An Account OF THE ROTULA ARITHMETICA Invented by Mr. George Brown, Minister of Kilmaures.  
Together with Instructions how to use it.  
EDINBURGH, Printed for the Author, M.DCC   

At Edinburgh, December 1. M. DC.XC.VIII.  
THE Lords of His majesty's Privy Council do hereby Grant to Mr. George Brown Minister, and His Heirs and Assigneys, the sole Privilege of Framing, Making, and Selling His Instrument, called Rotula Arithmetica, for the space of 14. Years yet to come, after the Day and Date hereof. And Discharges any other Persons to make or sell the said Instrument, dureing the space foresaid, without express Liberty and Licence from the said Mr. Geo. Brown and his foresaids, under the Pain of 500 Marks, besides Confiscation of the Rotula's made or sold.  
Extracted by Me, Sic subscribitur, Gilb. Eliot, Cls. Sti. Cons.   

TO THE READER.  
Courteous Reader!  
WHen first I applied my Mind to publish somewhat concerning my Rotula Arithmetica; I designed only (without Preface or Apology) to set down, in the plainest and most homely Dress, such Rules as might render those, who should happen to have both a Book and a Rotula, capable by the Help of the one, to make Use of the other; not doubting, but that such an useful, Machine as it is, would be very acceptable to all sorts of persons; men that want Arthemetick being by this means, in the space of four Hours made capable to Add, Subtract, Multiply and Divide without any other previous Knowledge, than that of Reading Figures, tho' otherwise such Persons were not able readily to Condescend whether 7 and 4 were 11 or 12: and the ablest Masters, being by the help of this Machine rendered more able to perform the most tedious, and most numerous Operations of Addition, Multiplication and Division, with the greatest Certainty, and without all that Rack of Intention, to which they are, by all the other methods hitherto known, obliged. But before I was ready to appear in Public, I understood by several Documents, that it is as unfit for new Productions, to go abroad, without the Helmet or cover of a preface, as it's unsafe for a Highland-man, to travel amongst those Nighbours, with whom he is at Variance, without the Protection of his Broad Sword and Target.  
Indeed some Persons have been so unjust as to spread a report that the Rotula is no new thing, but an old Invention of one Delamain, an English man, who obtained from King Charles I. a Privilege for his Mathematical Ring Anno 1630. Now this Report, as false as it is, was at first no small prejudice against the Rotula, in the Opinion of those, who knew no better, and who had a great deference for the sentiments of those, who were the Raiser's and Spreaders of this report: and all I myself could say at first, (having never seen Delamain's Book nor Instrument) was, that if the Rotula had ever been known in the World before, it had never been out of fashion, but had very soon after publication become very near as common as a Balance or an elln-wand, every one, who hath any thing considerable to measure or weigh, having likewise some Account to cast.  
But tho' this seemed to me a sufficient Demonstration concerning the Novelty of my Invention, yet I was obliged for the Satisfaction of others to procure some Copies, of De la main's Book and Projection, that such as were under a mistake might be convinced of the falsehood of the foresaid Report, by Comparing both Projections together.  
I should be loath to Charge the usage, I met● with in this affair, upon the Score of malice; that base Vice being as far below men of their Generous and Liberal Education, as it lies out of the Road of my Actions to give them any other Provocation, than that of being an Inventor: Nor can it be Imputed to Ignorance; for they are unquestionably men of Profound Learning, and Knowledge; It must then be chargeable merely upon the Score of Rashness, for had they but been at the Pains to Compare both Instruments together, they might have easily perceived the Difference to be as great as that betwixt a Line of Logarithme Numbers, and a Scale of equal parts: and for this cause I think I may Justly Charge them with Injustice, according to that known Maxim.  
Qui statuerit aliquid, parte inaudita altera, licet aequum statuerit, haud tamen aequus fuit.  
How reasonably then may the man be charged with Injustice, who passeth not only a harsh, but an unjust Verdict and Censure on a thing he hath not been at the pains duly to examine and consider.  
But to proceed; Our worthy Countryman the Lord Naper Baron of Merchistoun, Invented the Logarithm-Tables; and Mr. Gunter, an English man, Converted these Tables into a strait Lined Scale; after him Mr. Delamain, who was Gunter's Scholar converted Gunter's Scale into a double Circle, merely to ease men of Compasses: But then his Ring like Gunter's Scale, at that time, did only consist of one Line of Numbers, Sins and Tangents: and as he never dreamt of performing Addition, or Substraction by the help of his Machine, so he Ingenously acknowledges that it was not capable to perform even Division and Multiplication to an Arithmetical exactness; But many times a man might come short of very near an Unite; Nay He might have added that in Numbers of many Places a man may be some times to seek for Units, and Ten, if not for Hundreds. Nay to render it more certain the author requires That, which he proposes as a portable Instrument, to be made of several Foots or Ellns Diameter, which would render it unweeldy, and consequently less useful, either by Sea or Land: and it is not Improbable, that for these defects, that Instrument hath been antiquated, and hath given place to the Double Scale of Proportion, now so much in use.  
Now the Rotula performs all the four Arithmetical Operations Arithmetically, and to an Arithmetical Exactness, not only of the Integers, but even of the decimals, whither finite or Infinite. So that a man, who can but work by the Rotula, may within a little Time and Practice, learn to work by the Pen; if he should chance to want his Machine; Nay I believe, when the Rotula's are once become common, the mother may teach her Children at home, as much Arithemetick, as may serve them all their lives.  
I should now close this tedious business of a Preface, but that I am obliged to give some account of Mr. Glover, and his Invention called his Ro●e Ariehmetique.  
This Mr. Glover is a Scotish Gentleman, whose Elder Brother Thomas, who was this John's Master, was my Scholar about the Year 74. at which time he learned from me that Skill in Numbers and other things which he afterwards taught this Gentleman, and by which both of them have since become famous abroad.  
Now tho' this Invention of John Glover's be Posterior to my Rotula, as appears by the Date of his Privilege, Granted by his Majesty of France, which is of the 13. of March 1699. whereas my Privilege is granted in Scotland on the first of December 1698. Yet his comes so far short of mine, that I Verily believe, had he seen or gotten a perfect Account of mine before he proposed his own, he would have spared the pains of Publication.  
I must Confess that for any thing I yet know, his Tables or Circles for Multiplication and Division (which indeed are very Ingenious, and have cost him much Thought) are his own; as also his Tables for the Reduction of Pence to shillings, & shillings to Pounds. But in that Part which is common with his Rove and my Rotula, he seems to have got some hint of mine; and this I am the more apt to belive because about the time that I was busy in contriving the Rotula, there was a very smart Gentleman a near friend of his, Scholar with me at Stirline.  
But that which gives me greater evidence in this particular, is some expressions in his own Book which makes me fancy that he hath, at least, got some imperfect Description of mine, before he contrived that part of his which serves for Addition and Substraction.  
For, whereas there is on my fixed Plate 3. Circles, he speaks of three, and yet Immediately he takes away two of his, & turns them into Tables for Reduceing of Pence into shillings, and shillings into pounds, & these not exceeding the limits of 120. and instead of the third on the fixed he gives us nothing but a little segment, about a fifth part divided into parts beginning at 0. and ending at 24. which he calls his fixed Index: as also whereas my Circle is divided into 100 parts; he Chooses, (to make his differ from mine) 120. as being a common Product of 10. 12. and 20. These Numbers (as he alleadges in the beginning of his 1st. Chapter) being preferable to all other Numbers whatsoever; and yet near the close of the same Chapter he acknowledges that it would be better to divide the Circle for Addition and Substraction into 100 parts or some Power of 10. and so the Instrument would become universal: All which give me suspicion that in this part, he hath gotten, at least some lame account of mine.  
Moreover his Instrument is Defective and comes far short of mine, even in Addition; For in his, the Practitioner is obliged to mind or mark down how many Revolutions his movable Plate makes, and every one being 120, he hath 120 to Multiply by the Number of Revolutions, which is not only troublesome, but likeways dangerous especially in real Bussiness, where a man whose mind is bussied both about the figures of his Column, and the points of his movable plate, is obliged at the same time to mind the several Revolutions of his movable Plate, of which for every one he forgets, or overlooks, he loses 120 for his Pains; whereas in mine a man is not tied to any such intention above once for 1000 (which is more than any Column does ordinarily contain) the movable Plate, at every Revolution both marking and giving notice of the number of Revolutions.  
But, besides this, in my Rotula the same Circles that serve for Addition and Substraction serve likeways for Multiplication and Division, but in his Rove he hath one for Addition and Substraction and ten or eleven for Multiplication and Division and yet tho' the Circles were twice as large, and tho' they contained near twice as many figures as they do, they would be no more than what is necessary to do, what I am able to perform by mine.  
Lastly whereas his Tables are confined only to shillings and pence, and these of limited Number not exceeding, 120. There are on the waste, on the middle of my movable Plate, Tables for the Reduction of shillings, Pence, Farthings, Weights and Measures, be the Numbers never so large; Besides the decimal Tables for Money, weights Measures and the most ordinary common Fractions: By the help of which six last sort of Tables, the Multiplication and Division of Complex Numbers does become just as easy as that of Integers; without all that tediousness which Mr Glover proposes in his Book.  
To conclude, what I have said here is no more than was necessary for the Vindication of my own Invention, and to satisfy those, who already are, or hereafter may be misinformed either by the Story of Delamain, or Mr. Glover's Rove Arithmetic, who for what is Peculliarly his, deserves a good degree of commendation and Encouragement.  

CHAP. I. Concerning the Rotula, and the Rectification thereof.  
ALbeit in Books of this Nature, it be usual to prefix a Scheme of the Machine of which they treat; Yet I have thought fit in this, to omit that; because such as have a Rotula, need not a Scheme, and such as want one, have no use for a Book; I shall therefore (as briefly as I can) describe the Rotula, and then show You how to use it.  
The Rotula Consists of two Principal Parts, to wit, a Circular Plain moving upon a Center-pin, this we call the Movable Plate; and a Ring, whose Circles are described from the same Centre, this we call the Fixed Plate; Because it is fixed to the Box, to secure it from moving about the Centre, as the other does.  
The Fixed Plate is divided into three parts by two Circles; the Innermost of which is doubled, with a little Interstice for Peg-Holes.  
Near the Circumference of the Movable, there is another Double Circle, with a small Interstice also betwixt them, for Peg Holes.  
The space without the double Circle, on the Movable, and within that on the Fixed, are both of them equally divided into 100 Parts: and both are Numbered, beginning at 0. 1. 2. 3. and so proceeding in a Natural Order to 99 all the Divisions being drawn straight from the Centre.  
On the Fixed many of these Divisions are protracted, some only to the middle part; and others run over both: for confining the several single Coefficents' of the Respective Tabular Numbers, to which they are Prefixed, with this Caution when the Coefficients are the same, they are set down in the uttermost part; and when any Number admits of two pair of Coefficients, the one Pair is set in the Midmost and the other in the Out-most Part. Thus against 18. on the Fixed, You will find in the midmost Part, 2 × 9 (that is two times Nine or Nine times two; This × cross signifying the word times) and 3 × 6 in the outmost. Also on the Movable there is a Segment of a Circle, within the Peg-hole Circle, beginning at 9 of the Natural Numbers and ending at 72. This Segment is likewise Divided by the same lines that Divide the outmost Circle of Natural Numbers into equal parts.  
On the Fixed Plate at the Division betwixt 99 and 0 there is a little bit of Metal Screwed or Riveted, reaching likeways a lit le further than the Peg-hole Circle, on the Movable, this piece of Metal we call the Stop: and must always be placed next Your left hand, with the Number 25, or 30. toward your Breast. There is also on the Fixed, over against the Numbers 89. or 90. a little Circle Divided into to equal Parts; with a little Palm, which shifts one part at every revolution of the Movable; So that the figure, at which this Palm Points, Signifies the Number of Revolutions or Hundreds You have in your Account, or the Column last added  
When the Figures of the Movable Plate move towards that side of the Stop, which is next the cipher on the Fixed, we call the motion forward; but when they retire from it, the motion is backward.  
In Rectification, be sure not to touch the little Palm, till the Unites or Numbers beginning with a cipher on the Movable, be some of them against the Nynties on the Fixed; and then turn the little Palm to the cipher of its Proper Circle; after which turn back the Movable, till it will Move no further, and so when the Ciphers of the Movable are just at the Stop, as well as the Palm of the little Circle at its cipher; The Rotula is Rectified and ready for Operation.  
I have filled up the Vacant Space in the middle of the Movable with Decimal Tables, and Tables of Common Divisors, very useful for those, that have much Bussiness, or are in haste.  
Only observe carefuly that the Figures, on the right side of the Movable, are Top-●●e turvy; So that You must always take that which appears to be a 9th. for a 6th. and on the Contrary the 6th. for a 9th. and mind well, that the Unites are always next the Centre, and the Ten next the Circumference: thus 61. will appear 19 and 91. will look like 16.   

CHAP. II. Concerning Addition.  
AFter You have Rectified the Rotula, You may easily Perceive that all the Numbers on the Movable, are the same, with those directly against them, on the Fixed; but if You turn any other figure of the Movable to the Stop, then will all the Numbers on the Movable be just so many more than these directly against them on the fixed; for instance, if you were to Add any Number whatsoever to 7. first, bring that point of the Movable (which is not only directly against 7 on the Fixed, but indeed the same with it) to the Stop, and then you'll find all the Numbers on the Movable, 7 more than the respective Numbers on the Fixed, so that lib. ss. d. 345 17 10 976 13 8 158 18 9 746 16 11 843 19 6 977 15 7 865 14 10 743 18 11 896 15 8 739 16 7 478 13 4 287 14 8 958 19 11 549 15 5 378 17 9 486 14 8 297 18 10 684 14 8 1956 18 9 2768 11 6 9875 19 10 against 1. on the Fixed you have 8. on the Movable which is the Sum of 1. and 7. and against 9 on the Fixed you have 16. on the Movable, which is Just the Sum of 9 and 7. and on this Theory depends the Certainty of all your operations: wherefore you must take care in bringing up any number of Figures to the Stop one after another, to look for all the Items on the Fixed Plate, but not on the Movable; Otherwise you will miscarry in your Operation: Example, were you to add the Account on the preceding Page, you must begin on the Top of the Column of Pence, covering the fifth Number with a bodle Peg, or any other little thing, & then with your Peg bring up the First Four Numbers. To wit: 10 8. 9 11. thus, bring first up the point against 10. on the Fixed, which is ten on the Movable) to the Stop, and then that Point on the Movable which is against 8 on the Fixed, and then that Point of the Movable, which is against 9 on the Fixed, and Lastly that point of the Movable, which is against 11, on the Fixed, and you will find the Sum of these four Numbers on the Movable, at the Stop, to be 38. but you need not regaird the Sum, till you have done with the whole Column, wherefore shifting your Cover and Proceeding with the next four Figures as you did with the First four Numbers, and so forward till you come to the foot of the Column you will Find the Sum of your Pence to be just 175. of which 1 is found at the Palm, and 75. at the Stop on he Movable; these Reduced to shillings (as you are taught in the Chapter concerning Division) do veiled 14 shillings and 7d. set down your 7 under the Column of Pennies; and having rectified the little Palm, (which must never be Forgot, before you begin a new Column) set 14. of the Movable 〈◊〉 the Stop, for the 14. shillings you have to carry.  
Then proceed, beginning at the Top of the Column for shillings, bringing up 17, 13, 18, 16, and so forward by four and four, as You did with the Pennies, and Your shillings will amount to 3●7 shill.  
These reduced to Pounds, conform to the Directions contained in the Chapter of Division, do yield 17lib. 7sh. Set down your 7sh. under the column of shillings, and having rectified the Palm, put 17 of the Movable to the Stop, for the 17lib. you have to carry.  
Thence proceed to the Unites of the Pounds, and bring the Figures of that Column up, as you did those of s●illings and Pence, and you'll find the Sum of it just 151. the last Figure of which being I must be set down under the Integer Unites, and the other two Figures, to wit, 15 must be, after Rectifying the Palm, carried as before.  
After the same manner You will find the second Column, or Column of Ten of Pounds amount to 142, where setting down the last, to wit, the Figure 2, and carrying the other two, to wit, 14. You proceed as before to the Column of Hundreds, the last three Numbers of which having Thousands annexed to them, You may bring up altogether, to wit, 19, 27, 98, and save Yourself the Labour of a new Rectification or carrying. This last Sum amounting to 260, must be all set down together in Order, to wit, the 0 under the Hundreds, the 6 under the Thousands, and the 2d. Figure before all.  
After the same manner, You may Add all other Species whatsoever; providing always You Divide the lesser Species by their proper Denominators, in Reduceing them to the next greater Species.  
In Your first Practice of Addition, satisfy Yourself with Examples of Integers, where there are no Reductions, till after You have learned Division; and than You will find no Difficulty.  
If at any Time you Add two Columns in Integers, you must set down the two Figures at the Stop, under the two Columns; and carry that only, at which the Palm points. Observe, that how many soever of any one Species are requisite to make one of the next greater: That Number, I call the Denominator of the lesser Species.  
Thus, in Integers, 10 is always the Denominator; Because 10 in the Column of Unites, makes but one in the Column of Ten; and 10 in the Column of Ten, makes but one in the Column of Hundereds, etc. Nay, the Denominator of Ten is 100 etc.  
Also, 4 is the Denominator of Farthings, Lippies, Firlots: and Pecks in relation to Firlots; but 16 is the Denominator of Bolls and of Pecks in relation to Bolls; and 12 the Denominator of Pence, and 20 of shillings, etc. This is well to be minded, because we may have frequent use to speak of the Denominators of Species.   

CHAP. III. Concerning Substraction.  
SVbstraction finds the Difference betwixt two unequal Numbers.  
The greater of these two Numbers is called the Charge; and the lesser, the Discharge.  
In Substraction, you must always bring the several Figures of the Charge, one after another, together with the respective Figures of the Discharge; the one on the Movable, and the other on the Fixed, directly against one another; and if the Figure of the Charge be equal to, or greater than the respective Figure of the Discharge, you have the Remainder at the Stop, on the Movable.  
But if the Figure of the Discharge be greater than that of the Charge, then against the Denominator of the Species, on the Movable, you have the Remainder on the Fixed.  
Thus, were I to take 8 from 8, or 7 from 7, having set the one against the other, you have 0 at the Stop.  
So likeways, if I were to take 5 from 8; having brought 8 on the Movable, against 5 on the Fixed: I have at the Stop, on the Movable, 3 for a Remainder. But, if I had been to take 8lib. from 5lib. the Remainder is on the Fixed, against 10 (the Denominator of Integers) on the Movable. And 8ds. from 5 pennies, the remainder is on the Fixed, against 12 (the Denominator of Pennies) on the Movable. And 8sh. from 5sh. the Remainder is still on ehe Fixed, against 20 (the Denominator of shillings) on the Movable. And 8 Ounces from 5 Ounces, the Remainder is on the Fixed, against 16 (the Denominator of Ounces) on the Movable.  
And the Peason of all this is plain; because the Discharge, in this case, cannot be taken off the Charge, but off the Denominator, which is equivalent to a borrowed one of the next greater Species, and the Overplus, by the very Position of the Instrument, is added to the Charge.  
Only mind carefully, that as often as the Remainder is found on the Fixed, (which always happens when any figure of the Discharge is greater than the respective Figure of the Charge) you must, in that case, esteem the next preceding Figure of the Discharge an Unite more than really it is; Taking 1 for 0, and 2 for 1, and 3 for 2, and so of others.  
lib. ss. d. 25123478 11 4 Charge, 23254906 14 8 Discharge, 01868571 16 8  
Thus, in this Example, I bring 8 on the Movable to 4 on the Fixed; and because the pennies of the Discharge, are greater than the Pennies of the Charge; I look for 12, the Denominator of Pennies on the Movable, and against it, I find 6 on the Fixed, for my remainder. These I set down under the Pennies.  
Again, because I found my last Remainder on the Fixed, I esteem 14sh. in the Discharge to be 15. for which cause I bring 15 on the Movable to 11 on the Fixed: and against 20 the Denominator of shillings, I have 16 on the Fixed. These I set down under shillings,  
Thereafter, for the same Reason, esteeming the 6lib. of my Discharge to be 7, I bring 8 (the respective Figure of the Charge on the Movable) to it; and (because the Figure of the Charge, is greater than that of the Discharge) I have 1 on the Movable at the Stop, for the Remainder. Thence, because the last Remainder was found on the Movable, I must not change my 0 but bring 7 on the Movable to 0 on the Fixed, and the Remainder at the Stop, is 7. And proceeding, conform to these Directions, with the rest, I perfect the Operation, finding always the Remainder, on the Movable, at the Stop, when the Charge Figure is greater than that of the Discharge, or equal to it, but on the Fixed against 10. the Denominator of Integers on the Movable, when the Discharge Figure is greater than that of the Charge.  
After the same M●nner, and by the same Directions, You may Subtract any other Species whatsoever, if You do but carefully mind the Denominators of the several Species.   

CHAP. IU. Concerning Multiplication.  
MVltiplication supposes two Numbers, called Coefficients, to find a third, called the Product, which Product contains any one of the Coefficients, as oft as the other contains an Unite.  
Any one of the Coefficients, especially the greater, may be called the Multiplicand, the other the Multiplier; thus, 3 times 4 is 12. of which 3 and 4 are the Coefficients, and 12 the Product, which Product contains 3, as oft as 4 contains 1.  
When one of the Coefficients is 10, 100, 1000 You need no Instrument for Multiplication in Integers, for this is done merely by adding the Ciphers to the right hand of the other Coefficient.  
Thus, 10 × 64 is 640, and 100 × 64 is 6400, etc.  
But when the Coefficients are all, or many of them, signifying Figures, set the lesser Number under the greater, thus,  
Having Rectified the Rotula (with the Pen in your Right Hand ready to Write, and your Left Hand at the Rotula, to turn the Movable as shall be necessary) because 8 is the last Figure of the Multiplier, you must look for 8 × everv Figure of the Multiplieand, one after another, beginning at the last, but you must not regard the Products on the Fixed, but only upon the Movable (nameing always the Ten of the Product, first, as a single Figure; putting that Figure on the Movable immediately to the Stop, and then the Unites, setting them down on the Paper thus,  
First, I look for 8 × 2. and against that, on the Movable, I find 1 and 6. for which cause I set 01 on the Movable to the Stop, and 6 I writ on the Paper below the Coefficient 8.  
Then I look for 8 × 4. and against that, on the Movable, I find 3 and 3. for which cause I turn 03 to the Stop, and write down 3 on my Paper.  
Then I look for 8 × 5. against which I find 4 and 3. here I put 4 to the stop, and 3 again to the Paper.  
Thence at 8 × 6. I find 05 for the Stop, and 2 for the Paper.  
And 8 × 9 I find 07 for the Stop, & 7 for the Paper.  
And lastly, at 8 × 7. I find 63, all for the Paper, because it is the last Product.  
After which, I Rectify again.  
Now, as I have gone over all the Figures of the Multiplicand with 8, the last Figure of the Multiplier, so may you do by 7 for the second Product, and 5 for the third, and 9 for the fourth; carefully observing only to set the first Figure, or that next the right hand of every particular Product, under that Figure of the Multiplier, by which it is produced: these particular Products summed up do yield the total Product.  
I shall subjoin another Example, and so end with Multiplication.  
 
In this last Example the two Ciphers of the Multiplier are set to the right of the Unites of the Multiplicand, and then multiplying by 9 I set the two Ciphers behind the Product; and so, what was but 9 times before, does now become 900 times the Multiplicand.  
You see also the Unites of the Product made by 7, set under 7 of the Multiplier, and the Unites of that made by 3, under 3 of the Multiplier, all the rest duly observing Rank and File.  
To conclude this Chapter, and make You prompt in finding your Coefficients: Observe, that all the Products of any Coefficients are contained within Ten Times the least of the two, so that all the Products of 2 are within 20, and all of 3 within 30, etc.   

CHAP. V Concerning Division.  

