores called S. Marry, and there having tarried until your hourglass be clean run out, and seeking to know at that instant by your Astronomical ring, or by some universal Dial what hour it is, you find that it is a 11. of the clock and 10. minutes past, which wanteth 50. minutes of just noon, and that is the difference betwixt the two Meridian's, that is to say, the Meridian of Cape S. Vincent, and the Meridian of the isle S. Mary, both which places being in one self Parallel must needs have one self Latitude or Altitude, which is 37. degrees, which altitude being found in the Table in the second collum on the left hand, you find there also hard by it on the right hand, 209. leagues and ¾. of a league, for every hour of that Parallel, which number of leagues, if you multiply by the foresaid 50. minutes, the product shallbe 10/487. ½ which if you divide by 60. you shall find in the quotient 174. leagues and 19/●4. parts of a league, which is the just quantity of your whole voyage.   

Another example of counting the way of your ship in sailing right East. Chap. 36.  
SVppose then that you have to sail from the new found land right East in the Parallel of 50. degrees: and having caused your hour glass to begin to run just at noon, you set forth and sail the space of 15. days, not forgetting every day once to turn your hour glass. Now if at the 15. days end, you would know how many leagues you have sailed, to find in your Card in what place you are, you must first tarry until your hourglass be clean run out, and at that instant seek by your Astronomical ring, or by some universal Dial, to know what hour it is, which because you have sailed right East, you find to be two hours and twelve minutes afternoon, then resorting to the Table aforesaid, you find in the third colum the 50. degrees of altitude, together with the number 168. leagues and a half, annexed to the said degree for one hour of that Parallel, which sum being multiplied by two hours, maketh 337. leagues, then multiply once again 168. by the odd 12′·S and divide the product thereof by 60. so shall you found in the quotient 33. leagues and somewhat more, which being added to the former sum 337. leagues will make in all 370. leagues and a little more, and that is the true quantity of your voyage. And Cogniet saith that by practising this way of counting, you may know every day, yea every hour, what way your ship maketh in sailing right East or West.   

To know how much you go out of your way in sailing by one wrong rombe or by more. Chap. 37.  
IF in sailing to any place you direct your course either too high or too low by one rombe you lose in every 100 leagues 19 leagues and ⅗. of a league, so as you go out of your way almost the fift part of your voyage. And if by two rombes, then in every hundred leagues you lose 39 leagues, that is to say you go so far out of the right way: And if you fall three rombes out of your right course, then in every hundred leagues you lose 58. leagues: and if you fall four rombes, than you lose in every hundred leagues 76. leagues, and the more rombes that you mistake in your direction, the more you wander out of your right course. Thus much touching the Mariner's compass and his Card: now we will speak of the North star, and of his guards, and then of the sun and of the Moon, and so end.   

Of the North star, otherwise called the lodestar, and of his guards, and how to know the same. Chap. 38.  
THough every Mariner knoweth the North star so soon as he seeth it in the firmament, because it is his chief guide to direct thereby his ship in the night season in all the North parts of the world: yet every man that is no sailor knoweth him not, and therefore minding here to treat thereof, I think it not amiss first to teach him how to know it, and in what part of the heaven it is placed, and how far it is distant from the North Pole.  
The Poets do feign that of the 48. Images of the fixed stars that be in the heaven, which are otherwise called Constellations, there be two Bears having tails, whereof the one is called the great Bear, and the other the little Bear, in every of which are seven principal stars: And that bright star which is in the very tip of the little bears tail, is commonly called of our English Mariners the Lodestar, of the Latins Stella polaris, of the Greeks Cinosura, and of the Arabians Alrucuba. But to know this star, you must first found out the seven stars of the great Bear by looking towards the North part of the firmament, of which stars four being placed in his body, do make (as it were) a 4. square, and the other three stars being placed in his tail, which is somewhat rising in the midst, do represent the portion of a Circle, the little Bear hath also the like shape saving that her feet & belly do turn upwards, so as the tip of her tail is answerable to the two hindermost stars of the great Bear, as you may see in these figures following.  
Some do call the great bear great Charles Wain or Wagon, and the little Bear little Charles wain, because in each figure the four square stars do signify the four wheels, & the other three stars the 3. horses: Now to find out the Lodestar, imagine a right line to pass through the two hindermost stars otherwise called the guards or pointers of the great Bear, as you see here in this figure, and that line will rightly direct your eye to the Lodestar which standeth in the tip of the little bears tail, for there is no bright star that can be seen betwixt the two guards of the great Bear and the Lodestar. Notwithstanding some do affirm that there is another star nigher unto the Pole then the foresaid Lodestar, and that star is as they say no more but 50. minutes distant from the Pole: I have heard that there is a Priest and ginger with the King of Demnarke, who is wont to find out this star by the help of an instrument whereby he getteth first the sight of six other stars at one instant, which way I fear me is more troublesome than profitable, and if there were any such star indeed, and therewith so needful and meet to be used as is the Lodestar, I doubt not but that some of our learned Pilots would have found it out, and also have had some use thereof long ere this time: But leaving this matter, I will return to the description of our Mariners common Lodestar, and show of what bigness it is: also what Longitude, Latitude, declination, and motion it hath, and finally the chiefest and most necessary uses thereof.  
This star according to the prutenicall Tables is of the third bigness, and hath in Longitude 53. degrees 30. minutes, counting from the first star of the Ram's horn, whose place in these our days is in the 27. degree 30. minutes of Aries, and by this means the place or Longitude of the Lodestar is in the 21. degree of Gemini, & his Latitude, counting from the Ecliptic line towards the North Pole of the zodiac is 66. degrees 0. minutes, and his declination counting from the Equinoctial towards the North pole of the world is 86. so as he is in these days distant from the North pole 4. degrees, some say 4. degrees and 9 minutes, but Cogniet saith, that he is distant from the pole but 3. degrees and a half, who findeth by the Astronomical Tables that this star at the birth of Christ was 12. degrees 36. minutes distant from the Pole, and ever since hath gone decreasing, and shall decrease every day more than other, until it come to be but 26. minutes distant from the Pole, which is as nigh as it can approach to the Pole, for then his distance shall begin again to increase. This star maketh his daily revolution from East to West in 24. hours, as all other stars do by virtue of the first movable, but his circuit is so small, and his gate so slow, as in the space of 24. hours he goeth not much more than 24. degrees of the Equinoctial, and in his turning round about he maketh as it were a little Circle, in the midst whereof is the North pole itself, which is invisible & can not be seen.  
And though the image or constellation of the little Bear containeth many stars, yet the seven before mentioned and set forth in figure are most observed, of which seven stars in the little bear, two stars called the guards of the North star are to the Mariners most familiar, for divers respects hereafter declared. And of those guards the one is Northern & the other Southern, and both are said to be of the second bigness, & yet the Southern guard seemeth to the eye both lesser and darker than the other. Of which two stars I think it not amiss to set down here the Longitude, Latitude, and declination as I have done before of the Lodestar.  
The Longitude of the South guard is 100 degrees 30. minutes, whereby his place is in the ninth degree of Leo. And his latitude is 72. degrees 40. minutes, and his declination is 73. degrees, so as he is distant from the North pole 17. degrees.  
The longitude of the North guard is 109. degrees, 30. minutes, so as his place is in the 20. degree of Leo, and his latitude is 74. degrees, 50, minutes, and his declination is 76. degrees, so as his distance from the North Pole is 14. degrees. Thus having showed the greatness, the longitude, the true place, the latitude, and the declination aswell of the Lodestarre, as of his two guards, I think good now to set down the chiefest uses that sea men have of them.   

The uses of the North star, and of his guards. Chapter. 39  
THe first use is to know thereby the variation of the Mariner's Compass.  
The second is to know by his guards when the North star is above or beneath the Pole.  
The third is to know by the North star and his guards with the help of an instrument called a Nocturlabe the hour of the night.  
The fourth is to know the elevation of the Pole. And you have to note that the North star and his guards are always above our Horizon, and do never go down in any place whereas the North Pole hath any elevation, be it never so little. But besides these stars there be also in this our latitude divers other images of stars that are always above our Horizon, and do never go down, as the Dragon, the great Bear, the image of Cepheus, of Cassiopeia, the image of Auriga, having the Goat at his back which is a fair bright star of the first bigness and many others, which I leave to name, because I have heretofore described them all at large in my treatise of the Globes, by which stars by reason that with us they never go down, the latitude of any place and also the hour of the night may be known so well as by the Lodestarre.  
But now as touching them uses of the foresaid stars before mentioned, the first whereof showing how to found out by the Lodestarre the variation of the Compass, is taught before in the 25. Chapter, according to William Boorne his rule set down in his Regiment, and as for the other three uses they are plainly taught in the two chapters next following.   