SECTION 1.  
DIvision serves to find a Number showing how oft the great of the two given Numbers contain's the lesser.  
The greater of the given Numbers, we call the Dividend: The lesser the Divisor; and the Number demanded, or found the Quotient.  
When as many Figures taken from the left of the Dividend; as there are Figures in the Divisor, are equivalent to the Divisor, or better than it; then we set a Point over the last of these, to Determine the first particular Dividend, which for Brevity, I shall call the first Dividual.  
But if as many taken, from the left of the Dividend, be less than the Divisor, the Point must stand over the next Subsequent Figure of the Dividend, for Determining the first Dividual.  
Having Determined your Dividual, you must refer the first of the Divisor, (when they are equal in Number of places) to the first of your Dividual; but if they are unequal to the first two of the Dividual, and so forward, the second, third and fourth Figures of the Divisor to the Subsequent Figures of your Dividual as they lie in Order; So that in subduction, where you begin at the last of the Divisor, you must refer, or Subtract the Product of it, from the last of the Dividual.  
The Remainder of the first Dividual with the next following Figure of the Dividend Yields you a 2d. Dividual.  
So Soon as you have Determined your first Dividual, you presently understand, how many Figures you are to have in the Quotient; to, wit, one for the Point, or first Dividual; and one for every subsequent Figure of the Dividend.  
Wherefore, if the 2d, 3d, or any other Dividual should happen to be less than the Divisor; you must put a cipher in the Quotient for that Dividual; And so (as if it were but a new Remainder) bring down another Figure from the Dividend; to wit, the next following for a new Dividual.  
I shall first show you how to Divide by one Figure, and then by two, and after that by as many as you please.  
In Division by any one Figure, you have nothing to do, but to bring the Dividual on the Movable to the first Cell that occurs, in which the Divisor is a Coefficient; the other Coefficient in the same Cell, is the Quotient, and that (having first drawn a Line below the Dividend.) You must set down under the last Figure of your Dividual; and the Figure, at the Stop on the Movable, you must set over the same last Figure of the Dividual, for a Remainder. And so proceed Rectifying every time before you apply to the next Dividual.  
Example.  
 
Here you see, that 8 (the foremost Figure of the Dividend,) being less than 9, the Divisor; the Point, for Determining the first Dividual stands over the second fiigure of the Devidend: So that my first Dividual is 88: Which being thus Determined, I understand that I am to have in my Quotient 7 Figures; to wit one for the first Dividual, and one for every Subsequent Figure of the Dividend.  
These things considered, and the Rotula Rectified, I bring the first Dividual, 88 on the Movable, to the first Cell, that occurs on the Fixed, in which 9 is a Coefficient; and because the other Coefficient in the same Cell is 9; I set that down under 8, the last Figure of my Dividual; and, having 7 on the Movable at the Stop, I set 7 over the same last Figure of my Dividual for a Remainder; then I Rectifiie.  
Now the first Remainder and next Subsequent Figure being 76 I bring 76 to the first 9 Coefficient, & there I find 8 for my Quotient, and 4 at the Stop for my Remainder; these I set down as before, the one under the other, over the last Figure of the 2d. Dividual, and then Rectify.  
The 3d. Dividual being 45, and having without any Motion a Cell of 9, directly against it, I sinned 5 for my Quotient, and 0 for my Remainder.  
So that the 4th Dividual becomes 04, which being less than 9; I set 0 in my Quotient and then  
The 4 still Remaining with the next Subsequent Figure of my Dividend, making 43; I bring 43 on the Movable to the first 9 Coefficient, and there finding 4 for my Quotient, and 7 at the Stop for my Remainder.  
Having set down these and Rectified, I find my next Dividual 72, against a Cell. of 9, in which I have 8 for my Quotient, and 0 for my Remainder,  
So that my last Dividual being oniie 01, which is less than my Divisor I set nought in my Quotient; and 1 the the last Remainder, I set over 9 the Divisor at the end of the Quotient, with a little Line betwixt them for a Fraction. Thus 1 1/9 9  
If any Divisor consist only of one signifying Figure & Ciphers, you must Divide only by the signifying Figure, & from the Quotient cut off as many Figures towards the right-Hand, as there are Ciphers in the Divisor, observing, that if the signifying Figure be only an Unite, You have no use for the Rotula, or any other Instrument; but merely to write down the Dividend below the Line in the Quotient; & then cut off from it conform to the Number of your Ciphers. Example first; 

Divisor 1000) 976583 Dividend.  
976583 Quotient.    
In this Example, you see the Figures in the Quotient are the same with those in the Dividend, because the signifying Figure of the Divisor is but an Unite. But because there are three Ciphers to the Right of the Divisor, I have cut off three Figures from the Right of the Quotient, where you see that (as your Dividual Point Intimates) You have only three Integers in your Quotient, namely those to the lefthand, and the Remainder is a Decimal Fraction; or if you will, the Numerator of a common Fraction, whose Denominator is the Divisor; thus:  
 
Example 2d.  
In this Example, I first Divide as if my Divisor were only 8, so that I have 5 Figures in the Quotient, just as if the Dividual Point had stood over 7, the Second of the Dividend: But because of the two Ciphers in the Divisor, I cut off two from the Right of the Quotient, and so I understand, that if 800 Men had to receive, or pay out equalie amongst them 478552 lib. each Man's share would come to 598lib. 3. sh 9 d. and about an half Penny.   

Section 2d. Showing how to Divide by two Signifying Figures  

FIRST METHOD.  
In this and all Operations, where the Divisor consists of more signifying Figures than one, You must set the Quotient to the Right of the Dividend, and the Remainder under the several Respective Figures of every Dividual.  
Observe in all Subductions, that if the Remainder be either equal to, or greater than the Dividual you have taken the Quotient too little, or not set down your Figures right.  
When you are to Divide by any two signifying Figures, having first determined your first Dividual, & Rectified, look for your Divisor on the Fixed, & at that, put in a little Peg to mark it.  
Then with your other Peg or style, bring up that Point of the Movable which is against the standing Peg in the Fixed, once, twice, thrice, etc. till the Number against the Peg, with regard to the Palm, exceed the Dividual, minding only how many times you have brought up that Point, and these times are the Quotient; & wherever you find the Dividual on the Movable, (which is either at the Stop, or betwixt the Stop and the Point, against the Peg in the Fixed) there you have the Remainder on the Fixed.  
As also you must remember, that when ever the Number on the Movable against the Peg in the Fixed, is less than that on the Fixed, that then you are to esteem your hundreds, one more than that Figure is, at which the Palm Points; because if you should bring up that Number to the Stop; the Palm would certainly cast another hundred: Nay, 00 on the Movable, is always to be reputed 100, when it is not precisely at the Stop; and so you may Judge of every other Number on the Movable, when it stands against a greater on the Fixed.  
Example.  
 
Having set my Dividual point, I understand that I am to have 4 Figures in my Quotient.  
I put in a Peg at 15 on the Fixed for my Divisor and (the Movable and Palm being first rectified) I bring up constantly that Point of the Movable, which is Directly against this Divisor; saying once, twice, thrice, and so forward till, at 5 times, I find the Number against my Divisor exceed my Dividual, and then I put in 5 in my Quotient: After which, beginning at the Stop, I search betwixt it, & my Divisor-Peg, for my Dividual 75, and I find it just at the Stop; and against it on the Fixed, finding a cipher for my Remainder; I understand that 5 times 15 is just 75, and therefore I set down 0 under 5, the last Figure of my Dividual, and to it I bring down 9 the next subsequent Figure of my Dividend, for the second Dividual.  
Now because this 2d. Dividual is less than the Divisor, I put 0 in my Quotient; and to the 2d. Dividual as a mere Remainder, I bring down the next Figure of the Dividend, to wit 8, and so I have 98 for a third Dividual.  
Now you may either Rectify, and begin de Novo for 98, as you did for 75: Or, because this Dividual is greater than the last, to wit 75, you may proceed: Bringing up the Divisor once more, and saying 6 times; so have you 6 for your Quotient. And againct 98 on the Movable, you have 8 on the Fixed for your Remainder.  
When you have set down 8 on the Paper under 8, the last Figure of the third Dividual; you bring down to it 6, the last Figure of the Dividend, so have you 86, for a new Dividual, and having Rectified; you find the Quotient 5, and the Remainder 11, after the same manner as you did the first.  
If after this manner, you Divide the Sum of your Pennies in Addition by 12, and that of your shillings, by 20, you will reduce the first to shillings, and the second to Pounds, in the Quotients; and the remainders are pennies, or shillings, according to the Nature of the Sums Divided: The same may be said of every other Species, if it be Divided by its Proper Denominator.   

SECOND METHOD.  
I shall here likeways show you how to do the same by the Tables of common Divisors on the middle of the Rotula; which Method will likeways be sometimes very useful in long Divisions, either by two, or more Figures.  
Suppose then that the Sum of your pennies were 798. You must in reducing them to shillings, Divide them by 12, because 12 is the Denominator of pennies, wherefore having set them down as in the Margin, 
 & Marked your first Dividual, You must look for your Dividual 79 in that Table, whose first Number is 12, & if you can not find 79, you must take that which is next to it, but less; and that you will find to be 72, against which you have 6 for your Quotient; as you find in the Column on the Left of the Tables, under the Letter N. That 6 you put in your Quotient; and then Substracting 72, the Tabular Number, from 79, the Dividual, you set down the 7 that remains under the 9 of the Dividual. And then bringing down 8, the next of the Dividend, to 7 the Remainder, you have 78 for the 2d. Dividual; the next to which in the Table is 72, which gives another 6th. Figure for your Quotient, and 6 for your Remainder; So that in 798 pennies; you have just 66 shilling, and 6 d. The last to set down under pennies, and the first to carry to your shillings.  
After the same manner you reduce shillings to pounds, by the Table, which gins with 20: and Weights & Measures by the Tables of 16, 4, and 28, according to their several Denominators  
You may likeways in long Divisions, to prevent many turn of the Rotula, at one Fetch set down the Nine Multiplies of any two Figures, and so Divide as by the Table,  
Thus.  
In This lesson I put 19 to the Stop, and then against 19 on the Fixed; I have 38 on the Movable, and against 38 on the Fixed, I have 57 on the Movable. These I set down, and proceed, seeking 57 on the Fixed I find 76 on the Movable, and at 76 on the Fixed, I find 95 on the Movable; then I set down 76, and 95, also I find at 95 on the Fixed, 14 on the Movable, and at 14 on the Fixed, I find 33 on the Movable, and at 33 on the Fixed I find 52, these likeways I set down & then in the Last place at 52. on the Fixed I find 71 on the Movable, which having set down, I distinguish my nine Numbers into three by two Lines.  
Now because my 6th. Number is less than my fifth; I understand by that, That here I must add one hundred, which hundred continues invariable till the Unites and Ten, of a following Number grow less than those of a preceding, & then I must add one more to the hundreds,  
Having thus made my Table I search in it, for the nearest Number to 79; my first dividual, and finding 76, The fourth Number I set 4 in my Quotient; and substracting the Tabular Number 76, from 79 I set down the Remainder 3 under the 9, of my dividual.  
To this bringing down 6, I have 36 for my second dividual; for which looking in the Table I find 19, the nearest, which being the first number, I set 1, in my Quotient; and taking 19 from 36, I set down 17 the Remainder under 36, then bringing down 8 of the dividend to 17; I have 178 for my next dividual; the next to which in the Table is 171, which because it is the 9th Number, I put 9 in my Quotient, and the Remainder 7, I set under 8, and so proceeding after the same manner with the two subsequent Figures of the dividend; I find my Quotient to be 41939, & my Remainder to be 16.  
After the Method here, proposed for making your little Table, You may Examine the exactness of the Tables of common Divisors, and so understand, whether upon Occasion you may trust them or not.    

Section 3d. Showing how to Divide by any Number of Figures whatsoever.  
In this part, having first Determined the first Dividual observe whether the divisor and dividual be equal in Number of places. For if they are equal, you must Refer, or Compare, the first two of the Divisor to the first two of the Dividual: But if the Dividual have one Figure more than the Divisor; you must refer the first two of the Divisor to the first three of the Dividual.  
Then you must (either by the first or 2d. Method, of Dividing by two Figures) Find out how oft the foremost two Figures of the Divisor is contained in the Respective Figures of the Dividual, (whether the foremost two or three;) And having found your Quotient, you must set it under the Divisor (at convenient distance before the following Product) with the sign of Multiplication (to wit, ×,) after it, and after that the Letter d, and after that the sign of equality to wit (=) for instance, suppose the Figure found for the Quotient were 7, you must under the Divisor, as you see in the following Example, Set it down Thus, (7 × d =) which signifies that 7 times the Divisor is just equal to the Number that follows the sign of equality: This done, Multiply the Divisor by the Figure found, and set the Unites of the Product under the last Figure of the Dividual, and the rest in Order.  
If this Product do not exceed the dividual, than you are sure that the Figure found is the true Quotient Figure, for which cause you must write it in the Quotient, and then having Substracted the Product from the dividual; you must bring down the next Subsequent Figure of the Dividend to the Right of the Remainder, so have you a Second Dividual.  
But if the Product should happen to be greater than the Dividual (which will sometimes fall out) do not Expunge it; (for it may be afterwards useful) but abate an Unite from the Coefficient already found; so have you the true Quotient, by which when you have found it, Multiply the Divisor, & Subtract this last Product from the Dividual, & Proceed conform to the Directions already given. This is the far shortest Method of any I know for finding the true Figure for the Quotient; & does abundantly compense the little Trouble a Man is at in making sometimes two Products, by freeing a Man from all that tedious Chain of thought, by which he is Obliged to compare every several Figure of his Divisor, with the Respective Figures of his Dividual, especially, when they consist of many Figures. Only when you see the first Remainder, or the Difference betwixt your Tabular Number, and the Respective Figures of the Dividual) palpably too little to make the next Subsequent Figure of the Dividual equal, or answer as many times the third of the Divisor, you may abate an Unite from the Quotient Figure, already found before you make your first Product.  
EXAMPLE.  
 
In this Example, because the Divisor and first Dividual, have an equal Number of Places, I refer 15 the first two of the Divisor, to 76 the first two of the Dividual: And then by the Rotula, or little Table made by the help of the Rotula, I find 5 × 15, or 75 in 76: But then the Remainder, which is but 1, makes the third Figure of the dividual but 12, which being less than 5×9 (the third Figure of the Divisor,) I abate an Unite from the first Quotient, & so take it only 4 times; then I Multiply the Divisor by 4 and disposing the Product duly under the Dividual; I Subtract the Product from the Dividual, & to the Right of the Remainder, I bring down the next Subsequent Figure of the Dividend; so have I a second Dividual. Now the Divisor having but 4 Figures, and the second Dividual 5, I refer the first two of the Divisor to the first 3 of the Dividual, to wit, to 122. Wherefore searching the little Table for 122; I find 120, the nearest to it, which would yield me 8 for my Quotient, but the Remainder 2, makes the fourth Figure of the Dividual, only 28 which is much less than 8 × 9 the third Figure of the Divisor, wherefore I content myself with 7 for the true Quotient and proceed as before.  
In the third Dividual, I find by the foremost 3 figures 109 that I may have 7 times 15 (the foremost two of the Divisor) in that part of the Dividual, but looking back on 7 × d under the 2d Dividual I find, That that Product exceeds the 3d Dividual, for which cause I take 6 instead of 7 for my true Quotient.  
In the 4th Dividual, I have showed you how to do, in case you should chance to take your Quotient an Unite bigger than it ought to be: For finding 9× d greater than the Dividual, I Substitute 8× d under it, & Substracting that from the Dividual; I have 1211 for my last Remainder.  
I have now gone through the 4 common Rules, and if what I have said be well understood, a Man may propose as many Examples as he pleaseth, and perform them with the like ease: And I am confident, that a Man of a very ordinary Capacity may learn what concerns Addition and Substraction of all sorts or Species: Multiplication in Integers; And Division by one or two Figures in the space of one Hour, or at most of one Hour and a half, so that he hath remaining two hours and a half, for accomplishing this last lesson of Di●i●on by more Figures, which I doubt not but he will be able to do in shorter space, so that conform to what I undertake, the Scholar learns the 4 Com-Rules in the space of 4 Hours.  
But before I leave this, & to show you how Copiously useful the Instrument is, I will set you a Method of Dividing by many Figures, in which you may (without Table or setting down a Product) come both at your Quotient and Remainder, after the same manner as you ought to do, if you were Dividing by the Pen.  
To this purpose you may provide your self at first with a Label of Cart or Paper, writing on the right edge of it, the Figures beginning at 0, 1, 2, and ending at 9, 
PREFIXES 0 1 2 3 4 5 6 7 8 9  This we call the Label of prefixes as you have it in the Margin.  
Now suppose the following Example were proposed, having first rectified the Rotula, & with a Point determined the first Dividual proceed after this manner.  
 
First I bring 15, the foremost two of the first Dividual on the Movable, (with a little Peg to stay there till I have found my Quotient,) to the first Cell, in which 1 (The foremost Figure of my Divisor) is a Coessicient, and there I have in the same Cell, 9 for a Quotient, and at the Stop, 6 for a Ptefix; wherefore I lay 6 on the Label of Prefixes over 15, and so (0) the third of my Dividual, becomes 60: Now because 9 is my Quotient, and 9 is also the 2d. Figure of my Divisor, I look for 9 × 9 on the Fixed, and finding it exceeds 60 the 2d: Number of my Dividual. I therefore shift the Point with the Peg in it forward, to the next Cell of (1) and there having 8 for my Quotient, and 7 for a Prefix, I altar my prefix to 7, and then consider, whether this Quotient 8 × 9 the 2d. Figure of the Divisor, exceeds 70, the respective number of the Dividual, & because 8×9 is greater than 70, I shift my standing Peg once more to the next Cell of (1) & there having 7 for a Quotient, and 8 at the Stop for a Prefix; I am confident, (that the Remainder being greater than the Quotient) 7 will serve for all the following Figures of the Divisor, and so I put 7 in my Quotient, then laying aside your Label till you come to seek a new Figure for the Quotient. You must in the Subduction observe to take 7 times every Figure of the Divisor; beginning at the last from the respective Figures of the Dividual, that is to say the last from the last, and the last save one, from the last save one, and so forward till you take 7×, the foremost of the Divisor, from the foremost one or two Figures of the Dividual, according as they fall out to be one or two. Thus, I first rectify, and take out the Peg, & because I must take 7×6 of the Divisor from 6 of the Divial, I with my Peg bring up 7×6 on the Movable to the Stop, and searching for a Number on the Movable, having 6 Unites; I find the first that occurs to be 46 by which I understand, that I am to carry 4 to wit, the ten of 46, & because I have 4 on the Fixed, against 46 on the Movable, I understand, that I must set down 4 as a Remainder under 6 the last Figure of my Dividual, and so turning back the Rotula till 4 on the Movable appear against (0) at the Stop; I thereafter write down 4 below the 6 of my First Dividual.  
Then I bring up to the Stop, that Point of the Movable, which is Directly against 7×8 of my Divisor, and looking as before for 3 Unites I find 6 ten to, carry, and 3 on the Fixed to set down, these carried, and set down as becomes I next bring up 7×9 of my Divisor, & at (0) Unites on the Movable, I have to carry 7 ten, and on the Fixed (1) to set down under (0) of my Dividual, of these the 7 carried, and the (1) set down.  
In the last place I bring up 7×1S of 〈◊〉 Divisor, to the Stop, and on the Fixed I ●●ve 1 (against 15 the foremost two of the Dividual) for my Remainder, 〈◊〉 1, I set under 5, the Unites of the 〈◊〉 ●nd so having finished the first Subduction, I rectify and bring down 9 of the Dividend, to the right of the first Remainder, for a second Dividual.  
If you comprehend what I have said on the first, you may easily, and after the same manner go through with the other there Dividuals, and so perform the whole business yourself.  
If the Practitioner curiously observe the several Operationes by the Rotula, he will discover them to be so Natural, that with a carrefull Practice, he may come to such a Habit as will render him capable to do his business with the Pen when he wants his Rotula. Tho' I must confess; That the ablest Masters are not capable without it, to do any considerable business with that dispatch, Certainty and Exactness, and with so little Trouble to the Head; as by the help of the Rotula, they may perform with the greatest ease.  
If any difficulty Occur, in what hath been hitherto delivered, such as have the Opportunity, shall not want what help I can afford them. But because in what followeth, a Previous knowledge in Decimal Fractions, is supposed; If such as want that find any difficulty, they must be at the pains, or use a Master, as they find most convenient, for the attainment of that knowledge before they make any further or Considerable Progress.    

CHAP. VI Concerning COMPLEX NUMBERS.  
I Call those Numbers Complex, that consist of Intigers and Fractions, or small Denominations, such as lib. sh. d. or Stones lib. of Weight and Ounces. Chalders, Bolls and Pecks, etc. Elnes, Half quarters, or Eight parts, etc.  
If instead of these Fractions, or Denominations, you annex the Decimal to the right of the proper Integer distinguishing betwixt them with this mark (〈◊〉) called a Decimal Line, you may Multiply or Divide, as if the whole Number represented by all them Figures were an Integer. In adding Decimals, You must take care that those Figures next the Decimal Line make one Column, & the rest in order. Thus were I to Join the Decimal of 7sh. which is to the Decimal of 9d. which is I must state them Thus  
And so the Decimal of 7sh. 9d. is  
Observe that the Decimals for Pence and Farthings, where the last Figure is ● or ● as also those for third parts or sixth parts (whose last Figures are likeways ● or ●) are all Infinites; Thus the Decimal of one d. is the last Figure of which may be reiterated in Infinitum, or as oft as the Rules of Operation do require; for which I refer you to my Compendious, but complete system of Decimal Arithmetic: But least that should not come to your hands I shall here subjoin a few Rules.  
First you must not limit Infinites short of Decimal thirds.  
2ly You must not limit infinites unless they contain one of the Reiterated Figures; Thus because the Decimal of 4d. is etc. I may upon occasion satisfy myself with and 8ds. being, cann't satisfy myself with but as also the Decimal for one penny being etc. I can satisfy myself with no less than  
3ly Infinites in Addition and Substraction, must exceed the longest finite at least by one Step towards the right Hand, thus, were I to Join one penny viz.  
To three farthings viz.  
The Sum would be  
In which you may observe, that I have Reiterated the last Figure of the Decimal for one d. twice.  
4ly. Infinites must in Addition & Substraction, be equal in Number of places; Thus were I to set down a Decimal for 4½ d. tho' might serve for 4ds. yet because the decimal of ½ d. is  
The decimal of 4ds. is  
So that the decimal of 4½ d. is throwing away the cipher after the Decimal, as a thing of no value.  
Lastly, The Sum, difference and Product of infinites is infinite, unless they end in a cipher.  
Wherefore the Sum, Difference and Product of the last Column or Figure of an Infinite, must be reckoned on the Segment within the Peg holes of the Movable, and not on the whole Circle without; Thus in the last Example you see that 6 & 8, which make 9 on the ●hole Circle, make 10 on the Segment; & therefore I set, 0, and carry one; Put in all the rest of the Figures, I regard the whole Circle, but not the Segment; the ●eason, of which is manifest from the 1st. Chapter of my Decimal Systeme.  
In Cutting of your Decimals after Operation, you must observe.  
First that the Decimal of the Sum, or difference, must in Number of places, be equal to the longest Decimal of the Items.  
Secondly, That In Multiplication the Decimal of the Product, must be equal, in Number of places, to those of both Coefficients.  
Thirdly, That in Division, the Decimal of the Dividend alone, must equal those of the Divisor & Quotient both together.  
Fourthly, All Numbers, either Actually have a Decimal annexed to them, or must be supposed to have as many Decimal Ciphers as may be necessary, nay Finit Decimals themselves, must sometimes be supposed to have Ciphers following them; and on the other Hand Integers, must be supposed to have Ciphers to the left: For these Additional Ciphers do neither increase, nor diminish the true value of any Number.  
Multiplication by such pure Numbers as 10, 100, 1000, etc. Is done merely by shifting the Decimal Line of the Multiplicand, so many Steps nearer the Right, as there are Ciphers in the Multiplier; only if the Multiplicand be Infinite you must Reiterate the last Figure. Thus were it Demanded; How much would pay 100000 Men, to give them 3 Farthings a Piece? Now the Decimal of 3 Farthings is , and because there are five Ciphers in the Multiplier the Answer is So that 312lib. 10sh. will just pay 100000 Men at 3 Far things per Piece; here you have no Reiteration, because the Decimal is sinite.  
But at 4d. per Piece, the Decimal of which is an Infinite, to wit, You must Reiterate the last Figure, and then you'll find the Answer for ten Men to be That is 3sh. 4d. But for 100 That is 1lib. 13sh. 4d. and for 1000, . id est 16lib. 13sh. 4d. for 10000, id est 166lib. 13sh 4ds. after the same manner you may by the decimal Tables for pence and shillings, at one look turn any pure Number into their Proper Integers; Thus were 7000d. to be turned into Pounds, take the Decimal for 7d. to wit, & by shifting the Decimal Line one Step, you have the Value of 70d. to wit, which is 5sh. and 10d. But if you shift it two Steps to the right Hand, you have the Value of 700d. namely which is 2lib 18sh, 4ds. and if you shift it 3 Steps, you Reduce 7000d. to their proper Pounds and shill, namely that is 29lib. 3shill. 4d.  
I have contrived a little Pocket-Book whereby you may with the same ease convert any pure Number of the Species Current in this Kingdom, into pound Scots or Sterling, at one look.  
You may use the Table of Ounces to the same purpose, in turning Drops to Ounces, Ounces to Pounds, and Bolls to Chalders; All which requires an exact knowledge of Decimals. Example; The Decimal for 6, Ounces or Pecks, is and consequently 60 Pecks makes that is 3 Bolls, 12 Pecks, but 600 Pecks make that is 37½ Bolls & 6000 pecks are just Bolls.  
In Division by 10, 100, 1000, etc. Just contrary to Multiplication, you must remove the Decimal Line so many Steps nearer the jest, as there are Ciphers in your Divisor: Thus were 2916lib. 13sh. and 4ds. to be Divided amongst 10, 100, 1000, or, 10000 Men. You must first set down the Dividend and then the several shares of 10, 100, 1000, 10000, or, 100000, will appear as followeth.  
Divisor 10 is 291lib. 13sh. 4d 100 is 29 3 4 1000 is 2 18 4 10000 is 0 05 10 100000 is 0 00 7  
I must refer such as desire further Satisfaction in this particular to my Compendious, but complete System of Decimals   

CHAP. VII. Concerning the Rule of 3.  
IN the Questions of this Rule, there are always three Numbers, either expressly given or supposed, to find a Fourth.  
Of the given Numbers, there are always two of the same sort, Species or Notion, & these we shall call Relative Numbers, & the other or third Number, which falls under a different Notion, we shall call the singular Number: and the fourth Number demanded (which is always of the same Species, or Notion with the singular Number) we call the Answer; because, when it is found, it Answers the Question.  
Now to save you the trouble of two Rules, one Direct, and another Reverse: I shall lay down such easy Directions, as may render a Man capable to Answer Questions of either, without any such Distinction.  
In discerning the Divisor, (which is all the difficulty, and which must always be one or other of the Relative Numbers) you must carefully consider, whether the Answer ought to be greater, or lesser than the singular Number For,  
1st. When the Answer ought to be greater than the singular Number, than the least of the Relative Numbers must be the Divisor.  
2ly. But if the Answer ought to be less than the singular Number, than the greatest of the Relative Numbers must be the Divisor.  
Having discovered the Divisor, set it down next your Lefthand, and the other two, (which we now call Coefficients) at a convenient Distance; the one, (it matters not which) before, the other after [∷] which we call the sign of Proportion.  
The Numbers thus disposed, Multiply the Coefficients and Divide the Product, by the Divisor and the Quotient, yields you the Answer; Only observe, that when the Divisor is an Unite, there will be no use for Division; and on the other hand, when one of the Coefficients is an Unite there, will be no Multiplication.  
As also, if one of the Coefficients have an Infinite annexed to it, be sure to make that the Multiplicand.  
The following Examples, shall Illustrate what hath been said.  