To know by help of a little Table made according to the Mariners rule touching the 8. principal rombes, showing how and when the Lodestarre is either above or beneath the Pole, that you may know thereby the true altitude of the Pole in taking the height of the Lodestarre with your Astrolabe or cross staff. chap. 40. Chapter 40.  
TO know this you must always have due consideration of the two guards of the little Bear, for when those guards are just South-west from the Lodestarre, then is the Lodestarre at his highest in the very Meridian, and therefore it is right above the Pole, and when those guards be just North-east from the Lodestar than the Lodestarre is again in the Meridian at his lowest, and thereby right under the Pole, and in both places his distance from the Pole is but three degrees and a half, as Cogniet saith, according to which account he setteth down this Table following.  
This Table is divided into two collums, whereof that on the left hand containeth the eight principal rombes or winds, and that on the right hand containeth the degrees and minutes of distance of the Lodestarre from the Pole, being either above or beneath the Pole.  
If the guards be The rombes or winds.  The degrees and minutes of the declination of the Lodestarre from the Pole. 

West.  
Southwest  
South  
Southeast.   Than the Lodestar is 

1 ½  
3 ½  
3 0  
1 0   Above the Pole. 

East  
Northeast  
North  
Northwest   

1 ½  
3 ½  
3 0  
1 0   Beneath the Pole.  
This table differeth in one point from the Mariners common rule set down as well by Medina, and by Martin Cortes, as also by William Boorne touching the guards, for they all appoint but half a degree of declination of the North star from the Pole, the guards being either in the rombe Southeast, or Northwest, to both which rombes Cogniet appointeth one whole degreé of declination. The use of which table in seeking to know, the elevation of the Pole is thus. First, having taken with your Astrolabe the altitude of the Lodestarre above your Horizon, observe immediately in which of the 8. former rombes his guards be. For if they be in any of the 4. upper rombes of the Table, than you must subtract from the height of the Lodestarre taken with your Astrolabe, so much as is set down for that rombe in the collum on the right hand of the said table, and the remainder shall be the true altitude of the Pole, but if the guards be in any of the four neither rombes, than you must add so much to 〈◊〉 height of the Lodestar, and the sum thereof shall be the altitude of the Pole. But because this table serveth only for the 8. principal winds and for no more. Cogniet therefore setteth down the making of a twofold instrument, whereby you shall not only know (as he saith) how much the Lodestarre is above or beneath the Pole in every other rombe as well as in the 8. principal rombes, but also you shall know thereby the true hour of the night more exactly than the Nocturlabe of Munster or Appian doth show.   

How to make an Instrument which will show at any hour of the night how much the Lodestar is either above or beneath the Pole in every other rombe as well as in the 8. principal rombes, which are only contained in the former Table, and also the true hour of the night, the shape whereof followeth. Chapter 41.  
The shape or figure of the Rectifier of the North star.  
COgniet calleth this instrument Rectificatorium Stellae Polaris, that is to say, the Rectifier of the North star, joining thereunto a Nocturlabe differing nothing at all from the Nocturlabe of Munster and of others, but only in placing the 21. of October in stead of the 28. of the same month upon the line of North and South towards the handle of the instrument for such cause as is hereafter declared.  
The making of the rectifier of the Lodestarre, is thus: Upon a smooth piece of board of firm wood, or upon a piece of polished plate of brass or Latin being six or seven inches broad, having a handle, as you see in the former figure, draw a circle divided into 4. quarters by help of two cross Diameters cutting one another in the centre with right angles, the perpendicular whereof shall signify the Meridian line, that is to say, the line of North and South, at whose upper end set North, and at the neither end South, and the other overthwart Diameter shall be the line of East and West, having East marked on the right hand and West on the left hand, and in every quarter of the foresaid circle you may if you will place the like rombes that are in the Fly of the Mariner's Compass, as you may see done in the foresaid figure. Moreover, you must divide the upper quarter on the left hand into three equal parts, and having taken two of those parts with your Compasses, measuring from the North point down toward the West, and there make a prick, marking the same with a little black cross, and from that cross draw a right line that may pass through the Centre unto the circumference of the Circle, and there make another little black cross, and this line shall divide the circle into two equal parts, which line if you cross with another right line passing through the centre, and making thereby right angles, you shall divide the circle into other four quarters differing from the first four quarters, though not in quantity, yet in place, of which four last quarters you must divide that which is on the left hand into 90. equal degrees, beginning your account at the little black cross on the left hand, and so proceed downward towards the South point of the circle.  
Now to know how much the North star in every rombe mounteth or descendeth and how to place the same upon the instrument so as you may know how much to add to the altitude of the Lodestarre, or to subtract from the same, you must use this Table following, which Table consisteth The degrees and minutes of the declination of the loadstar from the pole. The degrees and minutes of the Quadrant, divided by diverse portions of numbers into 90 degrees.   Deg. Min.  ¼ 4 6  ½ 8 13  ¾ 12 22 1 0 16 36 1 ¼ 20 55 1 ½ 25 23 1 ¾ 30 0 2 0 34 51 2 ¼ 40 0 2 ½ 45 35 2 ¾ 51 48 3 0 59 0 3 ¼ 68 13 3 ½ 90 0 of two collums, whereof that on the left hand containeth the degrees and minutes of the declination of the Lodestarre, proceeding from one quarter of a degree to another, until you come to 3. deg. and a half, which is (as Cogniet saith) the greatest declination of the Lodestar, from the Pole: and the other collum on the right hand containeth the degrees and minutes of a quarter or quadrant, divided by divers and sundry proportions of numbers into 90. degrees which proportional degrees and minutes are to be reckoned in the first quadrant divided into 90 equal degrees, & next descending from the first cross on the left hand thus. First, beginning at the said cross, and descending downward, tell out 4. degrees, and 6. minutes, & right against that set down ¼. then tell in the same quadrant from the said cross 8. degrees, and 13. minutes, and right against that set down ½ and so proceed according as the Table directeth you, until you come to 90. against which you must set down 3. deg. & ½ And as this quarter is divided, so you may deal with the other 3. quarters, and in each of the 2 quarters which are beneath the 2 crosses forget not to writ this word Subtract, and in each of the 2. quarters above the 2. crosses writ this word Admetus, as you see in the former figure, Than there shall want nothing but only a ruler or index, which must be fastened to the centre of the instrument, so as it may be turned round about a pin having a round hole in it, through which you may see the Lodestar, and also at the same instant by lifting up and down the index you may see on the outside of the instrument the foreguard of the said star appearing even with the right edge of the index called Linea fiduciae, or the fiducial line, drawn from the centre of the instrument alongst the inward edge of the index, and so is the instrument for that purpose fully perfected. But if you would add thereunto a nocturlabe, than you must draw upon the centre thereof divers circles next to the space containing the winds: and the spaces betwixt those circles must be some wider, & some narrower For the first upper space must be narrow, containing the days of every month, and the next space somewhat wider, containing the number of those days set down in Arithmetical figures, and the third space wider than that, containing the names of the months than next to the lowest circle of the nethermost space place a little rundle fastened to the instrument with the foresaid hoslow pin, so as it may turn round about the saw. And this rundle must be divided into 24. hours, that is to say, 12 for the day, & 12. for the night, which rundle would be made with teeth whereof one must be longer than all his fellows signifying always the 12. hour of the night, which is always to be laid upon the day of the month wherein you seek to know the hour of the night. And remember in distributing the days of the month, that you always set the 21 day of October beneath towards the handle in the very line of North and South passing right through the midst of the handle, so shall the instrument show the hour of the night more truly than when the 28. day of October standeth beneath upon the line of South and North, as it doth in the common Nocturlabes, the makers whereof had respect to the Pole itself which is invisible, and not to the North star which is apparent to the eye, by means whereof the guards of the Lodestarre sometime do show sooner or latter by 7. degrees and 18. minutes, if you count from the Pole, which is almost half an hour difference, and for that cause also Cogniet maketh his account in his foresaid rectifier to begin at the little black cross, seven degrees more forward than it aught to do, if he should count from the Pole, and not from the North star.   

How to know by the foresaid twofold instrument as well the mounting and descending of the North star as the true hour of the night both at one instant and also the elevation of the Pole, Chapter 42.  
FIrst then having laid and stayed the great tooth of the movable rundle marked with 12. upon the day of the month wherein you seek, and holding the instrument by the handle with your one hand right before your face, leave not to put that hand forward from you, or to bring it backwards towards you, until you may see with the one eye, winking with the other, the North star through the hole of the pin, which is in the centre of the Instrument: and so soon as you see the North star, lift with your other hand the index up and down until you see also at that instant the North guard of the Lodestarre on the outside of the instrument appearing even with the fiducial line or inward edge of the said index. Than staying the index there, look upon what hour it falleth, for that shall be the hour of the night. And look also at that instant upon what degree of distance it falleth in the outermost border of the instrument wherein those degrees are set down together with these words add and subtract. For from the point marked with a little black cross nigh unto the Northwest descending through the neither moiety or half deal of the instrument until you come to the other black cross, placed nigh unto the Southeast, the North star is always above the Pole so much as the index showeth, which you must always subtract from the height of the North star to know the elevation of the Pole, having first taken with your Astrolabe the height of the North star. But if the index do fall upon any degree or part of a degree in the upper moiety or half deal of the instrument then to know the elevation of the Pole, you must add so much as the index showeth unto the altitude of the Lodestarre, as the words Add and Subtract written in the outermost border of the instrument do plainly show.  
But now if you would know at any hour of the day or night, in what rombe the foresaid guards be without seeing them and also how much the Lodestarre is declined from the Pole, you need do no more but to lay the great tooth of the movable rundle upon the day of the month, and then to bring the index unto the hour which you require, and the said index will show in the border of the instrument in what rombe the guards be, and how much the North star is above or beneath the Pole.  
I have found by often trial that this instrument will show the true hour of the night, and also in what rombe the guards be, and thirdly, how much the North star is at any time either above or beneath the Pole, and by adding the degree of distance found in the limb of the instrument, or by subtracting the same according to the rule before given from that altitude of the north star, which I have before taken with my Astrolabe, I have found at all times the true elevation of the Pole wheresoever I have made trial thereof. But sith other Stars may perhaps appear when the North Star with his guards shall be hidden, I would wish all careful Mariners to acquaint themselves with as many bright Stars as they can, and especially with those which do both rise and set, and also to learn by some Table the declination of every such Star and whether it be Southernlie or Northernlie, for by taking the Meridian altitude of any such star and by adding to, or by taking from the altitude thereof, his declination according as the same declination is either Northernlie or Southernlie (for if it be Northernlie, than you must subtract his declination, and if it be Southernlie, you must add the same to the altitude of the Star) you shall find thereby the altitude of the Equinoctial, which being taken out of 90. the remainder will show the elevation of the Pole. The meetest stars for this purpose in these our North parts of the world are these, Hircus the Goat, Canis minor, the little Dog, Canis maior, the great Dog, Dexter humerus Orionis, the right shoulder of Orion. Cingulum Orionis, the girdle of Orion. Cor Leonis, the lions hart. Bubulcus, the Bearward. Spica Virgins, the wheat ear in the hand of Virgo. Aquila volans, the flying Eagle. Caput Andromedae, the head of Andromeda. Ras Algol, the head of Medusa. Oculus Tauri, the bulls eye, and divers others,  
And in any case consider whether the declination of the star be greater or lesser than his Meridian altitude, for if his declination be greater than his meridian altitude, as of those stars which are nigh the Pole, than you must take his Meridian altitude with your Astrolabe at two several times, that is to say, when he is at his highest in the meridian, and also when he is at the lowest point of the same Meridian called the depression, and having added those Meridian altitudes together take half thereof, and that half shall be the elevation of the Pole, which way of finding out the elevation of the Pole is nothing meet for a Mariner that is under sail. But now to proceed according to Cogniet his direction in this matter, he saith, that if in sailing you approach so nigh unto the Equinoctial as the elevation of the Pole is not above 17. degrees, than the guards are not so easily seen, wherefore it shall be needful to take some other star whereby you may both know the hour of the night, and also how much the Lodestarre is either above or beneath the Pole, and Cogniet thinketh none of the stars more meet for that purpose than one of these two, that is, either the star called Caput Medusae, that is the head of Medusa called of the Arabians, Ras Algol, or else the star called Hircus, that is, the Goat, both which are fair bright stars and of the first bigness: and the cause why he appointeth the head of Medusa, is for that this star is directly opposite to the former guard, in such sort as this star is always above the Horizon, when both the guards are under the Horizon. Wherhfore having prepared your instrument, that is to say, having laid the longest tooth of the movable rundle upon the day of the month wherein you seek, and having found out thereby in such order as is before taught, the north star and also the star called the head of Medusa, both at one instant, mark where the index falleth, and immediately turn the index from that point to the point opposite, abating twelve hours, and so the Index shall show you three things at once. First, the rombe wherein the guards are at that present. secondly, the hour of the night. And thirdly how much the North star is above or beneath the Pole. But because this star is not known perhaps to all Mariners, Cogniet would have you to take the other bright Star called Hircus, which (as he saith) goeth 9 hours and ½ before the guards, in such sort, as when the former guard is East,, you shall find this Star, counting from the North star, to be Northwest, and almost 45. degrees distant from the Pole, with which star if you work, as you did before with the head of Medusa, saving that you shall not need to turn the index unto the opposite point, but only to rebate from the point on which it falleth 9 hours and ½ you shall know all the three things last mentioned. And this rule (as he saith) is so general as you may have your desire by working in like manner with any other star that is to you certainly known, & is at that time above the Horizon.   

What stars are observed by those that sail beyond the Equinoctial under the South Pole. Chapter. 43.  
THe ancient Astronomers, as Ptolomey, Timochares, Hypparchus and others did never describe any star to be more nigh unto the South pole, than that which is called Canopus, which is a fair bright star of the first bigness, and according to the Tables of Copernicus, is distant from the South Pole 38. degrees, and ¼. But those that have sailed in the South seas of later days, have found out other stars unknown to the ancient Astronomers, which are much nearer unto the said Pole. For Albericus Vesputius writeth of three Stars, making together a Triangle Orthogonall, that is to say, having one right angle, now called the southern Triangle, the middle star whereof is distant from the south Pole 9 degrees, ⅖.  
There be also lately found out divers images of other stars nigh unto the South pole, as that which is called Noah his Dove, or Pigeon, and another called Polophilax, made in the shape of a man, whose longitude and latitude hath not as yet been rightly set down by any that I have read.  
And he saith that these stars are neither any of those stars that are appointed to the twelve Signs in the zodiac, nor yet any of the 36. Images or constellations that be in heaven. Moreover he saith, that in taking the altitude of the great Star called the foot being in his right place, that is to say, when he is directly opposite to the head, and that you find his altitude to be 30. degrees, than you may assure yourself that you are right upon the Equinoctial. And if you find his altitude to be more than 30. degrees, than you are passed the Equinoctial towards the South Pole. But if you found it to be less than 30. degrees, than you are still on the North side of the Equinoctial.  
Besides the stars above mentioned, our Mariners in these North parts of the world are wont to observe divers other stars, to the number of 32. whose longitude and declination together with their bigness and also when they rise and set, and when they are mounted to the Meridian, that is to say, are just South, is plainly set forth by Tables collected of purpose out of the Astronomical Tables by William Boorne, which Tables you shall find in the 20. chapter of his book called the regiment of the sea, And Robert Norman doth also set down the like tables in his book called the Attractive, and therefore I think it superfluous to repeat the same again here, and specially sith I have described unto you all the stars that be in the firmament that were known to the ancient Astronomers, and have showed you how to found out by the Globe their longitudes, their latitudes, their declinations their greatness and all other accidents belonging to the Stars in my treatise of the Celestial Globe, which I wrote of purpose to further young sea men.  
The knowledge of the Stars serveth sea men chief to know thereby the latitude of any place, and also to know the hour of the night: And thirdly, to conicecture by their manner of rising and setting, and other their aspects what weather is like to follow, either foul or fair: the rules whereof to teach truly bebelongeth to Astronomers, yet many sea men by diligent observation do attain to right good judgement therein. Wherhfore leaving to speak any further of the Stars, I will now briefly speak of the Sun and of his motions, of his rising and setting in every latitude, and of his declination from the Equinoctial, and of other his like appearances. I say here briefly, because I have already spoken of him at large in the 1. part of my sphere whereas I treat of the zodiac.   