EXAMPLE, 1st. Concerning the Prices of GOODS.  
IF 3 els cost 10 L. what will 17¾ els cost? In this Example 3 and ●7¾ are the Relative Numbers, because they are both of the same sort, to wit, els, and 10 is the singular Number; ●ow because the Price 17¾ els, must ●●e greater than the price of three els to ●●it, than 10 L. that is the Answer; ●ust be be greater than the singular ●●umber; I thereby understand, that 3 ●●e least of my Relatives must be the Divisor, for which cause I set the Numbers as followeth. Taking instead of ¾ its Decimal .  
So that the Answer is 59 L. 3sh. 4d.  
You see I first Multiply the Coefficients, & then Divide the Product by 3 the Divisor, so have I the Answer: You may convert the Question, and so prove your Work, Thus,   

2d. EXAMPLE.  
If 17¾ els cost 59 L. 3sh. 4d. what will 3 els cost?  
 
You see in the Division, that I ought to have 3 Figures in my Quotient, as the Point intimates, of which one ought to be a Decimal, but that being a cipher I did not think it necessary to set it down.   

3d. EXAMPLE.  
At 3 sh. 4½ d. per lib. what will lib. of any thing come to?  
The Decimal for 3 sh. 4 pence is and that for ½ d.  
So that 3 sh. 4½ d. is  
Then conform to the Rule, the lesson will stand as followeth.  
 
You see in the lesson, there is no Division, because the Divisor is an Unite.   

4th. EXAMPLE.  
If 345½ els cost 58 L. 6 sh. o ¾ d. what will one Ell cost?  
 
 
In this Example, you see that one of the Coefficients, being an Unite: There is no Multiplication.   

5th. EXAMPLE.  
If ¾ els cost ⅘ lib. what will ⅚ els cost?  
 
Answ. 18 sh. 3 ¼. d. & some little more.  
Because the last Dividual is the same the Quotient is Infinite.  
In this last Example, the Decimal o● ⅚ being Infinite; The Product of the dash'● Figure, is reckoned on the Segment, but the Product of all the rest, is reckoned on the Integer-Circle,  
Item, because the Dividend is Infinite, in Prolongation of the Work I have Reiterated the last Figure.   

6th. EPAMPLE.  
At 5½ per Cent, what will 347 lib. 13 sh. 4 d. pay per Annum?  
In this lesson, observe, that tho' all the Numbers be of one Denomination or kind of things, yet (the 5½ L. falling under a different Notion, to wit, that of Interest; whereas the other two, to wit, the Cent. or 100, and 347 L. 13 sh. 4 d. are Principal Sums) the 5½ is the Singular Number, and so 100 is the Divisor.  
 
In this you have 6 Decimals in the Answer to wit, four for those of the Coefficients, and two for the Ciphers of the Divisor.   

7th. EXAMPLE.  
At 4 sh. 10½ per Crown, how many Pounds Sterl. must one have for 758½ Crowns.  
 
Answer. 184 lib. 17 sh. 8¼ d.   

8th. EXAMPLE.  
If 40 Men are able to finish a piece of Work in 8 Days, how many may do the same in 5 Days? here, because the Answer ought to be greater than the singular Number 5, the last of the Relatives is the Divisor, and so you are free from the cumbersome Reflections, on a Reverse Rule.  
  

9 EXAMPLE.  
If the penny Loaff, aught to Weigh 18 Ounces, when the Wheat sells at 10 sh. Sterling. per Boll, what ought the same to Weigh, when the Wheat sells at 15 sh. per Boll;  
 
In Questions of 5 Numbers, you have, for the most part, two pairs of Relatives, and but one singnlar Number, wherefore you may take any one of the pairs of 〈◊〉 with the singular Number, to find the first Answer, and that Answer will serve for a singular Number, for the other pair of Relatives in finding the last Answer.   

9th. EXAMPLE.  
At 5 per Cent per Annum, what will 197 L. 15 sh. 8 d. come to in 7 Years? In this the Relatives are 100 L. and 197 L. 15 sh. 8 d. for the first pair and, 1, Year and 7 Years for the 2 d. pair, and 5 is the Singular.  
 
So that the Answer is 76 L. 12 sh. 9½ d. and very little more.   

10th. EXAMPLE.  
At 2 Dollars per L. Flemish, Exchange at 35⅚ sh, Flemish per L. St. how many L. Stir. will 666⅔ Dollars come to?  
 
In this Division, finding my 3 d. Dividual, the same with the 2 d. and, (because of the Reiterated Figures of the Dividend) understanding that it will always be the same in Infinitum, I therefore Reiterate the 2 d. of my Figures in the Quotient; to wit, 6, till I have five Figures in all, as the Dividual-Point Intimates: And seeing I have 3 Decimal in my Dividend, and but one in my Divisor, I understand that the last two, of my five Figures in the Quotient, must be Decimals, and the dashed Figure is added, because, (as hath been already said) Infinites cannot be limited under Decimal thirds; so that my 666⅔ Dollars, makes just 266 L. 13 sh. 4 d. Flem. for my first Answer, and this must be one of the Coefficients for the 2 d. operation, because in this we have now two Parcels of Flemish Money, and but one L. Sterl.  
 
In this last Operation the third Dividual. viz.. 0000 being less than the Divisor, I put 0 in my Quotient, and so the Answer is 150 Lib. Stir.   

11th. EXAMPLE.  
At 5 L. 10 sh. per Cent per An what will the Interest of 756 L. 13 sh. 4 d. amount to in 7 Years, and 7 Months?  
Vide. my Compend. system anent Multiplicati-L. In. = on, with an Infinite in both Coefficients.  
The Answer is 317 L. 11 sh. 10 ½ d. very near   

12th. EXAMPLE.  
Interest at 5 ½ per Cent. What will 456 L. 13 sh. 4 d. (payable 3 Years hence) be worth in present Money?  
 
Add this Interest to its principal, and the 2 d. Operation will stand thus.  
Ans. 378 L 11 sh. 8 d. & less than ¼ d. more.  
I shall conclude with an Example of unequal Division, which may be very useful in fellowship.   

13th. EXAMPLE  
There is to be Divided amongst 14 Men 458 L. 5 sh. 11 d. with Proviso, that 9 of the Number (whose stocks or hazards were equull) have equal shares, but the other 5 are to have, one of them ½ share another ⅓ another ¼ another ⅙ another ⅚ of an equal share: The equal share, and consequently the several Fractions of the equal share, is demanded?  
In this you must add the Decimals of the several Fractions in the Question to 9, and so you will find your Divisor to be and not 14 thus  
The Sum of of all which is  
Hence the Question must be thus stated.  
 
If any Difficulty occur in these lessons, it may be easily overcome, and at a very Reasonable Rate, by a little converse with the AUTHOR.    
FINIS.   

ERRATA. Page 20, Line 4. read 8 for 6. P. 32. L. 8. R. 9276. for 8276. In the end of the Product. also the Example Page. 38. is only disorderly set ●r the 9 of the Quotient aught to stand under the 2d. of the Dividend. and the rest in Order.   

At Edinburgh, the Twenty eight Day of November, one Thousand, six Hundred Ninety Nine Years.  
THe Lords of His Majesty's Privy Council Having by their Act of the first of December, one Thousand, six Hundred, Ninety Eight Years, Granted to Mr. George Brown Minister, and his Heirs and Assign is the sole Privilege of framing, making and selling his Instrument called ROTULA AR●●METICA; For the Space of seven Years: Fre●● the Day and Date of the said Act. And Discharged any other Persons to make and sell the said Instrument, during the Space foresaid without express Liberty and Licence, from the said Mr: George and his foresaids: under the Pain of five Hun●●ed Marks; besides the Confiscation of the Rotvlas, Mice or S●ld. The said Lords of his Majestics ●ivie Conncil Do ●ereby Discharge the Imperting of the said Instrument, or ROTULA; During the Space foresaid, also well as the making or Selling thereof: Under the said Pain of Five Foundered Marks: 〈◊〉 the Cons●●cation of the Infir●ments or ROTVLA'S Imported. Also well as these made or Sold. And declares, this present Act, to the Commence from the Date of the former Act: Which is the first Day of December: One The●●●nd, s● H●nared, Nint●e Light Years.  
Extracted by Me, Sic subscribitur, Gilb. Eliot Cls. Sti. Cons.   

THE Principles of Geometry, Astronomy, and Geographie.  
Wherein is briefly, evidently, and methodically delinered, whatsoever appertaineth unto the knowledge of the said Sciences.  
Gathered out of the Tables of the Astronomical institutions of Georgius Henischius.  
By Francis Cook.  
Appointed publicly to be read in the Staplers Chapel at Leaden hall, by the Wor. Tho. Hood, Mathematical Lecturer of the City of London.  
AT LONDON Printed by john Windet, and are to be sold in Mark lane, oner against the sign of the red Harrow, at the house of Francis Cook.   

To the Worshipful well-willer and practiser of the Mathematics, M. Bernard Dewhurst.  
YOur courtesies (worshipful) inexpected, have brought me very fare in arrearage with you, and I have thought the time over long, until in some sort I might break silence, and say so. But better late than never, as the proverb goeth, and yet perhaps as good not at all, as not as it should be. I have intended many ways according to the courage of a scholar, to be even with you, or at least to show some sign of gratitude, but it would not be: and therefore have I rested so long upon hope of some sit opportunity, which now being fitly offered, I have accordingly taken hold of: I present therefore this little book unto your Wor. little I mean in body, but full of substance and matter: according to the commendation which the Poet giveth to Tydeus, the Father of Diomedes, and that was that Tydeu● Ingenio magnus, corpore paruus erat. A little fellow, but full of edge. I commend it unto your Wor● although the praise thereof shall rather proceed from itself, then from me, unless I could praise it worthily enough. I found it else where, and otherwise attired, and my labour and charge is only in this, that I have bestowed thereon a new coat after our English fashion, observing the matter, only altering the manner. Accept it I pray you none otherwise than I mean it, not as a gift worthy enough, but as some little sign of a thankful mind, which according as power shallbe answerable unto any good occasion, I will more manifestly declare in some greater matter.  
Your Worships wholly. F. Cook.   

To the learned and worshipful. Tho. Hood Mathematical Lecturer of the City of London.  
AMong many your scholars whose diligence have made them more able: I have adventured to put forward myself, being of all other most unfit and insufficient. My purpose herein is not the gaining of mine own commendation, but to communicate that with others, which I find beneficial unto myself. The commendation is wholly yours, from whom as from the Sun we receive, as in these exercises, all the light we have. And hereunto may be added another cause of this my bold enterprise, namely, that others, who are better furaished, may by mine example, being a novice in these studies, make proof unto the world, that your labours are not in vain, but that there are many which have greatly profited thereby, and which in due time (I hope) will break forth, both to the great commodity of the common wealth, and commendation and credit of the Mathematical Lecture. Take in good part (I pray you) these my first fruits of your own planting, and make favourable construction thereof, correcting the faults, and pardoning my boldness.  
Your Scholar in this ease more forward than able. F. Cook.   

To the loving and diligent Auditors of the Mathematical Lecture.  
GEntlemen and Goodsellowes, and to loin you both in one, men and school-fellowes: I have as you see adventured to she, although both fitted with the feathers of another, and lying but a low pitch, for want of wings. If I shall see me to any of you too forward, I crave l●s pardon, it is my first salt, I meant no ●urt: If any man will charge me of defects, I will confess my wants, and submit myself ●nto any reasonable censure. Howbeit I look for no hard measure from any, frequenting these exercises, either in the school or abroad, masmuch as they are for the most part tame beasts that belong to this fold: what the wild and savage fort can or will do I fear not, I can appeal from them, as in this case, no competent judges. I have after my manner dealt plainly with the matter, and I pray you take it as it is, masmuch as howsoever it the serveth, I can set it out no better, For as one saith.  
They that are learned and have the gift,  
may make of matters what they will,  
But he that hath none other shift,  
must go the plain way to the mill.  
As in this I have done, Far you well.   

¶ The descriptions of Geometrical and Astronomical terms.  

Of Magnitude. Chap. 1.  
THere be 3. kinds of Magnitudes, a line, a surface, a body.  
In a line, called by the Greeks' 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, we are to consider the definition, the terms, the sorts.  
The definition of a line is two fold: For it is defined to be either a length without breadth, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, or else the flowing of a point in length.  
The terms of a line are points: and here we are also to note, the distinction of points, and their denomination.  
They are distinguished into points Geometrical, and Physical.  
Geometrical points are such as have no part.  
Physical points be such as may be apprehended by our sight, of which sort are motes in the Sun shine.  
Points are denominated either centres and poles, whereof the principal are those of the world, of the Zodiac and of the Horizon: or else the Equinoctial, and the Solstitial points.   

For the seuerall sores of lines look in the 2. Chapter.  
In a Surface, in Greek 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 we are to attend the definition, and the terms.  
It is defined to be either an extended and broad extremity, or else longth, and breadth without depth.  
The terms of a surface are lines, whereof in the Chapter following: as also of the several sorts of surfaces in the 3. Chapter.  
A body is defined to be breadth, having depth adjoined thereunto: or a figure consisting of 3. dimensions.  
The bounds or limits of a body are surfaces.  
There be many sorts of bodies: whereof in the latter part of the book.   

Of Lines. Chap. 2.  
Alone is either right or crooked.  
In a right line note, the definition, and the divers kinds thereof.  
A right line is defined to be, either the shortest extension from one point unto another: or else, the shortest of those lines that have the same limits.  
There are 4. sorts of right lines.  
The first divideth a figure into 2. equal or unequal parts.  
Right lines dividing a figure into 2. equal parts, are either diameters, namely in a square, or in a circle: or Axes, as in a Sphere: or else diagonales, as in Polygons.  
Right lines divide a figure into 2. in equal parts, as chords in circles, the halves whereof are called Sines.  
The second fort of right lines are those that bond the figure, whereof those that limit the upper part, are called Corausci: those that bound the nether part, are called the Races: those that include the lateral parts, are called Costa, the side lines.  
The third sort of right lines are such, as are elevated either perpendicularly, or not perpendicularly.  
Right lines perpendicularly elevated, are called perpendiculars, plumb lines, squire lines, orthogonall lines, or lines at right angles.  
Right lines not perpendicularly elevated, are called Hypothenusalles or subtendent lines, also obliqne lines: likewise visual lines, or visual beams.  
The fourth fort of right lines, are such as are equidistant, as parallels.  
In a crooked line consider the definition, and the sorts thereof  
It is defined to be the running of a point unto a point, not by the shortest, but by a longer way.  
The sorts thereof are many: whereof some are simple, and some mixed.  
Simple crooked lines are such as are made by a point running round, as in the circumference of a circle, of a semicircle, and of an ark of a circle.  
Mixed crooked lines are such as compass not a circle: and they are either winding, or spiral.  
Winding lines are such as in their parts are inequallie elevated from the midst.  
spiral lines are such as being wound round about either some centre in a place surface, or some pillar, as we may see in cylinders and the screws of presses, do never return unto the same point from whence they began.   

Of Surfaces. Chap. 3.  
SVrfaces are either plane, or Spherical.  
Plane surfaces are such as are limited equally either with right, or else with crooked lines.  
Plane surfaces limited equally with right lines, are such as have either 3. or 4. or more right lines for their limits.  
Plane surfaces equally limited with 3. right lines, are triangles, which have their denominations either from the lines that enclose them, and then they are said to be of equal sides, of equal feet, of inequal sides and feet: or else from their angles, and so they are said to be right angled, obtuse angled, acute angled.  
Plane surfaces equally limited with 4. right lines, are either paralle og●ammes, whereof some are squares, some long quadiangles, some Rhombes, some Rhomboides: or else, they are Trapezia, or Tables.  
Plane surfaces equally limited with more them four right lines, are of many sorts, as a Pentagon, an Hexagon, etc.  
Plane surfaces equally limited with crooked lines, are circles, whose parts are called arkes, portions, or sections, a semicircle, a quadrant, a segment, or a sector of a circle.  
Arkes are parts of the circumference of a circle separated by chords.  
Portions or sections are the greater and less surface of the circle, distinguished by a chord.  
A semicircle is that which is contained under the half circumference, and the Diameter.  
A quadrant is the fourth part of a circle, included within 2. semidiameters.  
A Segment or sector of a circle, is a figure contained under an ark of the circumference, and 2. right lines drawn from the centre.  
Spherical surfaces are such, as are limited and contained under inequal, that is, under depressed and elevated lines.  
Spherical surfaces are either convexe, or concave.  
Conuexe spherical surfaces are such, as do bound the body on the outward part.  
Concave spherical surfaces are such, as do limit the body on the inward side.   

Of Angles. Chap. 4.  
AN Angle is made by the alternate or cross meeting of lines or surfaces.  
An Angle is either superficial, or .  
In a superficial angle there are to be considered, the definition, and the division.  
A superficial angle is defined to be the touch of two lines, the one inclining to the other in one surface.  
Superficial angles are divided two ways, for they are considered either by themselves, or reatively.  
Superficial angles considered by themselves, are either plane, or spherical.  
Plane superficial angles are such as are drawn upon a plane surface, and they are either right lined, or crooked lined, or mixed.  
Right lined plane superficial angles are such as are made by right lines only, and those lines are either perpendiculare, whereof a right angle is made, which is always equal unto the right angle next adjoining unto itself: or not perpendiculars, whereof is made either an acute angle, less than a right angle, or an obtuse angle, greater than a right angle.  
Crooked lined plane superficial angles are such, as are made of crooked or bowed lines only.  
Mixed plane superficial angles, are such, as one right and one crooked line doth make.  
Spherical superficial angles, are such as are drawn upon the convexe surface of the sphere, differing according to the compass of the greatest circle, described from the very top of the section.  
Superficial angles are considered relativelye when they are compared with others, in whose respect they are called joint angles, vertical, alternate, and opposite angles.  
joint angles are such, as a right line falling upon a right line, maketh on either side.  
Vertical angles are such, as the mutual joining together of two lines, doth make on contrary parts.  
Alternate angles are such as one line falling upon 2. doth make both on the right and least hand of either of them, aswell within as without.  
Opposite angles are those that can have no relation unto any one of the former.  
A  angle is that which is considered in  bodies, contemned under more than two plane angles, not situated in one and the self same surface.   

Of Bodies. Chap. 5.  
THe kinds of the third magnitude which is called a body, are divers: some be regulare, others are Irregulare.  
Regulare bodies are such as are limited by equal surfaces, the which surfaces are either turned round, or folded one toward another.  
The surfaces turned round, are either the equal sections of circles, or right lined figures.  
Equal sections of circles, are such as either sil up the whole plane, and by them the spheres are made, or else they are cut out hollow in the midst, whereof are made orbs, either uniformed or difformed.  
Right lined figures limiting regulare body's, are either right angled triangles, whereof Pyramids are made, whose upper part is called the Cone, or top, and the nether part and plane surface is called the base, or square or quadrangulare figures on the one side longer, from whence are derived figures long and broad, as pillars, or cylinders: or else they are of other sorts, which are infinite, from whence divers forms and kinds of bodies are drawn.  
The regulare bodies contained under surfaces, folded one toward another, are only these 5. the Tetraedrons', the Hexaedrons', the Octaedrons', the Dodecaedrons', and the lcosaedrons.  
Irregulare bodies are such, as inequal surfaces do limit and describe, the which surfaces are either turned round, or folded one toward another.  
The surfaces turned round and making irregulare body's, are either the sections of circles, or else they are inequal right lined figures.  
The sections of circles, are either greater than a semicircle, whereof the lenticulare bodies are made: or else they are less than a semicircle, and thereby are the Onalles made.  
The inequal right lined figures by whose conversion the irregulare bodies are made, may be of what sort soever, whereby divers kinds of vessels are framed, either wanting or exceeding the regulare form.  
The Irregulare bodies màde of inequal surfaces folded one toward another, may differ infinitely.   

Of the name and definition of the Sphere. Chap. 6.  
IN as much as we make often mention of the Sphere, and thereafter do entitle this present treatise the doctrine of the Sphere, it shall not be amiss to declare the name, and the definition thereof.  
The name is used in divers significations.  
1. It signifieth sometime any  regulare body, limited with one surface only.  
2. Sometimes it signifieth an instrument that doth instisie the apparences of heaven, and containeth the celestial circles, and is otherwise termed a ring or material sphere.  
3. Sometimes it signifieth the whole world, whereunto all the conditions of the sphere may be applied: For it is a  body, wherein nature abhorreth that any emptiness should be giu●: It hath a spherical form running daily about his own Axis without intermission: It hath a point placed in the midst thereof, namely the earth.  
The definition thereof, as it signifieth any body, is by Io. desacro baseo, set down two ways, the one after Encl. 11. Elem. the other out of Theodosius.  
The definition thereof taken out of Euclid, containeth the Geometrical description of the sphere: For the sphere is described by the fixed and unmoved diameter, and by the a●ke of the semicircle, which must be fully brought about.  
The definition of a sphere according to Theodosius, determineth, first the orbiculate form, every part whereof is equally distant from the centre: secondly the principal parts, as the convexe surface which is but one, and the centre, that is, the point in the midst equidistant from every part of the surface, and the Axis, about which the sphere is turned, and which is limited by the 2 poles. viz. The North pole or pole Arctic, and the south pole or pole Antarcticke: and thirdly, the solidity: For it is a complete body, having all the dimensions.   