Of the Sun, and of his motion, and of the chiefest appearances belonging to him. Chapter 44.  
THe Sun according to the moving of the first movable which is from East to West maketh his daily revolution in 24▪ hours, as all other Stars do, but according to his own motion, which is from West to East in going through the twelve Signs of the zodiac he spendeth a whole year, for his daily moving upon his own centre, called the eccentric, because it is out of the centre of the world, is little more than 59 minutes, 8. seconds, making thereby the whole year to consist of 365. days, five hours 49. minutes, 8. seconds, 19 thirds, 37. fourth's, 24. fiftes, and this is called of Copernicus, the equal tropical year, which taketh his beginning from the first point of Aries, otherwise called the Vernal Equinoxe, into which point the Sun entereth not every year at one self day of the month, for sometime he entereth into that point the tenth day, and sometime the eleventh day of March, which day of his entering is always truly set down in every Ephemerideses, and because the beginning of the tropical year is so uncertain, the Astronomers do make their year called the syderal year, to begin at the former star of the Ram's horn, and thereby do make the year to consist of 365. days. 6 hours 9 minutes and 39 seconds, according to which year they always rectify or bring to equality, aswell the equal as the unequal tropical year, of both which I have spoken in the first part of my Sphere, the 38. and 39 chapters.  
Moreover, the Sun hath three motions, that is, slow, swift and mean. His slow motion is when he is in the point called Auge or Apogeon, which is a point imagined to be nigh unto the outermost edge of the circle which carrieth the body of the Sun called Deferens Solis, and is furthest distant from the centre of the world, which point in these our days is in the 9 degree of Cancer, or there abouts. And being in that point, he goeth little more than 57 minutes in 24. hours. Again, his swift motion is when he is in the opposite point to the Auge, called Perigeon, which point in these days is in the 9 degree of Capricorn, and being in this point, he goeth one whole degree and almost two minutes in 24 hours, which is almost five minutes more than he maketh in his slow motion. His mean motion is when he is in the midst betwixt the two foresaid points, whereas in 24. hours he goeth one whole degree & somewhat more than two minutes. And these three sundry motions do cause the Equinoctial points not to be of equal distance. For the Sun spendeth seven days and somewhat more in going from the Equinoxe of March, to the Equinoxe of September, than he doth in going from the Equinoxe of September to the Equinoxe of March. For if in this present year 1592. which is leap year, you count the days from the Vexnall Equinoxe, which is the eleventh of March, unto the Autumn Equinoxe, which is the 13. of September, you shall found the number of the days to be 186. and the other number from the 13. of September to the eleventh of March, to be but 180. which is less than the first number by 6 days, and if it were not leap year the difference would be seven days, because that February in the leap year hath 29. days.  
It is necessary that sea men have some understanding of the three foresaid motions to the intent that they may the better know the true place of the Sun and thereby his true declination. And note that no calculation of his declination can continued without error above twenty four years. For as often as the leap year cometh about, which is every four years, the Sun is upon the Equinoctial sooner by half an hour. But as for the true place of the Sun, and especially every day at noontide, the Ephemerideses doth most truly show, and having his place, you shall easily find his declination by this Table following, which will serve for these twenty years and more, the like whereof, together with the use of the same is set down in the first part of my Sphere the 13. chapter.   

A Table showing the declination of the Sun every day throughout the year, and the use thereof. Chapter 45.  
Degrees of ●he Signs. ♈ ♎  ♉ ♏  ♊ ♐ Degrees of the Signs D M S  D M S  D M S 1  0 23 53  11 50 6●  20 22 57  29 2  0 47 46  12 10 56  20 35 7  28 3  1 11 39  12 31 34  20 46 55  27 4  1 35 30  12 51 59  20 58 20  26 5  1 59 20  13 12 12  21 9 21  25 6  2 23 8  13 32 12  21 19 59  24 7  2 46 54  13 51 58  21 30 13  23 8  3 1● 37  14 11 30  21 40 3  22 9  3 34 18  14 30 48  21 49 29  21 10  3 57 54  14 49 51  21 58 29  20 11  4 21 28  15 8 40  22 7 6  19 12  4 44 57  15 27 13  22 15 17  18 13  5 8 22  15 45 30  22 23 3  17 14  5 31 42  16 3 32  22 30 24  16 15  5 54 57  16 21 17  22 37 19  15 16  6 18 6  16 38 44  22 43 48  14 17  6 41 9  16 55 55  22 49 50  13 18  7 4 6  17 12 48  22 55 27  12 19  7 26 57  17 29 23  23 0 38  11 20  7 49 40  17 45 40  23 5 22  10 21  8 12 16  18 1 39  23 9 39  9 22  8 34 45  18 17 18  23 13 29  8 23  8 57 5  18 32 37  23 16 53  7 24  9 16 16  18 47 38  23 19 50  6 25  9 41 19  19 2 18  33 22 19  5 26  10 3 12  19 16 37  23 24 22  4 27  10 24 56  19 30 36  23 25 57  3 28  10 46 30  19 44 14  23 27 5  2 29  11 7 53  19 57 30  23 27 46  1 30  11 29 5  20 10 25  23 28 9  0   ♍ ♓  ♌ ♒  ♋ ♑    
THe use of this Table is thus: If the sun be in any of the signs set down in the front of the Table, then seek his degree (first found by the Ephemerideses) in the left collum, the degrees whereof do descend from one to 30. and the square Angle answerable to the sign and degree will show his declination. But if the sun be in any of the signs that are in the foot of the Table, then seek his degree in the right collum, the degrees whereof do ascend, and the square Angle answerable to that sign and degree will show his declination.  
There be other things also meet for Seamen to know, touching the sun as these: First to know the four seasons of the year, secondly to know by his declination, the length of both the day and the night in every latitude, and how it doth increase and decrease: Item to know in what rombe or wind, and at what hour he riseth and setteth, and also his Meridian altitude, that is, when he is right South every day, to find out by the help of that, and by knowing his declination, the true Latitude of any place, & by his shadow to know the hour of the day, which are two chief points that the Mariner hath most need to know. Of all which things I mind here to treat both plainly and briefly.   

Of the four seasons of the year, that is, Spring time, Summer, fall of the leaf, called otherwise Autumn, and winter. Chap. 46.  
IN this our Clime the spring is said to begin when the Sun entereth into the first point of Aries, which is about the xi. of March, and continueth unto the last point of Gemini, which time is said to be hot and moist, and therefore is likened to childhood: And summer beginneth when the Sun entereth into the first point of Cancer, which is about the 12. or 13. of june, and endeth when he is in the last degree of Virgo: and this time is said to be hot and dry, and therefore is likened to Adolescency. Than Autumn or fall of the leaf beginneth when the Sun entereth into the first point of Libra, which is about the 13. or 14. of September, and endeth when the Sun is in the last degree of Sagittarius: and this time is said to be cold and dry, and therefore is likened to manhood: Finally winter beginneth when the sun entereth into the first point of Capricorn, and endeth when he is in the last degree of Pisces: and this time is said to be cold and moist, and therefore is likened to old age: notwithstanding Galen in his first book de Elementis, saith that the spring is temperately hot and moist, and therefore a most wholesome time: And summer is more hot than cold, and more dry than moist, and therefore is said to be hot and dry: And Autumn is also said to be dry because it is more dry than moist, and yet neither hot nor cold, but unequally mixed, and thereby infective and causing sickness. And winter is said to be cold and moist, not because it is colder or moister than any other season, but because that in winter moisture exceedeth dryness, and coldness exceedeth heat. But you have to understand that these four seasons have not like qualities in all the 5. Zones: For in the burnt Zone, and specially to those that devil right under the Equinoctial the sun being in Aries or Libra causeth greatest heat, & thereby two summers because he is then right over their heads, & being in either of the Solstices, that is in the beginning of Cancer or Capricorn he causeth two winters because he is then furthest from them as I have declared unto you in the second part of my treatise of the Sphere the 20. Chapter, whereas I treat of the seasons and shadows incident to diverse Climes and Parallels whereunto I refer you, and so I end with this matter.   

How to know when the Sun riseth and setteth in every latitude, and thereby the length of the day and night, and also in what rombe or wind he riseth and setteth, and how much he declineth every day from the Equinoctial either Northward or Southward. Also how to know the elevation of the Pole, otherwise called the latitude of any place, by knowing the Meridian altitude of the Sun, and his declination. Chap. 47.  
THe most part of all these things have been taught before in my Sphere, in my Treatise of the Globes, and also in my treatise of the Astrolabes. And in the first two Treatises I show also how to find out the Longitude of any place, and therefore needeth not here to be rehearsed: But if you would know how to handle the declination of the sun being upon the sea, then read 7. 8. 9 and 10. chapters of William Borne his book called the Regiment of the Sea, and you shall be fully instructed therein: The whole effect of all which Chapters Robert Norman setteth down in few words in his new Attractive in this manner as followeth.  
First learn whether the sun have South declination or North declination, which you shall know by his being in any of the Northern or Southern signs: Than mark what shadow he casteth, and whether it striketh towards the Pole whereunto he is nearest, or to the contrary. For if the sun casteth his shadow the same way that he is from the Equinoctial, he shallbe betwixt you and the Equinoctial, & then having taken his Meridian altitude subtract the same from 90. & add unto the remainder the sun's declination for that day, and the sum thereof shallbe the elevation of the Pole or the distance of your Zenith from the Equinoctial otherwise called the Latitude which is always equal to the elevation of the Pole: But if the sun casteth his shadow to the contrary side of the Equinoctial, that is to say, being in his North declination casteth his shadow Southward, or being in his south declination casteth his shadow Northward, then either the Equinoctial shallbe betwixt you & the sun, or you in the Equinoctial, or else you shallbe betwixt the Equinoctial and the sun, which you shall know thus: Add the declination of the sun for that day wherein you seek unto his Meridian altitude, and if the sum of the addition be lesser than 90. degrees, than so much as it wanteth of 90. degrees shall you be distant from the Equinoctial on that side on which the shadow stréeketh: but if it amounteth just to 90. degrees, than you shallbe right under the Equinoctial. Again if it be more than 90. degrees, than so much as is the overplus, so much shall you be from the Equinoctial towards the sun, at which time you shall be also betwixt the Equinoctial and the sun.  
And if you find the Meridian altitude of the sun to be even with your Zenith, then look what declination the sun hath at that instant, and so much shall you be from the Equinoctial on that side wherein the sun is: But if the sun have no declination, then shall you be right under the Equinoctial line.   

Of the shadow of the Sun, and how to know thereby the hour of the day in any latitude by help of an universal dial. Chap. 48.  
MY former order now requireth that I should speak somewhat of the shadow of the Sun, and of the diversity thereof, according to the Clime or Parallel under which you sail. But for so much as I have showed you in the second part of my Sphere from the 20. to the 27. chapter of the same, what diverse shadows the Sun yieldeth in diverse Climes and Parallels, and also have showed what Vmbra recta and Vmbra versa is in the 40. Chapter of my Treatise of the Astrolabes: I mind not here therefore to make a new recital thereof, but only to show how you shall found by his shadow the true hour of the day in every Latitude by a general Dial, made to serve in all Latitudes, of which dials, though I have seen diverse and of diverse shapes, yet none liketh me better than that which William Borne setteth down in his Regiment the 21. Chapter, and calleth it the Equinoctial Dial, which serveth not only to know the hour of the day by the shadow of the sun, but also the hour of the night by the shadow of the Moon when she shineth clear, which Dial being of small charge, I would wish all Mariners to have: The making and use whereof is so plainly set down by himself in the foresaid Chapter as I think it superfluous to set it down again here. And thus ending with the the sun, I will now turn my pen to the Moon.   

Of the Moon and of all her divers motions. Chap. 49.  
THe Moon is a round thick and dark body, having no light of herself, but only such as she receiveth from the sun, and she maketh her daily motion from East to West as all other stars do in 24. hours, according to the moving of the Primum mobile. But according to her own motion, which is from West to East, the goeth but 13. degrees 12. minutes in the space of 24. hours, and that is according to her mean motion. And she passeth through the twelve signs of the zodiac in 27. days and eight hours: during which time the sun by his natural motion which is also from West to East, is removed from the place of conjunction almost 27. degrees, so as the Moon not finding the Sun there spendeth two days four hours, and 44. minutes more in overtaking him, which being added to 27. days and 8. hours do make in all 29. days 12. hours and 44. minutes. Notwithstanding by reason that the Moon hath aswell as the Sun three motions, that is, swift, mean, and flow, she may change sometime sooner and sometimes later than in 29. days 12. hours 44. minutes, and yet one change counted with an other shall make up the self same sum. And note that this her threefold motion dependeth upon two points, the one called the Auge, and the other the point opposite to the Auge.  
The point Auge of the Moon is when she is furthest distant from the earth, and the point opposite to that is when she is nighest to the earth: For wheu she is in the point Auge, she goeth little more than twelve degrees in 24. hours: but when she is in the opposite point she goeth almost 15. degrees in 24. hours. And in her mean motion which is in the midst betwixt the two foresaid points, she goeth 13. degrees and 12. minutes in 24. hours. Now because the Mariners do accounted the moving of the Moon by the points of their Compasses, they may thereby understand that she goeth not always in 24. hours one point and three minutes as they reckon, but sometime more and sometime less: For when she is in her slow motion she 〈…〉 more than 12. degrees in 24. hours, in which time the 〈…〉 one degree, so as the Moon is distant from the 〈…〉 degrees, which is but 44. minutes of an hour, which wanteth 4. minutes of a whole point, whereto is attributed 48′·S as hath been said before: and in her swift motion she goeth 15. degrees, in which the sun goeth one degree, so as she is distant from the sun 14. degrees, which is more than a point and 3. minutes of the compass. And you have to note that the point Auge of the Moon is movable and passeth through the zodiac in the space of 19 years, and thereby sometime causeth the full of the Moon to happen sooner or later.   

How to know in what sign the point Auge of the Moon is in any year. Chap. 50.  
IF you would know in what sign the Auge of the Moon is in any year, than you must consider the prime or golden number of that year for when the prime or golden number is one, than her Auge is in Aries. And if the Moon be also then in Aries, she is in her slow motion: and being in the point opposite which is ♎, she is in her swift motion. And sith this Auge of the Moon goeth through the 12. signs in 19 years as hath been said, it must needs fall out that in 9 years and a half her Auge cometh to be in Libra: And then the Moon being there she is in her slow motion: And being in the point opposite which is Aries, she is in her swift motion: Again when the prime is 5. then her point Auge is in Cancer: and the Moon being there she is in her slow motion, and when she cometh to be in the point opposite which is Capricorn, she is in her swift motion: And when the prime is 14. or 15. then her point Auge is in Capricorn, where if the Moon be also, than she is in her slow motion: and being in the point opposite which is Cancer, she is in her swift motion. And note that when the Moon is in her swift motion she maketh her change, or full, or any other aspect the sooner: And contrariwise when she is in her slow motion she maketh her change, or full, or any other aspect the later. Moreover you shall see her at the time of her change either sooner or later, according to the time of the year, for from january to june you shall see her within 24. hours after her change, because she hath during those months North declination from the sun, & maketh a greater arch than the sun doth. But from july to December you shall not see the Moon scant three days after the change. But you may see her within 24. hours before her change, because that during those months she hath South declination from the sun: And note that when the Moon is three days and 18. hours, which is the half quarter of the Moon, the sea men do call that time the prime day, because the Moon is then 4. points to the Eastward of the Sun, which is three hours, for to every point is attributed three quarters of an hour as hath been said before. Moreover it is necessary for sea men to know when the Moon riseth & setteth, and in what part of the Horizon in every Latitude: and how long she shineth and when she is full South, and also what Latitude she hath, and whether it be South or North every day and hour throughout the year. All which things are most easy to be found by the Globe, and by help of the Ephemerideses in such sort as is before taught in my Treatise of the Globes, and also by help of M. Blagrave his Astrolabe, her true place in the zodiac being first known by the Ephemerideses. It is meet also to know when she is in conjunction with the sun, or at the full, and the rest of her quarters, which is easily known by the Ephemerideses.   