The division of the celestial Sphere. Chap. 7.  
THe celestial sphere, according to Io. de sacro bosco, admitteth a double division, according to substance and according to accident.  
The celestial sphere considered according to the substance, is divided into several orbs, in the which we are to note the number, and the cause.  
The number is diversely set down: For the ancients contenting themselves with 8. orbs only, did distinguish them into the orbs of the 7. Planets, viz, of ♄ ♃ ♂ ☉ ♀ ☿ and ☽, and the firmament of the fixed stars: And the later Astronomers unto the time of Alphonsus, into 9 orbs: but the Modern, among whom Purbachius was the first, added the tenth.  
The cause is considered either in the diversity of their number, noted both by the former and later Astronomers, or in their order.  
The ancient Astronomers noted their divers number, either by the brightness of the Stars, reckoning so many orbs as they perceived to contain any stars: or by the peculiar motion of each several orb, reckoning so many orbs as they found simple motions belonging thereunto.  
The later Astronomers, for instruction and the better reconing sake, added the ninth and the tenth, as circles necessary for the understanding of the motion of the 8. sphere, unknown unto the ancient Astronomers.  
The order is proved, 1. by the slower motion of the higher, and the swifter motion of the lower orbs: 2. by the occultation or hiding of the higher stars, by the lower: 3. by the diversity of apect, either great, or little, or insensible.  
The celestial sphere considered according unto accident, that is, according to the situation of the heaven, or the course of the stars, is distinguished into a right, or an obliqne sphere.  
The right sphere belongeth unto those that dwell under the Equinoctial, who (by reason that the poles of the world, about the which the stars are carried by the firste movable, have none elevation, as also for that the Horizon cutteth all the parallels, under which the stars do go, at right angles) perceive no reflection in the diurnal motion of the stars.  
The sphere is said to be obliqne, wherein the O and the rest of the stars are carried from the East into the West, by an obliqne motion, and it is Septentrional unto those that have the North pole elevated, and Meridional unto those under the Southern elevation.   

The partition of the whole world: and the comparison of the celestial, with the Elemental Sphere. Chap. 8.  
THe whole frame of the world is made of some certain and those more principal and notable parts, whereof there is first the number and the name, and then the difference to be considered.  
The number and name is double: For the parts are either ethereal, or sublunare.  
The aethereal that is, the celestial part (without the which Philosophy admitteth nothing to be, although the Divines do add the third, which they call Angelical, and the Platonics, intellectual) is that, whereof we entreated in the 7. Chapter.  
The sublunare is that which containeth the elemental bodies, and those either simple, as the fire, the aer, the water, the earth: or else mixed, which are divers and innumerable, engendered of the 4. elements, either perfect or imperfect.  
The difference or dissimilitude of the parts of the world is that, whereby they are distinguished one from another, either in respect of their situation, or of their dignity, magnitude, motion, or their office.  
They are distinguished according to their situation: For the celestial part hath obtained the higher place, the elemental the lower.  
Their distinction according to dignity, is noted in the parts contained by the celestial Region, which parts are bright and immortal, and by the elemental region, those parts being of their own nature obscure and decaying: or else, it is noted in the parts containing, whereof the one is altogether with out alteration, neither increasing nor diminishing, the other is continually subject unto generation and corruption, and is increased and diminished.  
Their distinction according to their magnitude is considered, in that the celestial part with the great compass thereof, doth cover all things, like a thing without measure and end: but the elemental part is covered within the compass of the heaven, the diameter thereof containing the diameter of the earth, 23. times.  
Their distinction according to their motion, is in that the celestial part hath a circular, and a spherical motion: the elemental, a right motion, more imperfect than the circular.  
Finally, they are distinguished according to their office? For of those things that are engendered in the elemental part, the heaven, working by a continual motion, is as it were the formal and efficient cause, from whence life is derived: and the elemental part, which is subject unto passion and alteration, is as it were the material cause, from whence nourishment doth proceed.   

The reason of the sublunare, or elemental Region. Chap. 9  
THe Elemental region, which the heaven encompasseth, comprehendeth within it the elements, wherein we are to consider the definition, the number, and the situation or order.  
The elements are simple bodies, aswell in respect of the mixed bodies which are understood to be compounded of them, as of the simple and least parts: as also in respect of the division, for that they cannot be divided into bodies of diverse kinds (if they be given pure and without mixture. For the use of living creatures, and things growing doth make them impure.)  
The elements are 4. in number, found so to be, by sense, and by reasons.  
The elements are found to be sour by sense (which the Physicians do sollow): First for that more simple bodies cannot be showed: 2, nature hath allotted unto them certain places, to the end that other things might by them be bred, and nourished: 3, nothing else can evidently be showed, whereof other things may be made: 4, in living cretures there are certain parts, agreeable unto the natures of the several elements.  
The Elements are found to be four, by two reasons: the former whereof is drawn from the number of the four prime qualities, and the four fold possible knitting together of them. For heat may be joined either with dryness, which two make fire, or else with moisture, which two do make up aer: and cold may be joined with moisture, as it cometh to pass in the water: or with dryness, as in earth. The later reason is taken from the four fold difference of the right motion: For the elements are directly moved, either upward or downward.  
Such things as move upward as light things do, are said so to do, either simply, as the fire, which is the lightest of the rest: or respectively, as the aer, which is lighter than the water, or the earth.  
Such things as move downward, as heanye things do, are said so to do, either respectivelie, as the water compared unto the fire, and aer: or simply as the earth, which is the heaviest of all the rest.  
The situation and order of the Elements, is found either by their motion, or else by the communication of their qualities.  
And first by the motion: For inasmuch as the fire and the aer do naturally move upward, the fire shall occupy the highest place: the aer, an upper place: and for that the water and the earth do naturally move downward, the water shall possess a lower place, and the earth the lowest.  
Again, the order of the Elements is found out by the communication of their qualities, for it were unfit that such things as are merely contrary, but such as in some sort can agree together, should be nigh one another. The fire therefore shallbe joined unto the aer, by reason of the heat common unto them both: the aer unto the water, by reason of the common moisture: and the earth unto the water, by reason of coldness common to them both.   

The two fold differences, of the celestial motions. Chap. 10.  
THe whole frame of the world is carried round about, with 2. motions, each of them being, distinguished from the other in name, and in reason.  
The one of them is called the first and universal motion: likewise the diurnal or worldly motion, because it bringeth the day unto the world. For in this motion the ☉ and all the celestial bodies do every day arise and set: they call it also the violent, and rapt motion, because by the violent swiftness thereof, it carrieth with it the rest of the Spheres.  
The other is called the second and particular motion, altogether contrary unto the former, as by which all the particular orbs do resist the universal motion. They call it also Sinister motus, the motion to the left hand, as the former is in like sort called dexter, that is, the motion to the right hand.  
The 2. motions are also distinguished according to the reason or the substance in the which they are inherent: For they differ the one from the other three ways.  
The first difference is in respect either of the whole: For the diurnal motion is common unto all the celestial bodies: or else of the parts, or stars either fixed or wand'ring, which have a motion peculiar and proper unto themselves.  
The second difference is either in regard of the situation of the Axes, For the diurnal motion is made upon the Axe and poles of the world, and therefore the Equator divideth it in the middle: but the proper motion is made upon the Axe and poles of the Zodiac, and therefore the Zodiac doth cut it in the middle: Or else it is in regard of the position of the terms: inasmuch as the diurnal revolution is made from the East unto the west, or as Plinte termeth it, from the right toward the left hand: but the proper revolution is from the West unto the East, or from the left toward the right hand.  
The third difference is in consideration of the swiftness: For the diurnal motion fulfilleth his course within the space of 24. common hours: but the proper motion in divers distances of time, according to the largeness of the orbs: namely, the orb of the fixed stars performeth his circle, in 36000. years: of ♄ in 30: of ♃ in 12: of ♂ in 2. years: of the ☉ in 365. days, and about 6. hours: of ♀ in 384. days, after Pliny: the orb of ☿ in as many days as the ☉: and the orb of the ☽, in 27. days & 8. hours.   

The circular form, and circular motion of the heaven. Chap. 11.  
THe Heaven is circular in motion, and in figure.  
The circular motion of the Heaven is proved as well by 2. experiments, as by 2. arguments.  
The one experiment is taken from the stars of the 8. orb, which both in their rising & setting, do always keep one & the same habitude, both in regard of the earth, and one to another: which thing can agree with none other than a circular motion about the centre.  
The other experiment is also taken from the stars of the 8. orb, always appearing, and retaining in divers places the same distance from the Poles, and onestom another: which also agreeth with the circular motion only.  
The first argument is derived from the consutation of 2. opinions, whereof the one supposeth thàt the motion of the heaven is direct and instnite: which if it were, the stars should vanish out of our sight: The other, that the stars in their setting are quenched, and in their rising are lighted again: as Heraclitus affirmed, which is absurd, both in respect of the motion, which is perpetual and constant in itself: and of the contrary power, which cannot be in the earth: as also in regard of our Antipodes, whose West is our East.  
The second argument is drawn from the dignity thereof: For the circular motion is the most worthy, and more perfect than the right motion, inasmuch as it breedeth no scission or cutting, and is made about the midst of the whole, not by displacing the whole body, but by the only unchangeable succession of the situation of the parts.  
The criculare figure of the Heaven is proved partly by similitude, and partly by reasons.  
The similitude is this: namely that this sensible world is the image of the first Archetype, or pattern of the world, who is without beginning or end.  
The reasons do contain either the commodity of the circular figure, or the necessity.  
The commodity consisteth either in the capacity, or else in the swistnes or aptness unto motion.  
The capacity was sit for the heavens, in that they were to comprehend all other things. For the circular figure is the grated of alother, circumscribed with equal conne●ities.  
The s●●st●es or aptness unto 〈◊〉, is either belonging unto the diurnal motion, called also the right hand monon, nature all unto the 〈◊〉 storels unto the second motion, which resisteth the former.  
Their asons drawn from the necessity of the circular figure, are either in respect of the whole world: For if the 〈◊〉 were of any other figure, there must needs be some empty place, and a body without a place for elsan regard of the celestial orbs, which gather could not be tarned about by divers motions, or else they should suffer a scission or cutting in sunder not without their great hurt.   

There is one sarface of the earth and water, and that is round. Chap. 12.  
THe earth and the water make one globe, and it is proved by causes either general, or special.  
The general causes belong unto both the elements made up in one form, and are derived from 3. heads.  
First from the signification of the worder For both in common speech, and in the scriptures, it is called Orbis terra, the Globe of the earth, or the round world.  
Secondly, from the Spherical form aswell of the Heaven, into whose round compass inasmuch as it is included, it cannot be but it must also be round: as also of the shadow that this globe doth cast forth: For the Masters of Perspective do teach us, that such is the dark body as is the shadow thereof.  
third, from the natural descending of the portions, either of the earth, the said portions coveting the centre of the Globe, and falling upon the surface of the earth at right angles: or else of the water, sinking also into the centre of the world, for the which their descending they gather themselves into a round form, and cannot abide upon a plane surrace.  
The special causes are such, as decern the roundness of the earth, and of the water particularly.  
The roundness of the earth is discerned two ways.  
The one is according unto Longitude, from the Last toward the West, or contrariwise, and that either by all the stars, which in divers places do not appear at the same instant: or else, che●sely by the ☽, whose Eclipse falleth out at one and the same time, but by those in the East reckoned one way, by those in the West, another.  
The other is according unto Latitude, from the Equator toward each pole, gathered by the unlike elevation of the Pole, and inequal quantity of the days, both which increase unto those that go from the Equator towards the North or South.  
The roundness of the water is discerned by tokens derived from the swelling of the drops, either hanging, or thrown upon the dust, or laid upon the moss of boughs: as also from the swelling of the Sea, by means whereof the Land cannot be seen from the Ship below, although from the main top it may: and again if any shining thing be sastened to the top of a ship saying fair from the shore, it deseendeth by little and little, according as the Ship runneth further off, and at the last is hidden from the sight.   

The situation, immobility, and magnitude of the terrestrial Globe. Chap. 13.  
THe earth or globe of the earth and water, hath situation, rest, and magnitude.  
The situation as being in the centre, or the place of the world farthest distant from the extremities thereof, is proved by arguments either direct, or indirect.  
The direct are derived from the nature of the broken parts, expressed either Physicallye, or Astronomically.  
Physically, because wheresoever they are about the earth, we alwates observe them, that of their own inclination they tend downward. But the centre is the lowest place.  
Astronomically, inasmuch as all the semidiameters of the world, by which heavy things descend, are continued through the centre of the world, and there they cut one another. So that where the section is made, there must needs be the place of the earth.  
The indirect arguments consist in two suppositions.  
The one, that the earth were in the Axis of the motion of the Heaven towardé one side & then should be taken away the apparent reason of the middle Heaven: the reason of the shadows Equinoctial, Solstitial, and plagall: and the reason of the universal Equinoctialles.  
The other supposition, that the earth were without the Axis, removed from the poles, either to the East or to the Westward, and then shallbe taken away, both in the rising and setting, the equal quantity both of the dares, shadows, and stars.  
The Rest of the earth excludeth all local motion, either right or circulate.  
The right motion is that which is made from the midst, and it is either natural and peculiar unto the earth: For otherwise it should come to pass that heavy things should ascend: or else it is violent, some outward thing insorcing it: For otherwise it should come to pass, that the earth should forsake the centre of the world.  
The earth hath no circular motion, neither from the West to the Eastward, as some have thought: For if it had, all things that are moved in the aer, should always be moved to the westward: Neither from the last to the Westward, by the d●●nnall motion: For than it should be an harder matter to travail toward the East, then toward the west.  
The magnitude of the earth is nothing, being compared either with the whole world, whereof it is the centre, which is proved by Mathematical instruments that agree with the centre of the world: For they at one time, and through the same sight hole, show two Stars opposite in the Diameter: or else being compared but with the orb of the ☉ which is proved by the equal spaces of the days and nights.   

The measure of the compass of the earth. Chap. 14.  
THe circumference of the Globe of the Earth and water is found out by the rule of four proportional numbers, in which rule 3. numbers are given, and the fourth is unknown.  
The three numbers given, which contain the proportion of a segment of a celestial circle unto the like space on the earth, are: 1. the difference of Latitude: 2 the viatorie distance: 3. the circumference of the whole heaven.  
By the difference of Latitude is understood, so many celestial degrees, as any terrestrial places are distant asunder.  
The viatorie distance, is that terrestrial space of way, that is answerable unto one degree, or any other difference of Latitude, and it is found out 3 manner of ways.  
First, by the distance of any two places upon the earth, situated under one meridian, the said distance being precisely tried.  
Secondly, by the latitude of both places, either observed by instruments, or taken out of Tables.  
third, by subducting the less out of the greater: for so the difference of latitude shall appear, whereunto the space of way known between the places given, shallbe answerable. Whereby unto each degree of a great circle in the heaven, there are answerable upon the earth after Ptolemee, furlongs 500 passes 62500. greater leagues 15. After Eratosthenes, surl. 700. pass. 87500 leag. 21. ⅞.  
The circumference of the whole heaven (containing 360. gr.) is the 3. number in the proportion: 1, for that of a little and of a great globe, there is the like reason: 2. because the terrestrial meridian hath the same centre with the celestial.  
The fourth number of the proportion, that is the circuit of the greatest circle in the earth, hath 2. considerations.  
The first is the manner of the searching thereof, and that is, first by multiplying the third number, that is, the circumference of the heaven by the second, which containeth the space of way upon earth: and then by dividing the product by the first, which is the difference of latitude.  
The second consideration is of the quotient, or manifestation of the content, which according to Prolemee is miles 22550. furl. 180000. pas. 22500000. greater leagues, 5400. according to Eratosthenes, miles 61250. furl. 252000. pass. 61250000. greater leagues 7875.   

The measure of the Diameter, and Semidiameter of the earth, as also of the Area and Surface thereof. Chap. 15.  
IN measuring the terrestrial Globe, we conder either the Diameter, or the Semidiameter, or the Area, or the connex surface thereof.  
The Diameter is measured by the proportion thereof unto the whole circumference, by the rule of four proportional numbers, wherein again three are given, and the fourth is unknowen.  
The three numbers given, being thoroughly known and understood, must be duly placed, and they must contain two things.  
The first is, the proportion of the circumference of a circle unto the Diameter thereof, which is tripla sesquiseptima: that is to say the circumference is unto the Diameter, as 22. is unto 7.  
The second is the greatest circuit of the earth in any measure, which was set down in the 14. Chapter.  
The fourth number of the proportion being unknowen, is the Diameter, which is sought, first by multiplying the thirde by the second, which is 7. & dividing the product by the first, which is 22. then by subducting the 22. part (which cometh forth of the division of the circumference by 22.) out of the circumference, & dividing the remainder by 3. whereupon ariseth the content of the Diameter, after Ptolemee containing miles 7159 〈◊〉. furl. 57272 〈◊〉. pass 7159090 10 10/11 11 greater leag 1718 〈◊〉. After Eratosih. m●es 19488 7 7/11 11 fur. 80181 〈◊〉. pass 1948863 〈◊〉. greater leag. 2505 〈◊〉.  
The semidiameter is the distance between the convex surface of the earth, and he centre thereof (which some do imagine to be the place of helly the said distance is found two wa●es.  
1 By the proportion of the circle unto the Semidiameter, which is s●●tuple over and beside 1 1/14 14: or as 44 is unto 7.  
2 By dividing the Diameter into two parts: by which means it shall be found to contain after Ptolemec, miles 3579 ½. furl 28636 passes, 3579545 greater leag. 859 1 1/11 11;. After Eratosthenes, miles 9744. furl 40090 ½. pass. 974431. greater leag. 12●2 ½.  
The Area or plane is found by multiplying half the circuit of the earth taken in any known measure, by the Semidiameter thereof.  
The convex surface that covereth the whole earth, is found by multiplying the terrestrial Area or plane, by 4.   

The general definition and division of the circles. Chap. 16.  
IN as much as the surface of the Heavens is spherical, and their motion circular, therefore for the better conceiving of the reasons of the celestial motions, they are distinguished into certain circles as parts, whereof we are to show the names & the division.  
In the names we are to consider their acception, and their diversity, being notwithstanding all one in signification.  
The acception of the name Circle, is of two sorts, Geometrical, and Astronomical.  
The Geometrical acception, is when a circle is taken for a plane figure, which one line equally distant from the centre doth encompass.  
The Astronomical acception is other as it signifieth a circular line, or a circumference wanting breadth: or else a circular surface, which hath breadth thereunto adjoined.  
The diversity of names all one in meaning, is when circles are called (amongst divers Authors) threads, compasses, orbs, segments, rings, parallels & equidistant lines.  
The division of circles is diversly delivered by the Greeks' and Latins, in three respects.  
First in respect of the material sphere, within the which some of the circles are not placed, & are therefore called , fised, and manisold, as the Orisons and the Meridian's: others are placed within the sphere, & are therefore called intrinsecall, movable, and singular, as are the two polares, the Equator, the zodiac, the two colours, and the two tropickes.  
Secondly, in respect of the poles of the world, or the twofold motion of the heaven: and in this case the Greeks' distinguish them again into three sorts.  
The first are parallels, in number 5. namely the 2. polares, the 2. tropicks, and the Equator, all which have the same poles with the world, are equidistant on all sides, and serve the fyrst or universal motion.  
The second are obliqne circles, in number 3. namely the Zodiac serving the second motion, the Horizon, the milk way, all the which lie obliqne between the poles.  
The thirde are those circles that are drawn through the poles, and they are also 3. in number, namely the Equinoctial and solstitial colour, and the Meridian  
The thirde division of circles is in respect of their quantity, according whereunto some circles are called greater, and some less.  
The greater circles are in number 6. namely the Equator, the Zodiac, the●. Colours the Horizon and the Meridian, all which are equal one unto another, and cut the sphere into equal pieces.  
The less circles are in number 4. namely the 2. polares, and the 2. tropicks, which are not all of them equal one unto another, neither divide they the sphere into two equal pieces.   

Of the greatest circle containing the measure of the first motion. Chap. 17.  
THe whole heaven or universal frame, turned round by the first motion, doth in the middle place between the 2. poles, desscribe a certain circular compass, whereof we are to consider the name, the definition, the commodity.  
The name thereof is divers: for it is sometimes called 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, Aequidialis, that is, if we may so term it, equidiall, sometimes the line or the orb of the equality or equation of the day: sometimes the Equinoctial and Equator: and sometimes the girdle of the first motion or movable.  
The definition thereof is that, wherein the magnitude, situation, and equal conversion thereof are contained.  
The magnitude thereof is considered, in that it is the greatest circle, & hath these 2. proprieties, the one, that it divideth the sphere into 2. equal parts, the other, that it hath the same centre with the world.  
The situation thereof is in the midst between both the poles of the world, in which respect it differeth from the rest of the parallels, and obliqne circles.  
The equal conversion thereof, is that perfect revolution which it fulfilleth within the determinate space of 24. hours.  
The commodity thereof is great & manifold, and it is either Astronomical, or Geographical.  
The use thereof in Astronomical matters, is seen chief in 4. things.  
First, by the help thereof we understand the measure of the first motion, and thereby recon the time, which is the measure of the first motion.  
Secondly, it helpeth us in the finding out of the equinoctials, and that in two respects.  
The one, in respect of the whole Earth. For every Horizon of every country divideth the Equator only of all the parallels, into 2. equal pieces, whereby it cometh to pass, that when the ☉ is in the Equinoctial, the day & the night throughout the whole world are equal.  
The other in respect of certain Regions. For those that dwell under the Equator, in what part of the heaven soever the ☉ is, have always the Aequinoctium, or the day and the night equal.  
Thirdly, thereby we find out both the situation of the Stars either toward the North, or toward the South, because it distinguisheth the North part of the world from the South: as also their declination, either Septentrional or Meridional.  
fourthly, through the help thereof, we search the length of the artificial day.  
The utility thereof in Geographie is seen in 3. things.  
1 Thereby we set every town in his due place, in the terrestrial Globe.  
2 It bringeth us unto the knowledge of all the parallels, aswell celestial as terrestrial.  
3 By the aid thereof we finish the description of the earth.   

Of the greatest circle measuring the second motion. Chap. 18.  
THe stars of heaven which are moved round about from the West toward the East, do describe in the midst between their poles, a certain circular surface common to all the planets, and a certain circular line proper unto the ☉ only.  
Concerning the circular surface, there are delivered by the Astronomers to be considered, the names, the definition, the measure, and the use thereof.  
The names are divers, drawn either from the Greeks', or from the Latins.  
The Greeks' call it the Zodiac, either of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, life, for that it is the path wherein the ☉, (taken to be the author of life) doth walk: or else of 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, a living creature, for that the ancient Astronomers, have beautified this circle with the figures of certain living creatures.  
The Latins term it Signifer, as carrying the signs, they call it also the obliqne circle, or the circle leaning a side.  
The definition containeth the magnitude, the obliqne situation, and the limits thereof.  
Concerning the magnitude thereof, this is only to be considered: that it is one of the greater circles.  
The obliqne situation thereof, is either in respect of the parallels, which it cutteth at inequal angles: or of the irregulare ascensions, and descensions, or of the poles of the world, from the which it is not equidistant.  
The limits thereof are the 2 tropicks, which it toucheth, and divideth the Equator into two equal parts, declining therefrom by little & little unto the distance of 31. gr. 28. mi.  
The measure thereof is either in regard of the length that it hath, or of the breadth.  
The length thereof is 360. gr. and is divided into 6. Northern signs, as ♈. ♉. ♊. ♋. ♌. ♍. & into 6. Southrens signs, as ♎. ♏. ♐. ♑. ♒. ♓.  
The breadth thereof is 16. gr as well in regard of the roaming of the planets from the cliptrck, and specially of ♀. & ♂. as also in respect of the principal consfellations, whereof the greater part declineth from the midst of the zodiac.  
The use is chief seen in the obliquity thereof: for thereby it falleth out, that the parts of heaven, do with the more ease maintain their course against the first and universal motion: as also that the stars may sometimes be in the South, and sometimes in the North, for the greater benefit of the inhabitants of the earth.  
The circular line proper unto the ☉ only, hath divers names, with the definition & commodity peculiarly appertaining thereunto.  
It is called the wheeling, the way, the course, the place of the ☉, the Ecliptic line, also the  line, and the division of the zodiac.  
The definition thereof, is that whereby it is called a greater circle, dividing the breadth of the zodiac into 2. equal parts.  
The commodity thereof is noted as well in designing the Eclipses of the ☉ & ☽, which never happen but when both of them are under or very near the Ecliptic line; as also in distinguishing of the 4. quarters or seasons of the year.   