When the Moon is said to be in Conjunction with the Sun, or to be at the full, and what her greatest latitude is aswell from the Ecliptic line, as from the Equinoctial. Chap. 51.  
SHe is said to be in conjunction with the sun when the sun and she be both in one self sign and like degree: But when she is at the full, than she is opposite to the sun, and distant from him 6. signs, which is the one half of the zodiac, containing 180. degrees: And in every quarter she is distant from the sun three signs which is 90. degrees: Moreover the Moon is said to have Latitude both Northern and Southern from the Ecliptic line, which line the deferent of the Moon crosseth in 2. points, and thereby maketh two intersections, whereof the one tending towards the North is called the head of the Dragon, and the other intersection towards the South is called the tail of the Dragon, so as when the Moon is passed 90. degrees from the Dragon's head towards the North, than her Latitude is 5. degrees Northward: And when she is distant 90. degrees from the tail of the Dragon towards the South, than her Latitude is also 5. degrees Southward, which is the greatest Latitude that she hath on either side of the Ecliptic line, whereof I have written more at large in the first part of my Sphere the 25. chap. in which you shall find a figure representing the said Dragon both head and tail: But her Latitude is to be considered in two respects, that is not only from the Ecliptic line, but also from the Equinoctial, for from the Ecliptic line her greatest Latitude is but 5. degrees on either side of the Ecliptic, as hath been said before. But from the Equinoctial her greatest Latitude is 28. degrees and a half on either side of the Equinoctial, which in mine opinion might be more rightly called her greatest declination, which exceedeth by 5. degrees the declination of the sun, for that is but 23. degrees and a half, or rather 28′· on either side of the Equinoctial. But this great Latitude of the Moon is only to be understood when the prime is one, and that her Auge is in Aries: For when the prime is betwixt 9 or 10. years or more, the Moon declineth not from the Equinoctial on either side above 18. degrees and a half at the most.   

How to know in what part of the zodiac the head of the Dragon is every year. Chap. 52.  
IF you would know in what part of the zodiac the head of the Dragon is, than you must consider the prime, for when the prime is one than the Dragon's head is in the first point of Aries, even as the point Auge is. And in 19 years it passeth through the twelve signs aswell as the point Auge of the Moon, but with contrary course for the point Auge of the Moon moveth according to the succession of the signs, that is, from Aries to Taurus, Gemini and so forth: But the head of the Dragon hath a contrary motion, that is, from Aries to Pisces, and so into Aquarius and so forth, so as in 9 years and a half it meeteth just with the point Auge of the Moon in the sign Libra. Thus you see that by knowing the prime, you learn also to know in what sign the Auge of the Moon, and also the Dragon's head is, and what Latitude the Moon hath, aswell from the Ecliptic line as from the Equinoctial. But one of the chiefest points to be had by the Moon, is to know thereby the tides, that is when the sea floweth and ebbeth in any place: whereof we come now to speak.   

How to know the tides in any place by the Moon. Chap. 53.  
BEfore that you enter into any Haven or River, it is necessary to know the true tides of that place, which tides are subject to the motion of the Moon, for she causeth at one place or other always in one certain rombe full Sea. As for example it is always full sea at Antwerp when the Moon is either right East or West: and it is ebb or low water there when she is North or South, and because the Moon doth pass through all the rombes of the Mariner's Compass in 24. hours, they allow to every rombe ¾. of an hour, which is 45. minutes, and that being multiplied by 32. do make just 24. hours, wherefore if the two rombes, North and South do yield each of them 12. hours, than the first rombe must needs yield ¾. of an hour, the second one hour and ½. and the third two hours and ¼. and so forth of the rest: For by adding to every rombe ¾. you shall found that the East and West do yield always 6. hours: but than you must note that according to the age of the Moon the tides do fall every day later and later, wherefore to know the true time of the tide in any place, you must first learn by some Rutter or by the relation of others that can tell in what rombe the Moon causeth full sea in that place, & then at what hour it is full sea, the Moon being either in the change or at the full, which you shall know by allowing to every rombe ¾. of an hour in such manner as is before set down. But if you would know at what hour it is full sea in that place every day, than you must first understand that the Moon in 30. days slacketh 24. hours, which amounteth to ⅘. of an hour which is 48. minutes for every day, for so much she declineth every day from the sun, then look how many days the Moon is old, and having multiplied the same by ⅘. that is to say by 48. minutes, add the product thereof to those hours at which it is full sea, and you shall have the true time of full sea every day. As for example, suppose th' at it was full sea at the last new Moon at some place here in England at three of the clock in the afternoon, and now I would know at what a clock it shall be full sea 5. days after the new Moon. Now if you multiply 5. being the age of the Moon by 48. minutes, and add to the product thereof three hours, which was the time of the last change, the full sea shall be at that place at seven of the clock. But if the sum of such addition be above 12. hours, than you must cast away the 12. and the remainder shall show you the true hour of full sea.   

How to know by help of an instrument the tides at any place. Chap. 54.  
THere is also an easier way to know the tides every day in any place by the help of an instrument set down by Cogniet, whereof both the making and use here followeth. First upon some board well plained and made smooth, draw a Circle, and divide the same into 30. parts signifying the days or age of the Moon, setting the number of 30. above in the top of the instrument, and place all the other numbers as 1. 2. 3. and so forth towards your right hand, that done, make a movable roundle which may turn about within the verge of the first Circle, and divide that into 24. hours, and also into 32. rombes, setting the North point marked with the Flower deluce at the twelfth hour above, & the South point at the twelfth hour beneath, and the East point at the sixth hour on your right hand, and the West point at the sixth hour on your left hand, and so shall your instrument be perfect, the use whereof is thus:  
First you must know what rombe of the Moon in that place which you seek maketh a full sea, and also the age of the Moon by some Almanac or some other rule before taught, then having these two things turn the inward rondle of the hours & rombes until the foresaid known rombe doth justly answer to the 30. day of the great rondle, and there staying it firm with your finger, seek in the outermost border of the greater rondle the age of the Moon, and that will show you in the rondle of hours the very hour of full sea that day in that place.  
But one chief thing touching the knowing of the tides is to be noted as Borne saith, which is that the sea will flow more by one point of the Compass in the spring tides, then in any of the quarters of the Moon called Nep tides in every river that hath any indraft, and is of some reasonable distance from the sea: As for example, it floweth at graves end at the change or full of the Moon when she is South Southwest: but in any of her quarters it scant floweth when she is South and by West, and this rule as he saith is general for ever.  

The Sexagenarie Table whereby the products in Multiplication, the quotients in Division, and the square Roots are found Astronomical Fractions.    59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31     58 1 57 2 56 3 55 4 54 5 53 6 52 7 51 8 50 9 49 10 48 11 47 12 46 13 45 14 44 5 43 16 42 17 41 18 40 19 39 20 38 21 37 22 36 23 35 24 34 25 33 26 32 27 31 28 30 29 59      54 4 55 6 54 8 53 10 52 12 51 14 50 16 49 18 48 20 47 22 46 24 45 26 44 28 43 30 42 32 41 34 40 36 39 38 38 40 37 42 36 44 35 46 34 48 33 50 32 52 31 54 30 56 29 58 58        54 9 53 12 52 15 51 18 50 21 49 24 48 27 47 30 46 33 45 36 44 39 43 42 42 45 41 48 40 51 39 54 38 57 38 0 37 3 36 6 35 9 34 12 33 15 32 18 31 21 30 24 29 27 57 1 0 1       52 16 51 20 50 24 49 28 48 32 47 36 46 40 45 44 44 48 43 52 42 56 42 0 41 4 40 8 39 12 38 16 37 20 36 24 35 28 34 32 33 36 32 40 31 44 30 48 29 52 28 56 56 2 0 2 0 4       50 25 49 30 48 35 47 40 46 45 45 50 44 55 44 0 43 5 42 10 41 15 40 20 39 25 38 30 37 35 36 40 35 45 34 50 33 55 33 0 32 5 31 10 30 15 29 20 28 25 55 3 0 3 0 6 0 9       48 36 47 42 46 48 45 54 45 0 44 6 43 12 42 18 41 24 40 30 39 36 38 42 37 48 36 54 36 0 35 6 34 12 33 18 32 24 31 30 30 36 29 42 28 48 27 54 54 4 0 4 0 8 0 12 0 16       46 49 45 56 45 3 44 10 43 17 42 24 41 31 40 38 39 45 38 52 37 59 37 6 36 13 35 20 34 27 33 34 32 41 31 48 30 55 30 2 29 9 28 16 27 23 53 5 0 5 0 10 0 15 0 20 0 25       45 4 44 12 43 20 42 28 41 36 40 44 39 52 39 0 38 8 37 16 36 24 35 32 34 40 33 48 32 56 32 4 31 12 30 20 29 28 28 36 27 44 26 52 52 6 0 6 0 12 0 18 0 24 0 30 0 36       43 21 42 30 41 39 40 48 39 57 39 6 38 15 37 24 36 33 35 42 34 51 34 0 33 9 32 18 31 27 30 36 29 45 28 54 28 3 27 12 26 21 51 7 0 7 0 14 0 21 0 28 0 35 0 42 0 49       41 40 40 50 40 0 39 10 38 20 37 30 36 40 35 50 35 0 34 10 33 20 32 30 31 40 30 50 30 0 29 10 28 20 27 30 26 40 25 50 50 8 0 8 0 16 0 24 0 32 0 40 0 48 0 56 1 4       40 1 39 12 38 23 37 34 36 45 35 56 35 7 34 18 33 29 32 40 31 51 31 2 30 13 29 24 28 35 27 46 26 57 126 8 25 19 49 9 0 9 0 18 0 27 0 36 0 45 0 54 1 3 1 12 1 21       38 24 37 36 36 48 36 0 35 12 34 24 33 36 32 48 32 0 31 12 30 24 29 36 28 48 28 0 27 12 26 24 25 36 24 48 48 10 0 10 0 20 0 30 0 400 0 50 1 0 1 10 1 20 1 30 1 40       36 49 36 2 35 15 34 28 33 41 32 54 32 7 31 20 30 33 29 46 28 59 28 12 27 25 26 38 25 51 25 4 24 17 47 11 0 11 0 22 0 33 0 44 0 55 1 6 1 17 1 28 1 39 1 50 2 1       35 16 34 30 33 44 32 58 32 12 31 26 30 40 29 54 29 8 28 22 27 36 26 50 26 4 25 18 24 32 23 46 46 12 0 12 0 24 0 36 0 48 1 0 1 12 1 24 1 36 1 48 2 0 2 12 2 24       33 45 33 0 32 15 31 30 30 45 30 0 29 15 28 30 27 45 27 0 26 15 25 30 24 45 24 0 23 15 45 13 0 13 0 26 0 39 0 52 1 5 1 18 1 31 1 44 1 57 2 10 2 23 2 36 2 49       32 16 31 32 30 48 30 4 29 20 28 36 27 52 27 8 26 24 25 40 24 56 24 12 23 28 22 44 44 14 0 14 0 28 0 42 0 56 1 10 1 24 1 38 1 52 2 6 2 20 2 34 2 48 3 2 3 16       30 49 30 6 29 23 28 40 27 57 27 14 26 31 25 48 25 5 24 22 23 39 22 56 22 13 43 15 0 15 0 30 0 45 1 0 1 15 1 30 1 45 2 0 2 15 2 30 2 45 3 0 3 15 3 30 3 45       29 24 28 42 28 0 27 18 26 36 25 54 25 12 24 30 23 48 23 6 22 24 21 42 42 16 0 16 0 32 0 48 1 4 1 20 1 36 1 52 2 8 2 24 2 40 2 56 3 12 3 28 3 44 4 0 4 16       28 1 27 20 26 39 25 58 25 17 24 36 23 55 23 14 22 33 21 52 21 11 41 17 0 17 0 34 0 51 1 8 1 25 1 42 1 59 2 16 2 33 2 50 3 7 3 24 3 41 3 58 4 15 4 32 4 49       26 40 26 0 25 20 24 40 24 0 23 20 22 40 22 0 21 20 20 40 40 18 0 18 0 36 0 54 1 12 1 30 1 48 2 6 2 24 2 42 3 0 3 18 3 36 3 54 4 12 4 30 4 48 5 6 5 24       25 21 24 42 24 3 23 24 22 45 22 6 21 27 20 48 20 9 39 19 0 19 0 38 0 57 1 16 1 35 1 54 2 13 2 32 2 51 3 10 3 29 3 48 4 7 4 26 4 45 5 4 5 23 5 42 6 1       24 4 23 26 22 48 22 10 21 32 20 54 20 16 19 38 38 20 0 20 0 40 1 0 1 20 1 40 2 0 2 20 2 40 3 0 3 20 3 40 4 0 4 20 4 40 5 0 5 20 5 40 6 0 6 20 6 40       22 49 22 12 21 35 20 58 20 21 19 44 19 7 37 21 0 21 0 42 1 3 1 24 1 45 2 6 2 27 2 48 3 9 3 30 3 51 4 12 4 33 4 54 5 15 5 36 5 57 6 18 6 39 7 0 7 21       21 36 21 0 20 24 19 48 19 12 18 36 36 22 0 22 0 44 1 6 1 28 1 50 2 12 2 34 2 56 3 18 3 40 4 2 4 29 4 46 5 8 5 30 5 52 6 14 6 36 6 58 7 20 7 42 8 4       20 25 19 50 19 15 18 40 18 5 35 23 0 23 0 46 1 9 1 32 1 55 2 18 2 41 3 4 3 27 3 50 4 13 4 36 4 59 5 22 5 45 6 8 6 31 6 54 7 17 7 40 8 3 8 26 8 49       19 16 18 42 18 8 17 34 34 24 0 24 0 48 1 12 1 36 2 0 2 24 2 48 3 12 3 36 4 0 4 24 4 48 5 12 5 36 6 0 6 24 6 48 7 12 7 36 8 0 8 24 8 48 9 12 9 36       18 9 17 36 17 3 33 25 0 25 0 50 1 15 1 40 2 5 2 30 2 55 3 20 3 45 4 10 4 35 5 0 5 25 5 50 6 15 6 40 7 5 7 30 7 55 8 20 8 45 9 10 9 35 10 0 10 25       17 4 16 32 32 26 0 26 0 52 1 18 1 44 2 10 2 36 3 2 3 28 3 54 4 20 4 46 5 12 5 38 6 4 6 30 6 56 7 22 7 48 8 14 8 40 9 6 9 32 9 58 10 24 10 50 11 16       16 1 31 27 0 27 0 54 1 21 1 48 2 15 2 42 3 9 3 36 4 3 4 30 4 57 5 24 5 51 6 18 6 45 7 12 7 39 8 6 8 33 9 0 9 27 9 54 10 21 10 48 11 15 11 42 12 9        28 0 28 0 56 1 24 1 52 2 20 2 48 3 16 3 44 4 12 4 40 5 8 5 36 6 4 6 32 7 0 7 28 7 56 8 24 8 52 9 20 9 48 10 16 10 44 11 12 11 40 12 8 12 36 13 4      29 0 29 0 58 1 27 1 56 2 25 2 54 3 23 3 52 4 21 4 50 5 19 5 48 6 17 6 46 7 15 7 44 8 13 8 42 9 11 9 40 10 9 10 38 11 7 11 36 12 5 12 34 13 3 13 32 14 1    30 0 30 1 0 1 30 2 0 2 30 3 0 3 30 4 0 4 30 5 0 5 30 6 0 6 30 7 0 7 30 8 0 8 30 9 0 9 30 10 0 10 30 11 0 11 30 12 0 12 30 13 0 13 30 14 0 14 30 15 0 30 31 0 31 1 2 1 33 2 4 2 35 3 6 3 37 4 8 4 39 5 10 5 41 6 12 6 43 7 14 7 45 8 16 8 47 9 18 9 49 10 20 10 51 11 22 11 53 12 24 12 55 13 26 13 57 14 28 14 59 15 30 31 32 0 32 1 4 1 36 2 8 2 40 3 12 3 44 4 16 4 48 5 20 5 52 6 24 6 56 7 28 8 0 8 32 9 4 9 36 10 8 10 40 11 12 11 44 12 16 12 48 13 201 3 52 14 24 14 56 15 28 16 0 32 33 0 33 1 6 1 39 2 12 2 45 3 18 3 51 4 24 4 57 5 30 6 3 6 36 7 9 7 42 8 15 8 48 9 21 9 54 10 27 11 0 11 33 12 6 12 39 13 12 13 45 14 18 14 51 15 24 15 57 16 30 33 34 0 34 1 8 1 42 2 16 2 50 3 25 3 58 4 32 5 6 5 40 6 14 6 48 7 22 7 56 8 30 9 4 9 38 10 12 10 46 11 20 11 54 12 28 13 2 13 36 14 10 14 44 15 18 15 52 16 26 17 0 34 35 0 35 1 10 1 45 2 20 2 55 3 30 4 5 4 40 5 15 5 50 6 25 7 0 7 35 8 10 8 45 9 20 9 55 10 30 11 5 11 40 12 15 12 50 13 25 14 0 14 35 15 10 15 45 16 20 16 55 17 30 35 36 0 36 1 12 1 48 2 24 3 0 3 36 4 12 4 48 5 24 6 0 6 36 7 12 7 48 8 24 9 0 9 36 10 12 10 48 11 24 12 0 12 36 13 12 13 48 14 24 15 0 15 36 16 12 16 48 17 24 18 0 36 37 0 37 1 14 1 51 2 28 3 5 3 42 4 19 4 56 5 33 6 10 6 47 7 24 8 1 8 38 9 15 9 52 10 29 11 6 11 43 12 20 12 57 13 34 14 11 14 48 15 25 16 2 16 39 17 16 17 53 18 30 37 38 0 38 1 16 1 54 2 32 3 10 3 48 4 26 5 4 5 42 6 20 6 58 7 36 8 14 8 52 9 30 10 8 10 46 11 24 12 1 12 40 13 18 13 96 14 34 15 12 15 50 16 28 17 6 17 44 18 22 19 0 38 39 0 39 1 18 1 57 2 36 3 15 3 54 4 33 5 12 5 51 6 30 7 9 7 48 8 27 9 6 9 45 10 24 11 3 11 42 12 21 13 0 13 39 14 18 14 57 15 36 16 15 16 54 17 33 18 12 18 51 19 30 39 40 0 40 1 20 2 0 2 40 3 20 4 0 4 40 5 20 6 0 6 40 7 20 8 0 8 40 9 20 10 0 10 40 11 20 12 0 12 40 13 20 14 0 14 40 15 20 16 0 16 40 17 20 18 0 18 40 19 20 20 0 40 41 0 41 1 22 2 3 2 44 3 25 4 6 4 47 5 28 6 9 6 50 7 31 8 12 8 53 9 34 10 15 10 56 11 37 12 18 12 59 13 40 14 21 15 2 15 43 16 24 17 5 17 46 18 27 19 8 19 49 20 30 41 42 0 42 1 24 2 6 2 48 3 30 4 12 4 54 5 36 6 18 7 0 7 42 8 24 9 6 9 48 10 30 11 12 11 54 12 36 13 18 14 0 14 42 15 24 16 6 16 48 17 30 18 12 18 54 19 36 20 18 21 0 42 43 0 43 1 26 2 9 2 52 3 35 4 18 5 1 5 44 6 27 7 10 7 53 8 36 9 19 10 2 10 45 11 28 12 11 12 54 13 37 14 20 15 3 15 46 16 29 17 12 17 55 18 38 19 21 20 4 20 47 21 30 43 44 0 44 1 28 2 12 2 56 3 40 4 24 5 8 5 52 6 36 7 20 8 4 8 48 9 32 10 16 11 0 11 44 12 28 13 12 13 56 14 40 15 24 16 8 16 52 17 36 18 20 19 4 19 48 20 32 21 16 22 0 44 45 0 45 1 30 2 15 3 0 3 45 4 30 5 15 6 0 6 45 7 30 8 15 9 0 9 45 10 30 11 15 12 0 12 45 13 30 14 15 15 0 15 45 16 30 17 15 18 0 18 45 19 30 20 15 21 0 21 45 22 30 45 46 0 46 1 32 2 18 3 4 3 50 4 36 5 22 6 8 6 54 7 40 8 26 9 12 9 58 10 44 11 30 12 16 13 2 13 48 14 34 15 20 16 6 16 52 17 38 18 24 19 10 19 56 20 42 21 28 22 14 23 0 46 47 0 47 1 34 2 21 3 8 3 55 4 42 5 29 6 16 7 3 7 50 8 37 9 24 10 11 10 58 11 45 12 32 13 19 14 6 14 53 15 40 10 27 17 14 18 1 18 48 19 35 20 22 21 9 21 56 22 43 23 30 47 48 0 48 1 36 2 24 3 12 4 0 4 48 5 36 6 24 7 12 8 0 8 48 9 36 10 24 11 12 12 0 12 48 13 36 14 24 15 12 16 0 16 48 17 36 18 24 19 12 20 0 20 48 21 36 22 24 23 12 24 0 48 49 0 49 1 38 2 27 3 16 4 5 4 54 5 43 6 32 7 21 8 10 8 59 9 48 10 37 11 26 12 15 13 4 13 53 14 42 15 31 16 20 17 9 17 58 18 47 19 36 20 25 21 14 22 3 22 52 23 41 24 30 49 50 0 50 1 40 2 30 3 20 4 10 5 0 5 50 6 40 7 30 8 20 9 10 10 0 10 50 11 40 12 30 13 20 14 10 15 0 15 50 16 40 17 30 18 20 19 10 20 0 20 50 21 40 22 30 23 20 24 10 25 0 50 51 0 51 1 42 2 33 3 24 4 15 5 6 5 57 6 48 7 39 8 30 9 21 10 12 11 3 11 54 12 45 13 36 14 27 15 18 16 9 17 0 17 51 18 42 19 33 20 24 21 15 22 6 22 57 23 48 24 39 25 30 51 52 0 52 1 44 2 36 3 28 4 20 5 12 6 4 6 56 7 48 8 40 9 23 10 24 11 16 12 8 13 0 13 52 14 44 15 36 16 28 17 20 18 12 19 4 19 56 20 48 21 40 22 32 23 24 24 16 25 8 26 0 52 53 0 53 1 46 2 39 3 32 4 25 5 18 6 11 7 4 7 57 8 50 9 43 10 36 11 29 12 22 13 15 14 8 15 1 15 54 16 47 17 40 18 33 19 26 20 19 21 12 22 5 22 58 23 51 24 44 25 37 26 30 53 54 0 54 1 48 2 43 3 36 4 30 5 24 6 18 7 12 8 6 9 0 9 54 10 48 11 42 12 36 13 30 14 24 15 18 16 12 17 6 18 0 18 54 19 48 20 42 21 36 22 30 23 24 24 18 25 12 26 6 27 0 54 55 0 55 1 50 2 45 3 40 4 35 5 30 6 25 7 20 8 15 9 10 10 5 11 0 11 55 12 50 13 45 14 40 15 35 16 30 17 25 18 20 19 15 20 10 21 5 22 0 22 55 23 50 24 45 25 40 26 35 27 30 55 56 0 56 1 52 2 48 3 44 4 40 5 36 6 32 7 28 8 24 9 20 10 16 11 12 12 8 13 4 14 0 14 56 15 52 16 48 17 44 18 40 19 36 20 32 21 28 22 24 23 20 24 16 25 12 26 8 27 4 28 0 56 57 0 57 1 54 2 51 3 48 4 45 5 42 6 39 7 36 8 33 9 30 10 27 11 24 12 21 13 18 14 15 15 12 16 9 17 6 18 3 19 0 19 57 20 54 21 51 22 48 23 45 24 42 25 39 26 36 27 33 28 30 57 58 0 58 1 56 2 54 3 52 4 50 5 48 6 46 7 44 8 42 9 40 10 38 11 36 12 34 13 32 14 30 15 28 16 26 17 24 18 22 19 20 20 18 21 16 22 14 23 12 24 10 25 8 26 6 27 4 28 2 29 0 58 59 0 59 1 58 2 57 3 56 4 55 5 54 6 53 7 52 8 51 9 50 10 49 11 48 12 47 13 46 14 45 15 44 16 43 17 42 18 41 19 40 20 39 21 38 22 37 23 36 24 35 25 34 26 33 27 32 28 31 29 30 59 60 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30    