Of certain terms whereby the stars have relation unto the aforesaid circles. Chap. 19  
THe whole number aswel of the sixed stars, as also of the planets, hath relation both to the Equator, and to the Zodiac.  
They have a twofold relation unto the Equator, either in regard of the orbiculare longitude of the Equator, or of the lateral position, thereof.  
In the orbiculare longitude of the Equator we are to note the names, and the definition.  
It is sometimes called the longitude of a star: and sometimes the right ascension.  
It is defined: The ask of the Equator comprehended between the head of ♈, and the section of a great circle passing through the poles of the world and the true place of the star.  
The lateral position hath also name, definition, and division.  
It is called the declination of a start.  
It is defined to be: The ark of a great circle, passing through the poles of the world, and the true place of the star, the said ark being intercepted between the Equator, and true place of the starie.  
It is divided into the Septentrional, and Meridional declination.  
The relation that the stars have unto the Zodiac, is also two fold, either according to the Longitude of the Zodrake, or else according to the transuerse distance toward either of the Poles.  
In the Longitude of the Zodiac we are to consider the name, and the definition.  
It is called Longitude: For that it is recoved longwaies on the circumference of the Ecliptic: it is also called the true motion of the Star.  
It is defined to be the Ark of the Zodiac, intercepted between the head of ♈, and the section of a great circle passing through the poles of the Zodiac, and the true place of the star.  
In the transuerse distance we are to note the name, the definition, and the division.  
It is called Latitude, because it is reckoned according to the position that it hath from some one side of the Ecliptic.  
It is defined to be the ark of a great circle, drawn through the poles of the Zodiac, and the true place of the star, the said Ark being intercepted between the Zodiac, and the centre of the Star.  
It is divided into the Septentrional Latitude, when the stars are under the northerly signs: and into the Meridional Latitude, when they are in the Southerly signs.   

Of the propertion and supputation of the declination of every point of the Ecliptic, or the regard of the parts of the Zodtake, unto the Equator. Chap. 20.  
IN the declination of any point of the Ecliptic, 2. things are to be observed: the proportion, and the supputation.  
In the proportion we may note also 2. things: For either they have none obliquation, or else their obliquations are equal.  
Those that have none obliquation, are the head of ♈ and ♎, as being the common intersections of the Equator and the Zodiac.  
Those that have equal obliquations, are such as are equally distant from the Equator, and they are either greater obliquations, or else the greatest.  
The greater obliquations are those that have any distance less than the greatest from either of the sections, and of that sort there are always four.  
The greatest obliquations are those that have the greatest distance from the Equator, as the head of ♋, that s, the Summer solstice: and the head of ♑, that is the winter solstice.  
The supputation is made either by the tables of declinations, or of Sines.  
The Tables of declinations are calculated in sundry places by Astronomers, and they consist of the 2. sides, the Area, and of the two extremities or ends.  
The sides are either at the right hand, or at the left: that at the left hand, to be entered into, when you have the sign in the top of the table: and that on the right hand, when the sign is in the foot thereof.  
The Area is that, wherein at the common angle the declination is found.  
The 2. extremities are those that contain the signs: of which extremities, the one is called the top, or upper part: the other the foot, or the nether patte of the table.  
The supputation that is made by the table of Sines, is performed by the help of the rule of 4. proportion all numbers, wherein 3. numbers are given, and fourth is to be sought out.  
The 3. numbers given, must contain the right sine of the whole quadrant, or of the semidiameter: the right sine of the greatest declination of the ☉: and the right sine of the distance of the point of the Ecliptic given, from the first section of the Zodiac and the Equator.  
The fourth number produced by multiplitation and division, is the right sine of the declination sought, whose subtended ark declareth the number of degrees.   

Of the 2. circles called the colours, distinguishing the Equinoctial and Solstitial points. Chap. 21.  
FOrasmuch as there are certain points of the Zodiac and the Equator more notable than the rest, therefore the Astronomers have thought good to fit unto those points 2. Circles, whereof we may consider the reason of their name, their definition, their number, their figuration or description, and their use.  
They are terméd colours, that is imperfect, in 3. respects.  
1. Because they appear always incomplete, or maimed, the which thing notwithstanding seemeth to be common with divers other circles.  
2. Because they have some parts that do never arise.  
3. Because they are carried about after an imperfect manner, & not according to the position of Longitude, as the motion of the Heaven is.  
The definition containeth their magnitude, their intersection, and their motion.  
As touching their magnitude, they are of the number of the greater circles.  
As touching their intersection, they cut one another in both the poles of the world, at spherical right angles.  
In their motion, they are moved together with the sphere.  
Their number is two: whereof the one passeth through the Equinoctial points and the poles of the world, and is called either the equinoctial colour, or the distinguisher of the equinoctials: the other passeth through the solstitial points, and the poles both of Ecliptic and of the world: and is called both the solstitial colour, the distinguisher of the Solstices, and also the circle of the greatest declinations.  
Their figuration is described by the semidiameter of the world, whose revolution being fullye performed through the poles of the world and the Equinoctial points, maketh the Equinoctial colour, but passing through the poles of the world, and the solstitial points, it maketh the solstitial colour.  
Their use is manifold, but principally in 3. things.  
1. In distinguishing the Equinoctial and Solstitial points.  
2. In reckoning aswell the quantity of the greatest declinations of the ☉, by the ark intercepted between the Equator and the Ecliptic: as the space comprehended between the poles of the world, and the poles of the Ecliptic, which is always equal unto the ark of the greatest declination.  
3. For better understanding of the ascensions and descensions of the signs.   

Of the Meridian. Chap. 22.  
THe ☉ carried about by the first motion, when it is at the highest, designeth a point of a circle, whose definition, variety, and office, we are to consider.  
The definition taketh hold both of the names thereof, and of the matter itself.  
It is called the circle Meridian, Meridional, and Merinoctiall, the circle of the midday and midnight, either because it divideth both the day and the night into 2. equal parts, the one ascending, the other descending: or else, because so often as the ☉, according to the first motion, is under the Meridian, it is then either midday, or else midnight.  
The matter itself is that, according whereto it is defined to be one of the greater circles, drawn through the poles of the world, and the vertical point of any place given, and standing still when the Sphere is moved.  
The variety of the Meridian, by reason of the round figure of the earth, is either none at all, or manifold.  
It is none at all, either in regard of reason, or of sense.  
It is none at all in regard of reason, when one place is distant from another in Latitude only, that is, from the North to the South, or contrariwise.  
It is none at all in sense, when one place is distant from another according unto Longitude, which is from the East unto the West, or contrariwise 36. scrup. that is, about 300. furlongs.  
The variety is manifold in regard also of reason and of sense.  
The manifold variety in regard of reason is, when examining the least distance toward the East or West, we conclude another Meridian: and by this means we may have so many meriridians, as there shallbe places at every small distance toward the East.  
The manifold variety according unto sense, is as often as any two places shallbe distant one from another, between East and West, more then half a degree, and by this means we may have so many meridians. as there are half degrees of the Equinoctial circle.  
The office of the Meridiane is twofold, either Astronomical, or Geographical.  
The Astronomical office thereof is executed two manner of ways.  

1. In pointing out the Noons tide, or Midday, either natural or artificial.  
2. The divers habitudes and positions of the stars, following the motion of the heaven itself, are ascribed unto the Meridian.   
The Geographical office thereof is also of two sorts.  

1. By the help thereof the Longitude of all places is calculated: and what places are more oriental, and which more occidental.  
2. By the aid thereof we describe in the terrestrial plane, a correspondent merdiane line, for divers uses of Astronomicalli Instruments.    

Of the Horizon. Chap. 23.  
THere is also another circle, which the ☉ by the firste motion doth point out in the East and West points, whose definition, division, and dignity, is to be considered.  
The definition stretcheth itself both to the names thereof, and to the matter.  
It hath divers appellations: and is sometimes called the Horizon, Finitor, Finiens, as limiting our sight, and sometimes the compass or circle of the Hemisphere of divers regions.  
The matter itself attendeth the description of the centre or pole thereof, the circumference and the magnitude.  
The centre or pole of the Horizon, is the vertical point of each place, distant from the Equator so much, as the poles of the world are distant from the Horizon.  
The circumference of the Horizon is that, which the semidiameter of the world in his full revolution through the points of the East and West, and the rest of the brim of Heaven, describeth.  
The magnitude of the Horizon is considered, in that it is one of the greater circles, dividing the world (in regard of sense) into 2. equal segments, whereof the one is called the upper, the seen, or the diurnal segment, the other, the lower, the hidden, or the nocturnal segment.  
The division of the Horizon is considered, in respect of the Equator as it is either a right, or an obliqne Horizon.  
The right, or orthogonall Horrizon hath 3. proprieties.  

1 With the Equator it hath equal angles.  
2 It hath the pole, or the vertical point in the Equator.  
3 It hath the poles of the world in his circumference.   
The obliqne, bending, or inclining Horizon, is in all things contrary unto the right.  
The dignity of the Horizon, by reason of the manifold use thereof, is great: For by the help thereof, we learn 6. things.  

1 The quantities of the artificial day and night, and consequently the time of the rising, and setting of the ☉.  
2 The equal hour of the day, the ☉ shining  
3 The degree of the Zodiac, wherewith any star given doth arise and set.  
4 What stars do always appear, or are always hidden.  
5 The rising and setting of the stars.  
6 The Eclipses of the ☉ & ☽, either seen, or not seen.    

The division of the Horizon, according to Proclus. Chap. 24.  
Moreover, the Greeks' deliver unto us a more subtle division of the Horizon: and it is twofold, the one to be conceived in mind, only, the other falling within the compass of our sense, or our sight.  
Concerning the Horizon to be conceived in mind only, we are to note the names, the description, the cause.  
It is diverfly named, as either rational, or conceived by reason, by the Greeks' called 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, and also natural.  
The description thereof is absolved by a semidiameter, and a circumference and the Area thereof.  
The semidiameter is that line, whereof the one extremity is in the eyes of the inhabitants of the world, the other extremity is in the or be of the fixed stars.  
The circumference and the Area is the space and compass, which the semidiameter maketh, being carried about by the brim of that part of the heaven, that is extant above the Horizon.  
The cause alleged is, that our sight being inhable to pierce unto the beholding of all the fixed stars, doth conclude that there is a certain circle in the heaven, that limiteth the things seen, from the things not seen.  
In the Horrizon apprehended by our sense, we are to note the names, the description, the variety.  
It is called the Horizon sensible, or perceived by our sense, also the apparent, and artificial Horizon.  
The description is performed by a semidiameter, a circumference, and a plane.  
The semidiameter is that line, whereof the one limit is in our eye, the other is in the end of our sight upon the surface of the earth, consisting of a thousand furlongs, which end we imagine in a free prospect, to join the heaven and the earth together.  
The circumference and plane is that space & compass, which the aforesaid semidiameter turning about, doth describe.  
The variety is common aswel unto the Rational, as the sensible Horizon, & it is either none at all, or else manifold.  
The variety of the sensible Horizon is said to be none at all, when the Horizon doth continue all one and the same, and it is either in reason, or in sense.  
The sensible Horizon is not varied in reason, when the places are not any whit, nor any way changed.  
The sensible Horizon is said not to be varied in sense, when the places distant about 400. furlongs one from another (that is 48 mi.) do not alter, either the climate, or the length of the days, or the apparences of the heavens.  
The variety of the sensible Horizon is manifold, when the places are varied more than 400. furlongs, and are situated either toward the East, or West: in which variety neither the climate, nor the length of the day, nor the apparences of the heavens are changed with the Horizon: or else they are situated toward the north or south, wherein together with the Horizon both the climate, and the length of the days, & the apparences of the Heavens are altered.   

Of the two Tropickes. Chap. 25.  
THe ☉ carried about by the second motion, in his greatest declination from the Equator by the violence of the first motion, describeth certain parallels, whereof the general reason, the number, & the offices are to be considered.  
The general reason is either in respect of their names, or their definition.  
They are named by the Greeks' 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, Tropickes, by the Latines Versiles, conversius, Vertentes, tourning, and the Solstitial parallels.  
Their definition containeth their quantity and their circumference.  
Their quantity is noted, either in respect of the other circles, these being counted in the number of the less circles, or in regard of themselues, whereby they are counted equal, in as much as they are equally distant from the centre of the world, being separated the one from the other by the double distance of the ☉ greatest declination.  
Their circumference is that round compass, which the ☉, passing through the 2 solstitial points, doth describe.  
They are in number 2. the one Septentrional, the other Meridional.  
The Septentrional Tropic is on this side of the Equator (in respect of us) which we call either the Summer tropic, for that it passeth through the point of the Summer solstice, or●ls the tropic of ♋, because it is described through the end of ♊, & the beginning of ♋.  
The Meridional tropic is situated on the other side of the Equator, and is called either the Winter tropic, as passing through the point of the Winter solstice, or the tropic of ♑, because it is drawn through the head of ♑.  
The offices and commodities of them, are in number 4.  
1 They show the Tropes, that is, the conversions, or tournings of the ☉, aswel in Summer, happening in our age the 3. and 2. of the Ideses of june, as also in winter, the 3. & 2. of the Ideses of December.  
2 They show in every situation of the sphere, both the longest day, which is as long as the diurnal Ark of the Tropic of ♋, containeth hours: and the shortest day, which is as long as the space of hours, contained within the diurnal ark of the tropic of ♑.  
3 They point out the limits of the course of the ☉, and his greatest declinations: which are 23. gr. 52. mi. as in the time of Aristarchus & Ptolemee, or 23. gr. 28. mi. as it is now in our time.  
4 They show the  zone which they separate from the temperate, and the midst of the second climate, which they call dia-Syenes, and Anti-dia-Syenes.   

Of the 2. polare circles. Chap. 26.  
The two poles of the Zodiac, carried about by the regulare revolution of the universal frame, describe about the poles of the world, two circles, whereof the general reason, the number, and the use is to be noted.  
The general reason offereth to our consideration, their name, their definition, and their accidents.  
They are called the Polare circles, either because they are described about the poles, or by the poles.  
Their definition by the Latins, 'tis made by their quantity, and their circumference and plane.  
Touching their quantity, they are in the number of the less circles, equal in all places.  
Their circumference and plane is described either by lines, drawn from the poles of the Zodiac, unto the Axe of the world, at right angles, and having by the daily motion a perfect revolution: or else they are described by certain semidiameters, drawn from the centre of the earth unto the poles of the Zodiac, and turned about by the diurnal and nocturnal motion.  
The accidents of the polare circles do determine either their equality, for they are parallels, compared either one with another, in as much as they are equidistant from the centre, or compared with the tropicks, & the Equator: or else they determine their distance, either from the next tropic, which is 43. gr. or from the poles of the world, which is equal unto the ☉ greatest declination.  
They are two in number: The one Septentrional, the other Meridional.  
The Septentrional Polare circle is described by the North pole of the Ecliptic: the Meridional, by the South pole thereof.  
The Septentrional polare circle is called boreal, North of the North wind called Boreas, and Arctic, and Septentrional, because of the 2. constellations, the one of the greater bear called Arctos, the other of the less bear called Septentriones, which are nigh thereunto.  
The Meridional polare circle is called austral, or Southern of the South wind called Auster, and Antarcticke, ●s opposite unto the Arctic, and Meridional, of the South part of heaven, called Meridies.  
Their use is noted in that they comprehend the cold and frozen zones, and the inhabitants of the earth called Perison, whose shadows go round about them, and on either side limit the the distances of the poles.   

The Polare circles otherwise described according to the Grecians. Chap. 27.  
THe polare circles are described, either according to the greatest declination of the ☉ or the altitude of the Pole above the Horizon.  
The greatest declination of the ☉, by means of the motion of inclination of the eight Sphere, is divers. For it was one in times past, and is found to be another now: and of such circles, the reason is declared in the Chapter before.  
The polare circles described according to the altitude of the pole, require the consideration of their definition, their variety, their offices, and the manner of their description.  
In defining the Arctic polate circle, we say: 1. That it is the greatest of those circles which are always in our sight, that is, of those which we may see at the same instant: 2. that it toucheth the Horizon in one point: 3. that it is altogether above the earth.  
In defining the Antarctick polare circle, we say: 1. that it is equal and parallel unto the Arctic: 2. that it toucheth the Horizon in one point: 3. that it is altogether hidden under the earth.  
The variety is manifold, according to the didiversitie of the climates. For either they are not at àll, as in a right Sphere it happeneth, where excluding altogether the polare circles, the Greeks recon 3. parallels only: or else they are, and those sometimes either less, equal, or greater than the tropickes, or else they are equal unto the Equator and the Horizon. For by how much the pole shallbe higher, by so much shall these circles be greater.  
The offices and use of the Arctic circle is, in that it showeth the Stars that never arise ●nor set: of the Antarcticke circle, the con●trary is to be conceived.  
The means of their description is by those Stars, that in any Region do touch the Horizon.   

Of the Milky circle. Chap. 28.  
OF all the circles, there is none to be seen, beside the Milky circle, which for that the Greeks' do recon among the other circles, we will express the names, the definition, the causes thereof, and the distinct Stars which make the same.  
The names are divers: as Galaxia, the Milky orb or circle, the Milky Zone or milky way. The Arabians dal it Matarati, as it were a broad, space, or ark that moveth.  
It is defined to be one of the greater circles, obliqne, drawn or stretched toward both the Poles, most brightly shining, apparent unto the sense, inequal, both in breadth and in colour.  
The causes are divers, and those either fabulous, or natural.  
The fabulous causes are in number 4.  
The first is taken from the scorching of the ☉, as if the ☉ had sometimes made his motion there, and by his scorching had caused that place to be white.  
The second is drawn from the milk of juno, that running plentifully out of her paps, painted this circle of that colour.  
The third is fetched from the seat and habitation of strong and valiant men, whom the Poets have placed in this circle.  
The fourth is derived out of the way of the Gods, as if they passed thereby unto the palace of jupiter.  
The natural causes alleged (although they be many, yet) are principally, but 3.  
The first by Theophrastus: who said, that it is that joining together, whereby the heaven being divided into two hemispheres, is as it were by a certain clay fastened.  
The second, by Aristotle: who took it to be a Meteore, set on fire in such sort as a Comet,  
The third is Astronomical: which affirmeth that it is a girdle caused by many little stars, as it were one touching another, in the which concurring in that Place, the light of the Sun is diffused.  
The distinct stars that make it, are chiefly these: The arrow: the Eagle: the bow of ♐: the Altar: the 4. feet of the Centaur: the ship Argo: the head of the Dog: the right hand of Orion: Erichthonius or the Wagoner, with the Goat on his shoulder: Perseus: Cassiopeia: and the Swan.   

Of the 5 principal Regions of the world, commonly called Zones. Chap. 29.  
THe Universal Globe aswell of the heavens, as of the earth answerable thereunto, is distinguished into certain orbicular tracts, which the spaces comprehended between the 4. parallels do make, of which tracts we may consider the names, the definition, the general number, and their distance one from another.  
Their names are divers: For they are called either Zones, or swaddling bands, or girdles, or Mashes, or coasts.  
They are defined to be the space either of the heaven, or of the earth, comprehended between two less parallels, or else included on every side with the polare circles.  
Their general number is twofold: For either they are celestial, and so the causes of the terrestrial, or else they are terrestrial, of the same proportion with the celestial.  
The celestial are either Mean, or Extreme, or between mean and extreme.  
The Mean is that Zone which is included between the 2. tropickes, and is cut in two equal parts by the Equator.  
The Extremes or polare Zones, are those whereof (being but 2) the one is called the Septentrional Zone, within the Arctic circle: the other the Meridional Zone, within the Antarcticke circle.  
The Zones between mean and extreme, are also 2. whereof the one is Septentrional, comprehended between the tropic of ♋, and the circle Arctic, and the other Meridional comprehended between the tropic of ♑, and the circle Antarctic.  
The terrestrial Zones have the same reason with the celestial, aswell in respect of their number, as in regard of their names.  
The terrestrial Zones are also 5. in number, answering proportionallye unto the 5. celestial Zones, conically marked out by the 4. celestial parallels.  
The terrestrial Zones have the same reason with the celestial, in respect of their names also: For that terrestrial Zone that is under the mean celestial, is called mean: those which are under the extremes or polares, are called extremes septentrional, or Meridional: and those which are under the Zones between mean and extreme, have their name accordingly, and are either Northerly, or southerly.  
The distance one from another is in this manner: the mean or  Zone, according to the Latitude reckoned in the Meridian, containeth 47. gr. or 705. miles: the extreme intemperate Zones do each of them according to the said reckoning contain, as many degrees and miles, as the mean: the temperate zones between mean and extreme, do each of them contain according to the former reckoning 41. gr, or 645. miles.   

The difference of the Zones, and the manner how all places Upon the earth, may be brought within their compass. Chap. 30.  
THe difference also of the zones as well celestial as terrestrial, and the reason how all places upon earth may be referred unto them, is worthy the noting.  
Their difference is to be considered either in respect of their figure, or their accidental nature.  
The figure of the mean is uniform, and for the most part alike.  
The figures of the extremes are either of them equal to other, yet such, as that they seem rather to carry the shape of circles, then of zones.  
The figures of the zones between mean and extreme, be either of them alike, and equal unto the other: yet about the tropicks their figure is limited with a greater compass, then toward the polare circles.  
The accidental nature of the zones is that, in regard whereof they are said to be mean, extreme, and between mean and extreme.  
The Mean or  zone is divided into 2. parts, whereof the one is situated under the Equator, the other about the Tropickes.  
That part which is situated under the Equator seemeth to be temperate, and that for three causes.  

1. By reason of the sudden and cross access, and recess of the Sun.  
2. By reason of the continual equality of the night and day in that place.  
3. By reason of the swift carrying about of the ☉, by the first motion.   
That part which is situated under the Tropickes is hardly to be inhabited, and that also for 3. causes.  

1. For the slow conversion of the ☉.  
2. For the doubled projection of the Sunbeams, upon those places.  
3. For the great increase of the Summer days above the nights.   
The extreme zones are both of them frozen, by reason of the too much cold that falleth out there, by means of the obliqne projection, and reflection of the Sun beams.  
The zones between mean and extreme are both of them temperate, and are divided into 3. Regions, whereof one is situated about the middle part thereof, which we judge simply to be temperate, by reason of the moderate heat of the ☉, namely, from 34. gr. unto 48. gr. distance from the Equator: the other 2. regions are about the extremes thereof, the one being about the tropicks, and so subject unto the intemperate heat of the  zone, the other nigh unto the polares, and therefore subject unto the intemperate cold of the frozen zone.  
The reason how all places upon the earth may be referred unto those zones, hath two considerations.  
1. If the places have Septentrional Latitude, and that less than the greatest declination of the ☉, they belong unto the  zone: if equal, unto the trop. of ♋: if greater, and yet not exceeding ♋, gr. 30. mi. they belong unto the temperate zone. If the said septentrional Latitude be equal unto the compliment of the greatest obliquation, they must be placed under the arctic circle: if greater, under the frozen zone.  
2. If the places given have Meridional Latitude, the same judgement is to be pronounced of them, as of the places under Septentrional Latitude.   