How a general Rutter showing the tides in all places should be made. Chap. 55.  
WIlliam Borne doth set down what Moon doth make a full sea, aswell here in the most parts of our English coasts as in some other parts of France & Spain, and so do many others whose Tables touching the tides are called Rutters, whereof some are truly set down and some are false: But sith such Rutters do serve but for a few particular places, I would wish that some learned Pilot that hath sailed many and sundry long voyages, to make a general Rutter that might serve for all places if it were possible, or at the lest for so many places as are known in these days: And I would wish such general Rutter to be made in manner of an Alphabet: and that every place might have his true Longitude and Latitude added thereunto, to the intent that every place might be the more easily found out in any Map or Card that is graduated with degrees of Longitude and Latitude: And then to show what Moon doth make a full sea in every such place, which thing who soever would perform, he should in mine opinion deserve great commendation. And thus I end this Treatise, praying all the learned Seamen not to be offended or grieved with me for that I do make young Gentlemen our own Countrymen partakers of their most worthy knowledge, whereof the ignorant are not able to judge, nor to yield them that praise which they deserve: yea rather I hope that they will help to perfect what so ever I have herein left unperfected, to make the young Gentlemen the more skilful, and thereby the more serviceable to their Country, and in so doing they shall procure to themselves great good will and infinite thanks.  
FINIS.