Of the fowerfolde rising and setting of the Stars. Chap. 31.  
THe Poets, and for the better part all other Authors, do periphrasticallye describe the times of things, worthy the noting, by the Stars of heaven, either rising or setting.  
In their rising is to be considered, the definition, the subdioision.  
The definition doth chiefly consist of the name, and of the matter.  
The name in this place signifieth their first apparition unto the e●e, or their Ascension.  
The matter is that according whereunto, the rising of a star is defined to be the apparition of any star given, which before could not be be seen, as either being under the Horizon, or hidden by the Sun beams.  
The subdivision also offieth 2. considerations.  
1. That the stars do ascend or rise by the uninersall motion from the lower hemisphere unto the Horizon, either in the morning with the ☉, and then they are said to have a morning, a diurnal, a cosmical, or worldly rising: os else in the Evening at the ☉ setting, and then they are said to have an evening, a nocturnal, a chronical, or acronychall rising.  
2. That the stars do rise by the 2. motion freed from the ☉ beams, either before the rising of the Sun, and then they are said to have an heliacal morning rising, which cometh to pass in those stars that are flower then the ☉, or else after the setting of the ☉, and then they are said to have an heliacal evening rising, and that is in those stars, that are swifter then the ☉.  
In the setting of the stars there is also offered the definition, and the subdivision.  
The setting is defined to be: the occultation or hiding of any star given, either by the depression thereof under the Horizon, or by the ingression thereof into the beams of the ☉.  
The subdivision consisteth in their setting and withdrawing from our sight, which is done two manner of ways.  
1. By the first motion they descend from the upper hemisphere unto the lower, either in the morning which is done cosmically at the rising of the Sun, and that setting, as the rising also, is referred unto the ☉, and those fignes of the Zodiac which the ☉ possesseth: or else in the evening, which is done chronically, at the setting of the ☉, and this setting, as also the rising, is referred unto all the stars generally.  
2. By their proper motion at their entrance into the beams of the Sun, either before the sun rising, that is, cosmicallye, which happeneth only unto the stars that are swifter than the ☉, or else after the setting of the Sun, that is, chronically, which belongeth unto those stars only, that are slower than the ☉.   

Another more easy and perfect distinction of the risings and settinge, with the exposition of certain principles which are to be Understood for the reading of Authors, concerning the rising and setting of the Stars, taken out of Ptolemee, and the later Astronomers. Chap. 32.  
FOr the easier understanding of the Poets and other Authors, which by the rising and setting of the stars do circumscribe the times, 4. things chiefly are to be known.  
1. The latitude of the place whereof the speech is made, which may be gathered out of the Tables of the Regions, set down in all Geographical writings.  
2. The place of the ☉ in the Ecliptic at any time, which the ancient Records do minister, where notwithstanding you must note, that our age doth differ from former times: and that the ☉ in our age doth entre into the heads of the signs, sooner almost by 6. days, then in the ancient times.  
3. What signs are opposite one unto another: viz. ♈ to ♎: ♉ to ♏: ♊ to ♐: ♋ to ♑: ♌ to ♒: & ♍ too ♓.  
4 The difference of the rising, or of the setting.  
The rising is either heliacal and of the Morning, or Acronychall and of the evening.  
The heliacal or morning rising, is either true or apparent.  
The true heliacal rising is when a star joined with the Sun, doth together and at the same instant arise with him in the morning.  
The apparent heliacal rising, is when the star doth ascend and begin to appear at the dawning, and before the Sun rising.  
The Acronychall or evening rising, is also either true, or apparent.  
The true Acronychall rising, is when a star precisely riseth, at the very instant of the Sun setting.  
The apparent Acronychall rising is when after the setting of the Sun, the star being freed from the beams thereof, shall make his first appearance in the twilight.  
The setting of a star is also either heliacal, or Acronychall.  
The Heliacal setting is either true or apparent.  
The true heliacal setting is when a star at the ☉ rising, doth at the same instant set in the opposite part of the world, which before was called the morning star.  
The apparent heliacal setting is when in the morning, somewhat before the ☉ rising, the star is newly seen to set.  
The Acronychall setting is in like sort either true or apparent.  
The true Acronychall setting is when at the ☉ setting, the star also setteth, which all the mean time was called the evening star.  
The apparent acronychall setting is when after the setting of the ☉, the star doth not set at the same instant with the ☉, but by reason it is hidden by the beams of the ☉, it appeareth no more until the morning that it arise again.   

Of the Astronomical rising & setting of the signs: or as the Greeks' call it, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉. Chap. 33.  
THe rising, the coming unto the Meridian, and the setting of the signs, or of any point of the heauen, is either poetical, or astronomical.  
The Poetical or vulgar, is when the reason of the apparition, or occultation of the signs, is only in their comparison with the ☉, which was handled in the 31. & 32. chapters.  
The Astronomical rising, culmination, & setting of any star or point of the heaven is that which defineth the proportion of the time and space, both when and how great it is, wherein the aforesaid things are performed, either in a right, or an obliqne sphere.  
In the rising are to be considered the definition, and the bipartite division.  
The definition is either of the name, or of the matter.  
The ascension is called the rising, which we measure by the coascendent ark of the Equator.  
The matter is that, according whereunto it is defined, to be the ark of the Equator, comprehended between the sign rising, or the East part of the Horizon that containeth the sign, & the head of ♈, the which ark is to be accounted according to the orderly succession of the signs.  
The consideration had of the division, is that either a greater portion of the Equator riseth with the sign, & then it is said to have a right ascension, because it maketh righter angles with the Horizon: or else that a less portion of the Equator doth ascend therewith, and then it is said to have an obliqne ascension, by reason of the more obliqne angles that it maketh with the Horizon.  
The culmination is defined, either the passing of some point of the Zodiac, or of the world by the Meridian circle, or else the degrees of the Equator, which with the portion of the Zodiac given, pass through the Meridian.  
The setting of a sign or of any point of the heaven, offereth 2. things unto our consideration, the definition, and the diversity thereof.  
The definition is either according to the name, or according to the matter.  
According to to the name, it is called the descension or setting, which we measure by the ark of the Equator descending therewith.  
According to the matter it is defined to be the ark of the Equator, comprehended between the sign or point setting, and the head of ♈.  
The consideration of the diversity of setting, is either that a greater part of the Equator descendeth with the sign or point of the heaven, and then it is said to have a right, or along, and slow descension: or else that a less portion of the Equator setterh therewith, and then it is said to have an obliqne, or a short and swift descension.   

Of the diversity of ascensions, descensions, and culminations, in a right sphere. Chap. 34.  
THe Zodiac in a right sphere is fitted unto the equal conversion of the Equator, and together with the parts thereof, passeth by the East, or the West, or the midst of heaven, both in the quadrants or quarters, and in the signs.  
The quadrants in equal spaces of time do ascend and descend, or do pass through the midst of heaven, beginning either at the Solstitial points, namely at the head of ♋ & ♑, and counting to the end of ♍ & ♓: or else, beginning at the Equinoctial points, which are the heads of ♈, & ♎, and counting to the end of ♊ & ♐.  
The signs applied unto the motion of the Equinoctial, are considered either whole, or in parts.  
The signs considered wholly, have relation either unto the Equator, or unto the Zodiac.  
The signs in their relation unto the Equator, do ascend inequally: For some of them do rise rightly, and some obliquely.  
Those that have right ascension are ♊. ♋. ♐. ♑. with the which there do coascende 32. gr. 11. mi. of the Equator.  
Those that have obliqne ascension, are ♈. ♎. ♓. ♍. wherewith there do coascende 27. gr. 54. mi. of the Equator: and ♉. ♏. ♌. ♒. wherewith there arise 29. gr. 54. mi. thereof.  
The signs in their relation unto the Zodiac, or considered severally apart, have ascensions, either equal or inequal one unto another.  
They have equal ascensions, that come forth in equal times, and they are either opposite in the diameter, or equally distant from the Equinoctial points, as are ♓ ♈: ♒ ♉: ♑ ♊: ♐ ♋: ♏ ♌: ♎ ♍.  
They have inequal ascensions that neither are opposite, nor equally distant from the aforesaid points.  
The signs considered in their parts, have also relation either unto the Equator, or unto the Zodiac.  
The parts having relation unto the Equator do (as before) ascend inequally, and that either rightly or obliquely.  
The parts having right ascensions, are comprehended within the four signs, nexte unto the 2. solstitialles.  
The parts having obliqne ascensions, are contained within the signs next unto the Equinoctial points on each side.  
Those parts of the signs, that have relation unto the zodiac, have their ascensions partly equal, and partly inequal.  
Parts having equal ascensions are these: the first degree is equal unto the first degree of the opposite sign: and the first degree unto the last of another sign equidistant from the equinoctial points.  
Parts having inequal ascensions, are those, in whom neither opposition falleth out, nor equidistancie.   

Of the diversity of ascensions, and descensions in an obliqne sphere. Chap. 35.  
IN the obliqne situation of the sphere we consider either the proportion of the ascensions, or of the descensions of the zodiac.  
The ascensions are compared and applied either unto the Equator, or one with another, or unto the ascensions of a right sphere.  
Being compared unto the Equator, they are either equal, or inequal unto the ascensions thereof.  
In their equality they are numbered either in the Northern semicircle from the head of ♈, unto the end of ♍: or from the head of ♎, unto the end of ♓.  
In their inequality, they are reckoned either in the whole semicircles, beginning not in the Equinoctial points, but else where: or else the reckoning is made, in some of their parts.  
In their comparison one with another, they are either equal, or inequal.  
When they are equal, they are reckoned in some 2. concordant arkes of the Ecliptic, as in ♈ ♓: 14. gr. 50. mi. ♉ ♒: 18. gr. 51. mi. ♊ ♑; 27 gr. 16. mi. ♋ ♐: 36. gr. 58. mi. ♌ ♏: 40. gr. 57 mi. ♍ ♎: 40. gr. 58. mi. in the latitude of 40. gr.  
When they are inequal, they are reckoned either in parts not equidistant, or in the semicircle either ascendent, or descendent.  
The semicircle ascendent is from the head of ♑ unto the end of ♊, and that ascendeth more obliqne and swift.  
The descendent semicircle is from the head of ♋, unto the end of ♐, & it ascendeth more right and slow.  
When the ascensions are compared unto the ascensions in a right sphere, they are either less, or more obliqne: or greater, or righter than the said ascensions in a right sphere.  
The less or more obliqne fall out in the North semicircle: the greater or more right happeneth in the South semicircle: the distance between the ascensions of each sphere, is called the difference of ascensions.  
The descensions of the Zodiac, are unto the ascensions thereof either equal, or inequal.  
They are equal either in regard of the moities of the Ecliptic comprehended between the equinoctial points, or else according to the equidistant, or opposite parts of the Zodiac.  
The descensions of the Zodiac are inequal, being compared either unto the right sphere, or unto the same climate.  
The descensions of an obliqne sphere are more obliqne, than the descensions of a right sphere whereunto they are compared, when as the ascensions in an obliqne sphere, are more right than in a right sphere.  
The descensions of an obliqne sphere are more right than the descensions of a right sphere, when as the ascensions in an obliqne sphere, are more obliqne then in a right sphere.  
The inequal descensions of the zodiac, compared unto the same climate are to be noted, either in the parts of the Zodiac which descending obliqne do rise right, such as are the parts of the descending semicircle: or else in the parts of the zodiac, which descending right do rise obliqne, and such are the parts of the ascending semicircle.   

Of the natural day, and of the inequality and difference thereof. Chap. 36.  
Out of the premises we may, not unfitly, derive some matter concerning the days, whereof there are two sorts, the one is called civil or natural the other artificial.  
In the civil or natural day, we may consider the definition, the distinction, and the cause of inequality.  
The definition respecteth either the name, or the thing itself.  
It is called either natural, as caused by the natural, or regulare motion of the whole: or 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 by Prolemee, as consisting of the night and day together: or else civil, because all nations naturally do term it a day.  
The definition respecting the thing is that, according to which it is defined to be the space of 24. hours and certain minutes, consisting of light and darkness.  
The definition thereof is in respect of the continuance and length of the day, and thereof one is called inequal, or different, also the true and apparent day (the Greeks' call it 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, irregulare:) another the equal, or mean day.  
The inequal or different day, is the space of 24. hours and so many minutes, as are answerable unto each portion of the zodiac, which the ☉ doth daily run over.  
The equal or indifferent day, is the space of 24. hours and so many minutes, as are answerable unto the quatity of the mean motion of the ☉ in one day, which is 59 gr. 8. mi.  
The cause of the inequality happeneth unto the true natural day, either in a right, or in an obliqne sphere.  
The cause of the inequality happening in a right sphere, is through the inequal augmentation, by means either of the Equinoctial ascensions inequally answering the same, by reason of the obliquity of the zodiac: or else of the motion of the ☉, which about the centre of the world is inequal  
The cause of the inqualitye of the day happening in an obliqne sphere, is through the inequal augmentation apperteining either to the Equinoctial ascensions inequally answering the same, by reason of the obliquity aswell of the Horizon, as of the zodiac: or else, to the eccentricke circle of the ☉ wherein the ☉ running doth in equal times, perform an inequal motion.  
The Artificial day is handled in the Chap. following.   

Of the artificial day and night, and the diversity belonging to them both. Chap. 37.  
THe ☉ carried about by the first motion, distinguisheth the naturaall day into two parts, whereof the one is called the artificial day, the other the artificial night.  
Concerning the artificial day, Astronomy delivereth the definition, and the proportion thereof.  
The definition containeth the Author, and the term thereof.  
The Author of the artificial day is the ☉, who carried about by the first motion, describeth in the day time a certain ark.  
The term is either from whence: that is, from the easterly part of the Horizon: or by what: that is, by the vertical meridian: or unto what: that is, unto the Westerly part of the Horizon.  
The proportion of the artificial day is delivered in so much as appertaineth unto the length thereof, either in a right, or in an obliqne sphere.  
In a right sphere it is always equal unto it self, and to the night, by reason of the equality both of the Ascensions (for the one half of the Equator doth always equally ascend, and descend with six signs of the zodiac) and of the diurnal, and nocturnal segments.  
In an obliqne sphere the days to themselves and to the nights are either equal, or inequal.  
The days are equal both to themselves and to the nights in the Equinoctial, by reason of the equality both of the ascensions (for look how great the ascension of the diurnal ark is, so great also is the descension of the nocturnal) & of the segments which the ☉ describeth, the said segments being incident with the Equator.  
The days are inequal both among themselues, and to the nights, when the ☉ hath passed the Equinoctial points, aswell by reason of the diversity of the ascensions of the signs, as also by reason of the Sun's inequal describing of the parallels by the motion of the world.  
The artificial night giveth us to consider the definition and the measure.  
It is defined to be the part remaining of the natural day, comprehending the space between the setting of the ☉, and the rising thereof.  
The measure thereof is either equal, or inequal.  
The equality of measure falleth out in the right sphere always, in an obliqne sphere two times in the year.  
The inequality of measure hath notwithstanding either a like diversity in the signs equidistant from the Equator: or alternate in opposite points.   

Of the reason of the equal and inequal hours. Chap. 38.  
Having thus set down the description of the days, it falleth out now, to entreat of their parts commonly called hours, whereof we must consider the general reason, and the division.  
The general reason attendeth their definition, their number, and their subdivision.  
They are defined to be that space of time, wherein the 24. part, or 15. gr. either of the Equator, or of the Ecliptic, do fully arise.  
They are in number 24. belonging unto every natural day.  
Every hour is subdivided into 60. minutes, every minute into 60. seconds, etc.  
The division of the hours consisteth in this, that either they are reckoned in the Ecliptic, or else in the Equator.  
Those that are taken in the Ecliptic, the ascensions whereof do , are called inequal hours, whereof the names, the definition, and the number are to be noted.  
They are called natural (by Io. de sacro bosco) and temporal, and artificial, and Planetary.  
They are defined to be the space of time wherein the moiety of a sign of the Zodiac, counted from the place of the ☉, or the opposite thereof, doth ascend.  
Their number is as much by day, as by night: For 6. signs of the Ecl. do always arise, aswell by day, as by night.  
The hours that are reckoned in the Equator, which ariseth uniformly, are called equal hours, whereof we are in like manner to note the names, the definition, and the number.  
They are called natural (by many) and equinoctial hours.  
They are defined to be that space of time, whe● in 15, gr. of the Equator do fully arise.  
Their number is always inequal, saving in the 2. Equinoctial seasons. For at other times, 6. signs of the Equator do not every day completely arise and set.   

Of the divers accidents of divers parts of the earth, according to the divers situation of the Sphere. Chap. 39  
THe situation of every place and region on the earth, is in the space either of the burnt, or temperate, or frozen zone.  
The places situated in the  zone, are either in the mean spaces, or between mean & extreme, or in the extremes.  
Their situation that are in the mean spaces, differeth from the rest: 1. In the 4. sorts of shadows which they have, viz. Septentrional, Meridional, Oriental and Occidental: 2. In their 4. solstices which they have, two being highest in ♈ & ♎, and two lowest, in ♋ & ♑: 3. In their continual Equinoctialles: 4. In their two Winters, and two Somers.  
Those that have their situation between the means and extremes, do (as the former) differ from the rest: 1. In the double passage of the ☉ over their heads, but not in the heads of ♈ & ♎: 2. In their four shadows and Solstices, although not happening at the same time, as in former situation,  
Those that have their situation in the extremes of the burning zone, do differ from the other: 1. In that the Sun cometh but once unto their Zenith: 2. in the length of their greatest day, which is 3. ho. ½.  
Those places that are situated within the temperate Zone, are either in the extremes, or in the mean.  
The extreme spaces are those, that are under either the trop. of ♋, (whereof we spoke before) or the Arctic circle.  
Those that are under the Arctic circle do differ from other: 1. In that they have the Zodiac coincident with their Horizon, and the pole thereof with their Zenith: 2. In that the signs do arise unto them either most swiftly, or most flowelye: 3. In the length of one day, consisting of 24 hours.  
Those that are situated within the mean spaces of the temperate zone, do differ from others: 1. In their vertical point, which the Sun never cometh unto: 2. In their shadows, which are only 3.  
Those places that are situated within the frozen Zone, are either in the mean spaces, or in the extremes.  
Those that are within the mean spaces of the frozen zone, do differ from other: 1. In the interfection of the Zodiac, and the Horizon in equidistant points: 2. In that some portion of the Zodiac, is always either above the horizon, or under the same.  
Those that are within the extremes of the frozen zone, are either under the Arctic circle, (whereof we speak a little before) or under the pole.  
Those that are under the pole do differ from other:  
1. In their Horizon, which is all one with the Equinoctial: 2. in their day, which is half a year, by reason that the one moiety of the zodiac doth always appear above the horizon.   

Of the diversity of the names of the inhabitants. Chap. 40.  
THe inhabitants of the earth compared one with another, have divers appellations, by reason aswell of the shadows of the ☉, as of the Horizon, or parallels and meridians.  
The shadows cast by the ☉ upon the earth at Noon, are either infinite, or none at all, or else they are finite.  
The shadows that are infinite or equidistant unto the beam, are cast in the frozen zones, whose inhabitants are called Periseti, that is shadowed round about, because their shadows do go in compass round about them.  
Those that have no shadows at Noon tide, are in the  zone, whose inhabitants are named either Aseu, because when the ☉ is in their Zenith, they have no shadow at all: or else, Amphiscii, having 2 shadows, the one Septentrional, when the Sun goeth from them toward the South, the other Meridional, when he passeth from them toward the North.  
Those whose shadows are finite, are named Heteroscii, as having but one of those shadows, either Septentrional, as in the Septentrional temperate zone, or else Meridional, as in the Meridional temperate zone, whereof Lucan maketh mention.  
As concerning the inhabitants of the world, whose comparison one with another standeth upon the Horizon, or the parallels and Meridian's, we have 5. things to consider.  
1. Some of them have the same sensible Horizon, whom Albertus calleth Simul habitantes, dwelling together.  
2. Some of them do dwell under the opposite points of the same parallel, and are called properly by the Greeks' Pertoeci, as if you would say, dwellers about, of the Latins Transuersi, dwellers on the other side.  
3. Some of them dwell under the same parallel, but not in the opposite points, having a divers Longitude, whom Albertus calleth circular dwellers.  
4. Some of them dwell under the points of the same semimeridian equidistant from the Equator, having a contrary Latitude, and are called Antoeci, or Antomis, also obliqne inhabitants.  
5. Some of them do inhabit an equal, or also the same parallel, but under the points of the Meridian diametrallie opposite, and are called Antipodes, Antichthones, and opposite.   

The distinction of the Surface of the earth, according to the length of the dates. Chap. 41.  
FOr the more exact knowledge of the longest days in every place of the world, sensibly changing themselves, the Astronomers have devised, the distinction of parallels, and of Climates.  
The parallels offer unto our consideration, their definition, and their supputation.  
They are defined to be circles distinguishing the climates, and distant one from another at the most, but quarters of hours.  
Their supputation is divers, delivered by 3. sorts of Geographers.  
1. By the common Geographers, which do distinguish the space of the earth from 12. gr. 45. mi. unto 50. gr. 30. mi. into 15. parallels, attributing unto each one, of an hour.  
2. By the Mariners, who in like manner do recon 14. parallels, distinguished by quarters of hours, from the Equator unto 45. gr: but then they proceed by half hours, unto the 19 parallel: and then by adding on whole hour, they come unto the 21. prrallele.  
3. By the more subtle Geographers, who make 48. several parallels, from the Equator toward the pole of the world, unto the 66. gr. 30. mi of elevation: and from thence augmenting them by days, they add 20. more.  
The climates are to be considered in their definition, in their division, in their number, and in their magnitude.  
The definition is thus: A climate is a space of the earth, included within 3. parallels, containing the difference of ½ an hour.  
Their division, is either general, or particular.  
Their general division it that, in regard whereof some of them are called Northern climates, and some Southern.  
The Northern climates have their proper names, derived from the places through the which they do pass.  
The Southern climates are those that are named by the contrary.  
Their particular division is that, in regard whereof every one of them is divided into 3. parallels, the first, the middlemost, and the last parallel.  
Their number is known through the supputation of the parallels.  
Their magnitude is inequal, aswell in respect of their Longitude, as of their Latitude.  
Their Longitude toward the Equator, is greater, by reason of the greater compass of circles: and toward the poles it is less, by reason of their less compass.  
Their Latitude is inequal, in respect of the space of degrees, that half an hour doth contain, and it is greater about the Equator, by reason of the almost equal compass of the degrees: and less about the poles, by means of the narrow inclination of the roundness of the earth.   

Of the light, and of the shadows, and their differences. Chap. 42.  
FOrasmuch as there hath been often mention made of the shadows, it shall not be amiss if we set before your eyes, the methodical description thereof: and seeing that contraties are by their contraries made more manifest, we will declare the nature of the light, and of the shadow.  
The nature of the light is showed by the definition, the division, and the cause thereof.  
It is defined to be the image, or the beam of the bright light.  
It is divided, either into the first and principal, or the secondary and reflexed light.  
The first and principal is that, which proceedeth directly from the light body, and is either perpendiculare, or obliqne.  
The perpendiculare light is that, by the fall whereof right angles are made.  
The obliqne light is that which falleth not at right angles.  
The secondary or reflexed light is that, which from one side spreadeth itself on all parts, without any falling of the beams.  
The cause of the light is either the Elemental bright light, whereof here we teach nothing, o● the celestial.  
The celestial bright light, is that which either causeth the shadow, as that of the ☉, of the ☽, of ♀: or else, which hath no power to make any shadow, as the light of the of the other Stars.  
The nature of the shadow is declared by the definition, and by the division thereof.  
It is defined to be a light diminished: or a certain form of a dark body, always contrary to the body casting the light.  
The division thereof is two fold, the one drawn from the coasts of the world, the other from the position of the dark body.  
The shadow taking the appellation from the coasts of the world, is of 2. sorts.  
The one is extended toward some coast, and it is either Oriental, or Occidental, or Meridional, or Septentrional.  
The other is perpendiculare, or a right shadow by a perpendiculare, which is not extended, as it is unto those, that have the ☉ in their Zenith.  
The division derived from the position of the dark body is that, in respect whereof one shadow is called right or extended, another reversed.  
The right shadow is that, which is caused by the dark body, perpendicularelie erected upon the terrestrial plane.  
The reversed shadow is that, which is caused by the dark body, that is parallel unto the Horizon.   

Of the Eclipses in general. Chap. 43.  
OF all the apparences of the heaven, the Eclipse is the principal: and therefore we will declare the general reason of the same, by the definition, and by the terms thereof.  
The definition is either barely and planelye propounded, or else it is more largely expressed.  
The plane definition thereof is that, whereby it is defined to be the taking away, or the hindering of the bright light, so that it cannot come unto the eye.  
The larger expression thereof is thus: unto every Eclipse there belong 3. things: a bright heavenly light, our sight, and a shadowy or dark body.  
The bright heavenly light was form by the Creator, for the expelling of shadows, and it is twofold, a greater and a less.  
The greater, is that of the ☉, shining of itself.  
The less is that of the ☽, casting about (as out of a looking glass) her light borrowed of the ☉.  
Our sight is divers, according to the divers position thereof, upon the round compass of the earth.  
The shadowy or dark body is also twofold, viz. the body of the ☽, the one moiety whereof the ☉ enlighteneth not: and the earth, whose shadow is always opposite unto the ☉.  
The terms of the Eclipse, which in this kind of doctrine the Astronomers do use, are in number three.  
The first is the quantity of the body either of the ☉, whose visual diameter (as a chord) doth subtende in the Auge of his eccentricke 31. mi. and in the opposite thereof 31. mi. or else of the ☽, whose apparent diameter doth in the Auge of her eccentricke and epicycle, subtende 29. mi. and in the opposite thereof 36. mi.  
The second is the quantity of the shadow, which the motion of the ☉ through either Absis, doth cause to , aswell in regard of the longitude from the surface of the earth, which for the most part containeth 276. semidiameters of the earth, as also in respect of the latitude, which also in the place of the overthwart crossing of the ☽ is diverse, both in respect of the ☉, being in either Absis, and of the ☽ which in her opposition is either in the Auge of her epicycle, and then it is 75. mi. or in the opposite thereof, and then it containeth 94. mi.  
The third is the quantity of the terms eclipsed, either of the ☽, which are 15. parts 12. mi. or of the ☉, by reason of the Parallux of the latitude of the ☽, being either about the South, and it is 11. gr, 22. mi. or about the North, being 20. gr. 40. mi.   

The particular description of the Eclipses, Chap. 44.  
THe beams therefore both of the ☽ and of the ☉, may be hindered from shining upon the earth.  
The beams of the ☽ being borrowed, may be hindered by the coming of the earth, and the shadow thereof, between the ☉ and her, and that maketh the Eclipse of the ☽, whereof we may consider the time wherein it happeneth, & the continuance thereof.  
The time of her eclipse is when she is at the full, when the ☉, being in opposition with the ☽, driveth the shadow either according to the longitude, as every month it cometh to pass, or else according to the latitude, which falleth out when the ☽ is either within or nigh unto the Nodi, that is, the head and tail of the Dragon.  
In the continuance it is to be considered, that the stay of the ☽ in her darkening, is either long or short.  
The long stay is with her whole body, when her opposition falleth out precisely in the Nodi.  
The short stay is when she is distant from the Nodi, and then her body is darkened either all, or half, or less than the half.  
She is darkened wholly, when she hath her latitude less than the semidiameter of the shadow, by the quantity of her apparent semidiameter.  
She is darkened half, when she hath her latitude equal unto the semidiameter of the shadow.  
She is darkened less than the half, when she hath her latitude greater than the semidiameter of the shadow.  
The beams of the ☉ are hindered by the interposition of ☽, and that is called the Eclipse of the ☉, wherein we may have consideration of the time wherein it happeneth, the diversity thereof, and the difference thereof from the Eclipse of the ☽.  
The time wherein the ☉ is eclipsed, is in the new ☽, at which time she seemeth to have a diametral conjunction with the ☉, aswell in respect of longitude, as of latitude.  
The diversity thereof is, in that it is eclipsed either wholly, or less than all.  
The ☉ is wholly eclipsed, when the ☽ being in visible conjunction with the ☉, is in the Nodi.  
The ☉ is eclipsed less than wholly, when as the ☽ being in visible conjunction with the ☉, hath latitude, but yet less than 35. mi. or else, when the semidiameters of the ☉ and the ☽ are joined together.  
The difference of the Eclipse of the ☉ from the eclipse of the ☽, is in regard of the time, the continuance, and the universality.  
The difference considered in the time, is in that the ☽ is darkened in the opposition, but the ☉ in the conjunction.  
The difference considered in the continuance, is in that the darkening of the ☽ falleth out to be long, but the Eclipse of the ☉ but short, by reason of the small quantity of the ☽, and the swift motion thereof.  
The difference considered in the universality, is in that the Eclipse of the ☽ is every where seen, but the Eclipse of the ☉, in one only part of the earth, namely in that, which is covered by the shadow of the ☽.   
FINIS.   

DARY 's MISCELLANIES.  
Being, for the most part, A Brief COLLECTION of MATHEMATICAL THEOREMS, From divers Authors upon these Subjects following.  

I. Of the Inscription and Circumscription of a Circle.  
II. Of plain Triangles.  
III. Of spherical Triangles.  
iv Of the projection of the Sphere in plano.  
V Of Planometry and the Centre of Gravity.  
VI Of Solid Geometry. And (therein) Gauging.  
VII. Of the Scale of Ponderosity, alias, the Stillyard.  
VIII. Of the four Compendiums for quadratique Equations.  
IX. Of Recreative Problems.   
By Michael Dary.  
London, Printed by W. G. and sold by Moses Pitt at the White-Hart in Little-Britain, Tho. Rookes at the Lamb and Ink-bottle in Gresham-Colledge, and Wil Birch at the Bibk in New-Cheapside in moorfield's. 1669.   

TO THE READER.  
Courteous Reader,  
THou hast here presented to thy view and censure a few Mathematical Notes, most whereof have lain by me many years; and the reason of their rushing into the Public in this homely dress, is, for that I and some others have been traduced and derided in a Book lately published, Entitled, A Guide to the Young Gager, put forth by several Authors: In the tail of which Book there is a whole Broadside (intricate, preposterous, inartificial, and most prodigiously erroneous) disgorged by the Lieutenant or Bringer-up against a Book, Entitled, The Art of Practical Gauging; which Book I am sorry to see so encumbered with Press-Faults, though they were none of mine: But most of these Guns we shall Charge again, and turn them upon this our Lieutenant, for we scorn to give an Answer to his invective Examination, till we have first examined him.  
In his Stereo. prop. pag. 6. pr. 2. he tells how to find the Solidity of a Frustum Pyramid whose bases are parallel but not alike: The word Frustum Pyramid I cannot understand, but if he had said Frustum of a Pyramid, he would have been understood by every judicious man: But the Solid there spoken of (as I am informed) might very fitly have been called a Prismoid, for it is a kind of a Prism; for although the Sides thereof should be continued, they would never be included or terminated in one point, as the Pyramid is; therefore, why Frustum Pyramid?  
For the construction of this his Frustum Pyramid, he bringeth (as an induction of particulars) four Pyramids, four Prisms, and a Parallelepipedon, that is in all five Prisms; (for a Parallelepipedon is a Prism) but in reality there is but two Prisms, and the four Pyramids, which are indeed but one Pyramid, in the largest sense it is capable of, both bases being parallel but not alike, each being a rectangle.  
In his Proposition pag. 105. (which he prosecutes in pag. 106, 107, 108.) the stress of his Argument is weak and infirm (I pass by the Press-Fault pag. 105. z+2 for it should be z − 2) for pag. 106. he saith z = ⅗ that is z is 3 and A, 5; these numbers make good the Question: But stay, though we should grant z = ⅗, it is yet to demonstrate, that z is 3 and A, 5; which I am not bound to tell him how to do: But it is evident the man made haste to the 108 page, to fling dirt in the face of Van Schooten (for he doth not care who he doth bespatter) and it is manifest, if you compare this last page with his Title page, which saith, Particularly intended for Gauging; let any man judge, whether this Proposition have any relation at all to Gauging. But we would have him to know, that we can perform this Proposition and two other, of good use for the rational Ordinates' of the Circle and Hyperbola, without casting dirt in the face of any person.  

Prop. 1. xx+yy = zz. ques. x? y? z? Sol. x = aa − bb. y = 2 ab. z = aa+bb.  
Prop. 2. xy+yy = zz. ques. x? y? z? Sol. x = 2 ab+bb. y = aa. z = aa+ab.  
Prop. 3. xx+xy+yy = zz. ques. x? y? z? Sol. x = aa − bb. y = 2 ab+bb. z = aa+ab+bb.   
In all these three Solutions you may put a and b = any two numbers taken at pleasure.  
Moreover in Page 107. his Note for Progressions is invalid, and of no force: For 

the Sum of the Sum and Difference of any two numbers is equal to the double of the— greatest.  
the Diff. of the Sum and Difference of any two numbers is equal to the double of the— lest.    
From whence the Argument is clear by assumption, for he assumes the same greatest number twice: Therefore the Sum of the Sum and Difference must be the same that it was before; to wit, the double of that same greatest number: So it is evident that there is no need of Unity for the first Term of this Progression as he intimates: If he shall say his Note is true notwithstanding my declamation, then let him show a reason why he doth amuse his Reader with such a foolery. Are these the Men of a sound judgement which they speak of in their Preface? The like general Theorem may be laid down in Geometrical Progressions, but no need of Unity for the first Term.  

The Fact of the Fact & quote of any two numbers is equal to the Square of the— greatest.  
The quote of the Fact & quote of any two numbers is equal to the Square of the— lest.   
But this I shall not insist upon.  
In his invective Examination, Pag 1. he saith, That our Equation is five times greater than it needs: If he could have said it had required five times more work in the operation, he had said something; but to see how full he is fraught with Envy, for this very Proposition hath been commended by divers Artists in this City, for the contrivance of it, because it doth accommodate both the Spheroid and the Parabolical Spindle with one and the same Divisor, and the Operation is very near as short as the old way for the Spheroid.  
From the first page to the twelfth he is wholly taken up with inveighing against the Table of Segments; in which I see there are many Press Faults, but these Faults (as I said before) are none of my Faults: For the making of the Table I gave this Rule, 6,2831853) ∶ 0,017453 A − S ∶ × Q (= K, which I shall demonstrate.  
It is manifest from Archimedes, Snellius, and other Authors, that if the Radius of a Circle be Unity, that then the Area of that Circle is 3,14159266 too much, or 3,14159265 too little: Also from the same Authors it holds, that the Area of a Sector is equal to the Fact or Rectangle of the Radius into ½ the Arch or Base of the Sector: Then if you put A = to the Arch, S = to the Sine of that Arch, it must hold 1 × ½A = the Area of the Sector, and 1 × ½S = the Area of the Triangle in the Sector; but the Sector less the Triangle is equal to the Segment of the Circle in that Sector: So then ½A − ½S = any Segment of a Circle whose Radius is Unity.  
And by the 11. and 12. Prop. Partis Cyclicae of Leotaud's Examen of the Quadratures of Greg. of St. Vine. it holds; As the Area of the Circle given 3,14159265, is to ½ A − ½ S; so is Q, the Quadrature or Area of any Circle proposed, to K, its like Segment. Then multiplying both the first and second terms by 2, in Jimbals (for so he calls the Symbols in derision) it will stand thus; 6,2831853) ∶ A − S ∷ × Q ∶ (= K.  
Now because A stands in Degrees and Centesmes, it must be reduced to the same parts that S stands in, which may be done thus; As 360 deg. the whole Peripheria in the Parts of A, is to 6,2831853 the whole Peripheria in the parts of S; so is A, to 0,017453A: therefore 0,017453A, comes in the room of A, and the Rule stands thus; 6,2831853) ∶ 0,017453A − S ∷ ×Q (∶ = K which was to be demonstrated.  
In page 12. this insulting Scosser saith, This may serve for instruction to Segment-makers' for the future, to inform them whether their Work goes on regularly or not.  
Reader, I hope thou hast more understanding than to take this Bringer-up for one that modestly makes public pure Geometry (as he speaks in his Preface.) But if thou dost, I shall undeceive thee: For it is apparently true, that if a Table of the Segments of a Circle shall be differenced never so often, they have no equal Differences; but he seems to intimate that they have: For if he mean otherwise, I would fain know of him the Habitude of that rank of Differences by which he will prove the truth of a Table of Segments. But methinks I hear the Reader object and say, That if a Table of Segments be differenced far enough, the Differences at last will be equal: Stay there, herein lies the deception; for those Figures that appear to be equal, are only frontier-Figures, and if you make the Table of Segments a good company of places larger, you will then see the ragged Regiment that stands behind these Frontiers. This he ought to have told his Reader, otherwise he publisheth (not pure, but) very impure Geometry.  
In page 13. he proceeds to the Trial of the Tables of Wine, and Beer and Ale; and the Table for dividing the Gauging Rod, and he concludes them also badly calculated, as may be proved (saith he) by taking the second or third Differences.  
What? the second, or the third? 'Tis a marvel when his hand was in, he had not put in the fourth Differences too: O ye Blind Guides! know ye not that the Tables for Wine, Ale and Beer, are capable but only of the first and second Differences; which I prove thus for the Wine: the construction of the Wine Table is the Square of any Diameter in Inches, divided by 883. Then if you put D = a Diameter proposed (do you see now) in Jimbals it will stand thus: DD+0D+ 0 1 Diff. 2 Diff. DD+2D+ 1 2D+1  DD+4D+ 4 2D+3 2 DD+6D+ 9 2D+5 2 DD+8D+16 2D+7 2  
All to be divided by 883. And if you should interpole never so often, you shall find no more but the second Differences. But it seems they will have the second and third Differences come in, for 'tis all one to Anthony who Kisseth Dorothy.  
But what if our Lieutenant should say he did intent the third Differences for the Trial of the Gaging-Rod Table: But certainly he did not, (out of doubt he would not be so unkind to his Young Gager, to leave him thus in the dark guideless,) if he did he is caught in an evil Net, for there he might as well have said the thirteenth as the third, for that Table is the ordinates' of a Parabola standing at equal distances. This may serve for an instruction to our Table-Tryers, how they burn their Fingers again: Let them learn how to be Table-makers' before they turn Table-Tryers.  
Courteous Reader, I am sorry I have held thee in this discourse so long: Now let me address myself to thee, that we may understand one another. In Chap. 6. Prop. 1. thou wilt meet with a Solid, which I call a Prism; by which word is there meant a Solid having two Bases, equal, parallel, alike, and alike situate, and in the Peripetasma a Right Line may be every where applied, from one Base to another.  
A Pyramid of the same Base and height with the Prism, is ⅓ thereof: And in the Peripetasma a Right Line may be every where applied, from the Base to the Vertex.  
A Pyramidoid of the same base and height with the Prism, is some certain portion thereof; as if it be parabolical, it is ½ if it be spherical, it is ⅔: And in the Peripetasma a Right Line may be no where applied from the Base to the Vertex.  
For my Division, it is such as is used by others: As for Example, 24) ∶ 37+41S ∶ ×4 (= 13. and may be read thus; 24 dividing 37 more 41, multiplied by 4, equal to (or quotes) 13: Or it may be read by analogy thus: As 24, is to 37 more 41; so is 4, to 13. If thou merest with some Divisions that stand double lined, they were things that had lain by me a good while, and I would not stand to alter them.  
So craving thy favourable construction, where any thing hath slipped amiss, for it was not the intent of him who desires, if he were able, to be  
Thine to serve thee, Michael Dary.   

ERRATA.  
THe five last Pages are wrong numbered, 42, 43, 44, 45, 46 they should be 44, 45, 46, 47, 48; and then pag. 24. lin. 20. for of it's read of all its pag. 26. lin. 15. for A = S read A − S p. 29. l. 5. for L. = read = L. p. 31. l. 5. for 0,159 read 0,16 p. 33. l. 1. for If read 8. If p. 36. l. 12. for Conjugate read rectangular Conjugate l. 16. for rectanguled read rectangular l. 18. for rectanguled read rectangular p. 37. l. 8. for = D read = D, p. 43. l. 3. & 4. for sides read lines p. 45. l. 14. for let be read let it be, and for terms in read terms: In p. 47. l. 16. for number read any number. Pag. 36. for Convex Begirter or Zone you may read Peripetasma.   

The Contents.  

CHap. 1. Of the Inscription and Circumscription of a Circle. pag. 1.  
Chap. 2. Of Plain Triangles. pag. 10.  
Chap. 3. Of Spherical Triangles. pag. 13.  
Chap. 4. Of the projection of the Sphere in plano. pag. 20.  
Chap. 5. Of Planometry and the Centre of Gravity. pag. 23.  
Ch. 6. Of solid Geometry. p. 29.  
Chap. 7. Of the Scale of Ponderosity, alias the Stillyard. p. 43  
Chap. 8. Of the 4 Compendiums for quadratique Equations. pag. 45.  
Chap. 9 Of recreative Problems. pag. 47.    

Dary's Miscellanies.  

CHAP. I. Of the Inscription and Circumscription of a Circle.  
1. FOrasmuch as the Ratio of an Arch line to a right line is yet unknown, it is absolutely necessary, that right lines be applied to a Circle for the Calculation of Triangles wherein Arch lines come in Competition.  
2. Right lines applied to a Circle are Chords, Sins, Tangents, Secants, and versed Sins.  
3. The Chord of an Arch is a right line extended from one end of that Arch to the other end thereof: The Sine is a right line drawn from one end of that Arch, Perpendicularly upon the Diameter drawn from the other end of that Arch: The Tangent is a right line touching one end of that Arch extended till it Concur with the Secant: The Secant is a right line extended from the Centre of the Circle till it Concur with the Tangent: The Versed Sine is a right line being a Segment of the Diameter, drawn from one end of that Arch till it be cut by a Perpendicular (i. e. the Sine) from the other end of that Arch.  
4. It is to be noted by this Definition in Prop. 3. that the Chord of an Arch is common to two Arches, one of them being the Compliment of the other to a whole Circle; and likewise the Versed Sine is common to two Arches, one of them being the Compliment of the other to a whole Circle: But the Sine of an Arch is common to two Arches, one of them being the Compliment of the other to a Semicircle.  
5. As the Sum of two Sines is to their difference, so is the Tangent of the ½ Sum of those Arches, to the Tangent of their ½ difference.  
6. As the Sum of two Tangents, is to their Difference, so is the Sine of the Sum of those Arches, to the Sine of their Difference.  
7. As the Sine of the Sum of two Arches, is to the Sum of their Sins, so is the difference of those Sins, to the Sine of their Difference.  
8. If you put R = The Radiu s of a Circle, A = an Arch proposed, C = the Chord of that Arch, S = the Sine of that Arch, T = the Tangent of that Arch, and Z = the Secant of that Arch.  
Then  
9 If twice three Arches equi-different be proposed, Then, as the Sine of one of the means, is to the sum of the Sins of its Extremes; so is the Sine of the other mean, to the sum of the Sins of its Extremes.  
10. And hence, if a rank of Arches be equi-different, As the Sine of any Arch in that rank, is to the sum of the Sins of any two Arches equally remote from it on each side; So is the Sine of any other Arch in the said rank, to the sum of the Sins of two Arches next to it on each side, having the same common distance.  
11. Three Arches equi-different being proposed, If you put Z = the Sine of the greater extreme, Y = the Sine of the lesser extreme, M = the Sine of the Mean, m = the Co-sine thereof; D = the Sine of the common difference, d = the Co-sine thereof, and R = the Radius.  
 
12. From the last before going, it is evident, that if two thirds (i. e either the former, or the latter 60 deg. or the former 30 deg. and the latter 30 deg.) of the Quadrant be completed with Sines: the remaining third part of the Quadrant maybe completed by Addition or Subduction only.  
13. If in a Circle, two right lines be inscribed cutting each other, The Rectangles of the Segments of each line are equal. And the Angle at the point of Intersection is measured by the Half-sum of its intercepted Arches.  
14. If to a Circle two right lines be ascribed from a point without; The Rectangles of each line from the point assigned to the Convex and Concave are equal. And the Angle at the assigned point is measured by the half difference of its intercepted Arches  
15. If in a Circle (or an Elipsis) three right lines shall be inscribed, one of them cutting the other two: Then the Rectangles of the Segments of each line so cut, are directed proportional to the Rectangles of the respective Segments of of the Cutter.  
16. If a plain Triangle be inscribed in a Circle, the Angles are one half of what their opposite sides do subtend.  
17. Therefore the Angles of a plain Triangle are equal to a Semicircle.  
18. And hence, if a Rectangled Triangle be inscribed in a Circle, the Hypothenuse thereof is the Diameter of the Circle.  
19 As the Diameter of a Circle is to the Chord of an Arch; so is that Chord, to the versed Sine of that Arch.  
20. And hence, if from the right angle of a rectangled Triangle, a Perpendicular be let fall upon the Hypothenuse, the Hypothenuse is thereby cut according to the Ratio of the squares of the sides.  
21. If in a Circle, any plain Triangle be inscribed, and a Perpendicular be let fall upon one of the sides, from the opposite angular point; Then as that Perpendicular is to one of the adjacent sides, so is the other adjacent side, to the Diameter of the Circumscribring Circle.  
22. If a Circle be inscribed within a plain Triangle, Then, as the Perimeter is to the Perpendicular; so is the Base on which it falleth, to the Radius of the inscribed Circle.  
23. If a Quadrilateral Figure be inscribed in a Circle, and Interfect with Diagonals, The Rectangle of the Diagonals is equal to the two Rectangles of the opposite sides.  
24. If a Circle be both inscribed and circumscribed by two like ordinate Polligons; Then, as the Co-versed Sine of the side of the inscribed is to the Diameter, so is the Area of the Inscribed to the Area of the Circumscribed.  
25. If an ordinate Polligon be both Inscribed and Circumscribed by two Circles; Then, as the Diameter of the Circumscribed, is to the co-versed Sine of the side of the Polligon; So is the Area of the Circumscribed, to the Area of the Inscribed.  
26. In any right lined Figure, if a Circle be Inscribed; Then, as the Peripheria of the Circle, is to the Area thereof; So is the Perimeter of the right lined Figure, to the Area thereof. Et Contempt  
27. But in all Circles, as the Peripheria is to the Area, so is 2 to the Radius.  
28. Therefore; In any right lined Figure, if a Circle be inscribed, as 2. is to the Radius; So is the Perimeter of the right lined Figure, to the Area thereof.   

CHAP. II. Of Plain Triangles.  

1. A Triangle is a Figure Comprehended of three sides; and is either Plain or Spherical.  
2. A Plain Triangle is that which is described on a Plain Surface, whose three sides are right lines; and it is either right Angled or obliqne Angled; and the obliqne, is either obtuse or acute.  
3. If a line drawn from the top or vertex of a Triangle equally Bisecting the Base, be equal to the Bisegment, the Vertical. Angle is a right Angle; if lesser Obtuse, if greater Acute.  
4. In a Plain Triangle, a right line equally Bisecting the Vertical Angle, cuts the Base directly according to the Ratio of the adjacent legs.  
5. Any one side of a Triangle is less than the sum, and greater than the difference of the other two sides.  
6. Any one side being continued, the exterior Angle is equal to the two interior Angles opposite.  
7. In any right angled Plain Triangle, the sum of the squares of the sides containing the right Angle is equal to the square of the Hypothenuse.  
8. In a Plain Rectangled Triangle any one of the sides may be put for Radius, and the other sides shall be Sines, Tangents, or Secant.  
9 In any plain Triangle the sides are directly proportional to the Sins of their opposite Angles, Et Contempt  
10. In any Plain Triangle, as the sum of any two sides is to their difference, so is the Tangent of the Half-sum of their opposite Angles, to the Tangent of their Half-difference.  
11. In any Plain Triangle, as the Base is to the sum of the Legs, so is the difference of the legs, to the difference of the Segments of the Base, cut by a Perpendicular from the vertical Angle.  
12. In any Plain Triangle, as the Base is to the sum of the legs; so is the Sine of ½ the vertical Angle, to the Sine of the sum of ½ the vertical Angle, and either of the Angles conterminate at the Base  
13. In any Plain Triangle, as the Diameter is to the versed Sine of the vertical Angle; 

So is the square of the sum of the legs less by the square of the difference of the legs  
To the square of the Base less by the square of the difference of the legs    
14. In any Plain Triangle, the fact of the legs and the Sine of their Angle, is equal to the fact of the Base, Petpendicular, and Radius.    

CHAP. III. Of Spherical Triangles.  

1. A Spherical Triangle, is that which is described on the surface of a Sphere.  
2. The sides of a Spherical Triangle, are Arches of three great Circles mutually intersecting each other.  
3. The measures of Spherical Angles, are arches of great Circles, described from the Angular Points as their Poles, and subtending their Angles.  
4. Those are said to be great Circles, which bisect the Sphere.  
5. Those Circles which cut each other at right Angles, the one of them passeth by the Poles of the other. Et Contempt  
6. The Distance of the Poles of two great Circles, is equal to the Angle comprehended by them.  
7. The 3 Angles of any Spherical Triangle being given, there are likewise three sides of another Spherical Triangle given, whose Angles are equal to the sides of the former Triangle.  
8. The sum of the Sides of a Spherical Triangle are less than two Semicircles.  
9 The sum of the 3 Angles of a Spherical Triangle, are greater than two right Angles, but less than Six.  
10. Two Angles of any Spherical Triangle, are greater than the difference between the 3 Angle and a Semicircle.  
11. Any side being continued, the Exterior Angle is less than the two Interior opposite ones.  
12. In any Spherical Triangle, the difference of the sum of two Angles and a whole Circle, is greater than the difference of the third Angle and a Semicircle.  
13. A Spherical Triangle, is either Rect-angular, or Obliqueangular.  
14. A Rect-angular Spherical Triangle, is that which hath one right Angle at the least.  
15. The Legs of a Rect-angular Spherical Triangle, are of the same affection with their opposite Angles.  
16. In a Rect-angular Spherical Triangle, if either leg be a Quadrant, the Hypothenuse is also a Quadrant; but if both be of the same affection, the Hypothenuse shall be less than a Quadrant; if of different affections, than greater: Et Contempt  
17. In a Rect-angular Spherical Triangle, if either of the Angles at the Hypothenuse be a right Angle, the Hypothenuse shall be a Quadrant; but if both of the same affection, it shall be less, if different then greater: Et Contempt  
18. In a Rect-angular Spherical Triangle, either of the Oblique-angles is greater than the Compliment of the other, but less than the difference of the same Compliment to a Semicircle.  
19 An Obliqne angular Spherical Triangle, is either Acute-angular, or Obtuseangular.  
20. An Acute angular Spherical Triangle, hath all its Angles Acute.  
21. An Obtuse angular Spherical Triangle, hath all its Angle; Obtuse or mixed, viz. Acute and Obtuse.  
22. In an Acute angular Spherical Triangle, each side is less than a Quadrant.  
23. In an Obliqne angular Spherical Triangle, if two Acute Angles be equal, the sides opposite to them shall be less than Quadrants; if Obtuse, greater.  
24. In an Obliqne angular Spherical Triangle, if two Acute Angles be unequal, the side opposite to the lesser of them shall be less than a Quadrant; but if Obtuse, the side opposite to the greater, shall be greater.  
25. In every Obliqne angular Spherical Triangle, if the Angles at the Base be of the same affection, the Perpendicular drawn from the top of the Vertical Angle shall fall within the Triangle; if different, without.   

In Obliqne angular Spherical Triangles, if a Perpendicular be drawn from the Vertical Angle, to the opposite side, (continued if need be.)  

COnsectary 1. The Cosines of the Segments of the Base are directly proportional to the Cosines of the sides of the Vertical Angle: Et Contempt  
Con. 2. The Cosines of the Angles at the Base are directly proportional to the Sins of the Vertical Angles: Et Contempt  
Con. 3. The Sins of the Segments of the Base are reciprocally proportional to the Tangents of the Angles Conterminate at the Base: Et Contempt  
Con. 4. The Cosines of the Vertical Angles are reciprocally proportional to the Tangents of their sides: Et Contempt    

Axioms for the Solution of Spherical Triangles.  
Axiom 1. IN Rect-angular Spherical Triangles having the same Acute angle at the Base: The Sins of the Hypothenuses are proportional to the Sins of their Perpendiculars.  
Axiom 2. In Rect-angular Spherical Triangles having the same Acute angle at the Base: The Sins of the Bases and the Tangents of the Perpendiculars are proportional.  
Axiom 3. In all Spherical Triangles, the Sins of the Angles are directly proportional to the Sins of their opposite sides: Et Contempt  
Axiom 4. In all Spherical Triangles, as the fact of the sides containing the Vertical angle, is to the square of the Radius; So is the fact of the Sins of the ½ sum, and the ½ difference of the Base and difference of the Legs, to the square of the Sine of ½ the Vertical Angle.  
Or thus. In all Spherical Triangles, having added all the 3 sides together, find the difference betwixt each side and their half sum: And then,  
As the fact of the Sins, of the ½ sum of all the sides, and the difference of the side opposite to the Vertical Angle, is to the fact of the Sins of the differences of the containing sides from the said ½ sum, so is the square of the Radius to the square of the Tangent of ½ the Vertical Angle.    

CHAP. IU. Of the Projection of the Sphere in Plano.  

1. Orthographically.  
IF a Sphere be by a Plain cut into two Hemispheres, and the Eye be placed at an infinite distance, Vertically to one of the Hemispheres; then a right line infinitely extended from the Eye, to any assigned point in the Spherical surface of that Hemisphere, shall porject the assigned point upon the Plain: And the distance upon the Plain, from the apex of the Hemisphere to the projected point, is equal to the sine of the Arch from the Vertex of the Hemisphere to the assigned point; the Radius of the Sphere being put for Radius.   

2. Steriographically.  
If a sphere be by a plain touched, and the eye be placed in the Spherical-surface Diametrically opposite to the touch-point; Then a right line infinitely extended from the eye to any assigned point in the Spherical surface shall project the assigned point upon the Plain: And the distance upon the Plain from the touch-point to the projected point is equal to the Tangent of ½ the Arch from the touch-point to the assigned point: The Diameter of the Sphere being put for Radius.   

3. Gnomonically.  
If a Sphere be by a Plain touched, and the Eye be placed at the Centre of the Sphere; Then a right line infinitely extended from the Eye to any assigned point, in the Spherical surface (whose distance from the touch-point is less than a Quadrant) shall project the assigned point upon the Plain: And the distance upon the plain from the touch-point to the projected point is equal to the Tangent of the Arch from the touch-point to the assigned point: The Radius of the Sphere being put for Radius.   

To project a Meridian line upon any Horizontal plain.  
1. Having prepared a piece of Metal or Wood, and made it a true Plain, and in some convenient point thereof (taken as a Centre) erected a Gnomon of sufficient length, at right angles to the Plain: This done, six the Plain truly Horizontal  
2. If you take the Sun's Co-altitude (i. e. his distance from the Zenith) three times in one day, and (according to the Steriographical projection (having a line of Tangents by you) set off from the Centre of your plain or foot of the Gnomon, the Tangent of ½ each Arch upon his respective Azimuth, or shadow (continued if need be) made by the Gnomon at that instant when the Co-altitude is taken, you will insert three points upon the Plain.  
3. If you find out the Centre to those three inserted points, than a rigth line infinitely extended by this Centre found, and the Centre of the Plain, or ●oot of the Gnomon, is the true Meridian line: Which was to be projected.    

CHAP. V Of Planometry, and the Centre of Gravity.  
1. IN any plain Triangle, the fact of the Base, and the Perpendicular, is equal to the double of the Area of that Triangle.  
2. In any Plain Triangle, as the Diameter is to the Sine of any one of the angles, so is the fact of the adjacent Legs to the Area.  
3. And hence, in how many soever Plain Triangles, having one angle equal in Common, the facts of their sides including the common Angle, are directly proportional to their Areas. Et Contempt  
4. In any plain Triangle, as the fact of the Diameter and the Sine of any one Angle, is to the square of the opposite side; so is the fact of the Sins of the other two Angles to the Area.  
5. If you put P = the Semi-perimeter of any plain Triangle, and D, D, d, = the respective differences accrueing by Subduction of each particular side from the Semi-perimeter, and the Area = A, Then: 1) √ D D d P (= A.  
6. A Triangulate (i. e. any right lined figure) is Composed of Triangles, and the Triangles are less by two than the number sides, and the Diagonals are less by three: And the Area thereof is equal to the Area of its Triangles.  
7. If you put 1 = the Radius of a Circle, than the Area (and also the Semiperipheria) shall be = 3, 1416 fere: according to Van Culen, Snellius, and Hugenius.  
8. If you put D = the Diameter of a Circle, P = the Peripheria, and A = the Area, then:  
9 If you put K = the Area of the Segment, or Kant of a Circle; V = the Versed Sine in the Segment, D = the Diameter of the Circle, or Q equal to its Area; also if you put D) 2 V (= u, a Versed Sine (to be found in a Table of Versed Sins, the Diameter being 2,000 etc.) whose respective Arch in Degrees and Decimals being doubled, you may call A, and the correspondent Sine of A you may call S, Then: 8) ∶ 0,017453 A − S ∶ × D D (= K Or 6,2831853) ∶ 0,017453 A − S ∶ ×Q (= K  
10. Again, if you put K or k = the greater or lesser Segment of a Circle, cut by a Chord-line, D = the Diameter of the Circle y = the difference of the Segments of the Diameter cut at right Angles by the foresaid Chord-line, also if you put D) y (= S, a Sine (to be found in the Common Table of Sines, the Radius being 1,0000 etc.:) whose respective Arch in Degrees and Decimals being doubled you may call A, and the correspondent Sine of A you may call S, Then:  
Another way:  
11. If you put Z = the Area of a Zone of a Circle intercepted between the Diameter and Chord-line parallel to it, D = the Diameter of the Circle, B = the breadth of the Zone, Then:  
12. If you put k = the Area of the Segment of a Circle (not greater than a Semicircle,) C = the distance of the Centre of Gravity from the apex of the Segment, r = the Radius, c = the Chord of the Segment, and u = the Versed Sine in the Segment, Then:  
13. If you put H = the Area of an Hyperbola, G = the distance of the Centre of Gravity from the apex of the Hyperbola, a = the Axis, B = the Base, r = the Semitransverse Diameter (between the Vertex of the Hyperbola, and the Centre of the assymtoptes) Then:  
14. If you put D = the distance of the Centre of Gravity of a Plain right lined Triangle from one of the Angular points, and I = the right line from that Angular point Bisecting the opposite side, Then: 3) 2 l (= D.  
15. If you put D = the distance of the Centre of Gravity of the Sector from the Centre of the Sector, c = the Chord, r = the Radius Bisecting A = the Arch, Then: 3 A) 2 r c (= D.  
16. If you put L = the distance of the Centre of Gravity of a, (= the lesser of two Superficial figures proposed) and I = the distance of the Centre of Gravity of A, (= the greater of two Superficial Figures proposed) from the common Centre of Gravity of both the foresaid Figures, and D = the whole distance of their respective Centres of Gravity, Then:  
A + a) D A (L. = and A+a) D a (= l.  
17. As an unite is to the Radius, so is the excess of the 3 angles above a Semicircle (in a Spherical Triangle) to the bossed surface of that Triangle. The excess is to be taken in the same parts, as is the Radius.  
18. If a Sphere be enclosed in a Cylinder, and that Cylinder be cut with plains parallel to its base; then the Intercepted rings of the Cylinder are equal to the Intercepted surfaces of the respective Segments of the Sphere.   

CHAP. VI Of Solid Geometry.  
1. If you put Pr = A Prism, B = one of its Bases, and P = the Perpendicular height of the Prism, Then: 1) B P (= Pr.  
2. If you put Fr = the Frustum of a Pyramid or a Cone intercepted between two Plains Parallel cutting the axis, B = the greater Base, b = the lesser Base, and P = the Perpendicular height of the Frustum Then, 3) ∶ B+ √ B b + b ∶ × P (= Fr.  
3. If you put Fr = the Frustum of a Spherical Piramidoid, Sphere, or Spheroid, Intercepted between two Plains parallels, one of them passing by the Centre, B = the greater Base, b = the lesser Base and P = the Perpendicular height of the Frustum, Then: 3) ∶ 2 B + b ∶ × P (= Fr.  
4. If you put Fr = the Frustum of a Porabolical Piramidoid or Conoid Intercepted between two Plains parallel, cutting the axis, B = the greater Base, b = the lesser Base, and P = the Perpendicular height of the Frustum: Then 2) ∶ B + b ∶ × P (= Fr.  
5. If you put S = a Sollid made by Rotation, R = the Radius or nearest distance, between the Centre of Cravity of A = the begetting Figure, and a right line in the same Plain assigned (without) for an axis, and p = the Peripheria of a Circle whose Radius is unity, Then:  
1) A R p (= S. or Thus 0,159) ARE) = S.  
6. If you put Fr = the Frustum of a Parabolical Spindle, intercepted between two plains Parallel, one of them passing by the Centre, B = the greater Base, b = the lesser Base, P = the Perpendicular height of the Frustum, Then:  
15) ∶ 8 B + 7 b ∶ × P (= Fr.  
7. In the 2, 3, and 4 Propositions: (which Propositions are general for Pyramids or Cones, Piramidoids or Conoids, of what Base soever) If it will serve your turn to find only the Cone, Conoids, and Parabolical spindle, when their Bases are Circles, it may be delivered thus,  
1. If you put Fr = the Frustum of a Cone incepted between to Plains parallel, cutting the Axis at right Angles, D = the Diameter of the greater Base, d = the Diameter of the lesser Base, P = the Perpendicular height of the Frustum, and make S = D + d, Then: 3, 82) ∶ S S − D d ∶ × P (= Fr.  
2. If you put Fr = the Frustum of a Sphere or Spheroid, intercepted between two Plains parallel, one of them passing by the Centre Cutting the Axis at right Angles, D = the Diameter of the greater Base, d = the Diameter of the lesser Base, and P = the Perpendicular height of the Frustum, Then: 3, 82) ∶ 2 D D + d d ∶ × P (= Fr.  
3. If you put Fr = the Frustum of a Parabolical Conoid, intercepted between two Plains parallel cutting the Axis at right Angles, D = the Diameter of the greater Base, d = the Diameter of the lesser Base, and P = the Perpendicular height of the Frustum, Then: 2,54) ∶ D D + d d ∶ × P (= Fr.  
4. If you put Fr = the Frustum of a Parabolical Spindle, intercepted between two Plains Parallel, one of them passing by the Centre cutting the Axis at right Angles, D = the Diameter of the greater Base, d = the Diameter of the lesser Base, and P = the Perpendicular height of the Frustum, Then: 1 9,1) ∶ 8 DD + 7 dd ∶ × P (= Fr.  
If you put Fr = the Frustum of a Sphere intercepted between two Plains parallel, one touching and the other cutting the Sphere, d = the Diameter of the Base, and P = the perpendicular height of the Frustum, Then: 2,54648) ∶ dd. ●. 1⅓ P P ∶ × P (= Fr.  
This Rule will also hold if it were the frustum of a Spheroid, putting d d: = the fact of the right angled Conjugates in the base.  
9 If you put Fr = the frustum of a Cone intercepted between two Plains parallel, one of them being fixed the other movable, D = the Diameter of the fixed base, p = the perpendicular height of the frustum, and d = the increment or decrement of any two Diameters at one Inch distance in the perpendicular, Then: 3,82) ∶ 3 DD±3 Ddp+ddpp ∶ × p (= Fr.  
10. A Cooper's common Cask, that is such as are round at their heads (and not eliptical as some Oil Cask are) being Proposed, if you put D = the Diameter at the bouldge, d = the Diameter at the heads, G = the diagonal from the middle of the bongue-hole to the bottom of either of the heads, L = the length of the Vessel, and make S = D+d, Then:  
11. According to the Equated Circle, now in use, if you put Q = the Quantity of Liquor in a Cooper's common Cask being filled totally or partly, the axis being posited parallel to the horizon, the Vessel being taken as the middle Frustum of a parabolical Spindle intercepted between two Plains parallel, equidistant from the Centre cutting the axis at right angles, D = the Diameter of the bouldge, V = that proportion of this Diameter which is wet, d = the Diameter at the heads, and P = the perpendicular height or length of the Vessel, also you shall put D) V (= N, which N abuts you to K = a Segment to be found in the Table of the Segments of a Circle whose Area is unity: Or if you have not by you a Table of Segments you may find K by Chap. 5. Prop. 9 and then if you divide by 19, 1 you shall have cubical Inches, if by 4412, Wine Gallons; if by 5386, Ale or Beer Gallons; and the Rule will stand thus: 19,1) ∶ 8 D D + 7 d d ∶ × P × K (= Q.  
But if the said Vessel be taken as the frustum of a Spheroid intercepted, etc. Then instead of: 8 D D + 7 d d: you shall put: 10 D D + 5 d d: And yet the foresaid Divisors hold true to all intents and purposes.  
12. Concerning the Cylindroid, etc. in its several kinds and several frustums.  
By the word Cylindroid (in this place) is meant a Solid contained under three Surfaces (i. e.) two Plains parallel and a Convex begirter, whereof the two Plains parallel are called the Bases, and are both Circles or both Ellipsis, or else one a Circle and the other an Ellipsis, and the Convex Surface is called the Zone; in which Zone there may be every where a right line applied from any point in one base to some point in the other: and if such a Cylindroid be cut with two Plains meeting in the Centres of both Bases, cutting (or rather inserting) conjugate Diameters in both Bases, Then: 

If you put C = the solid Content of a Cylindroid.  
A & B aloft = the two rectangled conjugate Diameters.  
G & H below = the two rectangled conjugate Diameters.  
A & G opposite = the two correspondent Diameters.  
B & H opposite = the two correspondent Diameters.  
Also P = the perpendicular height of the Cylindroid, Then:    
13. To cut the Cylindroid with divers Plains parallel, the Plains being of one common distance, and that distance being taken for Unity in the lineal Mensuration of the Cylindroid: To do this from that Base wherein G and H are Conjugates, you shall make P) A − G (= D and P) B − H (= d, and Then:  
All to be divided by 3,82.  
If it shall so happen that A = B and G = H, it is the frustum of a Cone, Then:  
All to be divided by 3,82.  
If it shall so happen that B = H, Then:  
All to be divided by 3,82.  
If it shall so happen that B = H = G, Then:  
All to be divided by 3,82.  
If it shall so happen that A = B = G = H, Than it is a Cylinder: to be divided by 3,82.  
14. Now instead of P = the perpendicular of the whole height of the Solid, if you shall put p = the perpendicular height of any part thereof, from that Base wherein G and H are Conjugates, and C = the solid content of the Cylindroid at that height, than it holds as in the margin:  
But if such a solid have not its Zone made by Circles or Ellipsies but by four flat sides at right angles to the foresaid Conjugates, than it is a Prismoid: Nevertheless the Rules before prescribed hold to all intents and purposes, if you take away the Divisor 3,82 and in the room thereof place the Divisor 3.  
15. This last Proposition to find the Content of the Cylindroid or Prismoid at any height or depth may be also performed by the Table of figurate Numbers following, thus: 

1. Having got the first Frustum, first, second, and third differences (if there be so many) of your Solid, multiply them by the respective numbers in the Table at that height or depth.  
2. Add all these Products together into one Sum, having respect to the signs + and − (if they shall be so signed) and this Total is the Content of your Solid at the height or depth proposed.    

A Table of Figurate Numbers for the speedy Collection and exact Correction of the first, second and third Differences, in finding the Content of the Cylindroid or Prismoid.  Depth. 1 Frus. 1 Diff. 2 Diff. 3 Diff. 01 1 00 000 0000 02 1 01 000 0000 03 1 02 001 0000 04 1 03 003 0001 05 1 04 006 0004 06 1 05 010 0010 07 1 06 015 0020 08 1 07 021 0035 09 1 08 028 0056 10 1 09 036 0084 11 1 10 045 0120 12 1 11 055 0165 13 1 12 066 0220 14 1 13 078 0286 15 1 14 091 0364 16 1 15 105 0455 17 1 16 120 0560 18 1 17 136 0680 19 1 18 153 0816 20 1 19 171 0969 21 1 20 190 1140 22 1 21 210 1330 23 1 22 231 1540 24 1 23 253 1771 25 1 24 276 2024 26 1 25 300 2●00 27 1 26 325 2600 28 1 27 351 2925 29 1 28 378 3276 30 1 29 406 3654 31 1 30 435 4060 32 1 31 465 4495 33 1 32 496 4960 34 1 33 528 5456 35 1 34 561 5984 36 1 35 595 6545 37 1 36 630 7140 38 1 37 666 7770 39 1 38 703 8436 40 1 39 741 9139 41 1 40 780 9880 42 1 41 820 10660 43 1 42 861 11480 44 1 43 903 12341 45 1 44 946 13●44 46 1 45 990 14●●0 47 1 46 1035 15●●0 48 1 47 10●● 16●●● 49 1 48 11●● 〈◊〉 50 1 49 11●● 〈◊〉  
3. The construction of this Table is very easy, for you may see it is nothing but a Collection of Unites: But to make the respective numbers in this Table for the first, second, and third Differences to any Depth proposed, without any gradual Collection, this is the Rule.  
If you put D = the Depth  
 
16. The symmetry of like Superficies and like Solids. Superficies are like, and Solids are like, if the Angles be equal in number and quantity,  
17. Like Superficies are one to another as the Squares of their Correspondent Sides: Like Solids are one to another as the Cubes of their Correspondent Sides.   

CHAP. VII. Of the Construction and Use of the Scale of Ponderosity (commonly called) the Stillyard.  
1. A Right Line resting on a fulciment equiponderate being proposed: Then if any ponderosity shall be applied to a point of Pendency in that ●ine; it ought to be understood, that that Ponderosity is transplanted from the Fulciment to that point of Pendency: But if any Ponderosity shall be withdrawn or taken away from a point of Pendency in that Line, it ought to be understood that that Ponderosity is transplanted from the point of Pendency to the Fulciment.  
2. A Right Line resting on a Fulciment equiponderate being proposed: If divers Ponderosities shall be pendantly applied on sundry points of that Line, so that the said line be equiponderate again: Then the facts (of each Ponderosity by its transplantation from the Fulciment) on this side the Fulciment, are equal to the facts on that side the Fulciment.  
3. A Right Line, resting on a Fulciment, equiponderate by divers Ponderosities, pendant in sundry points of that line, being proposed: If two of the Ponderosities pendant shall be transplanted, so that the said line be equiponderate again: Then the facts of each Ponderosity by his distance run in transplantation are equal.  
4. If a Stillyard or Scale of Ponderosity, or (as the Dutch call it) the Roman Beam, be true, (i. e. doth give the truth) in two points (the farther distant the better) it is true in all the points of pendancy throughout, the Divisions being equal.   

CHAP. VIII. Of the four Compendiums for quadratique Equations.  
IN these four Compendiums you have both the affirmative and negative Roots symbolically expressed, z being the unknown Symbol in each Equation, (but made known by each Solution) a being the known factor in the first term; b being the known factor in the second term; r being the resolvend.  
But if at any time an Equation shall be proposed, encumbered with vulgar fractions, let be reduced to its least terms in whole numbers, if possible; if not, let it be reduced to its least terms in Decimal●; and than it will fall under one of these four Equations following.  
In which Equations you must note, that if there be no known factor expressed in the first term, than a is understood to be unity: Furthermore, it is evident, that if any quantity shall be signed— that then t●● the Square Root, or the Root of any ev●● power of such quantity so signed, is in●●plicable: Therefore when this shall happen in the Solution of any Equation whatsoever, that Equation may be said to 〈◊〉 impossible.  
  

CHAP. IX. Of Recreative Problems.  
Problem 1. To find a, b, c, three numbers under this qualification. aa = bb+ cc. ques. a? b? c?  
Sol. A = nn+1s. B = nn − 1. C = 2 n. You may put n = any number.  
Probl. 2. To find a, b, c, three numbers under this qualification. ab+ bb = cc. ques. a? b? c?  
Sol. A = 2 n+1. B = nn. C = nn+ n. You may put n = any number.  
Probl. 3. To find a, b, c, three numbers under this qualification.  
a+ ab+ bb = cc. ques. a? b? c?  
Sol. A = 2 n+1. B = nn − 1. C = nn+ n+1. You may put n = number.  
Probl. 4. To find a, b, c, d, e, five numbers under this qualification.  
a+ b = c. cc − a = dd. ce − b = ec. Ques. a? b? c? d? e?  
Solution  
 
You may put n = any whole number. But note, that every Equation is to be reduced to its least terms, if need require.  
Probl. 5. To find the Content of any Solid made by Rotation, if you can get the ratio of the Squares of its Ordinates', to any rank of Rect●ngles: This is the Theorem.  
If two ranks of Quantities or Numbers be proposed (in any qualification or order whatsoever) having one common ratio between each pair of Correspondents: Then in both ranks their Correspondent Sums o. Differences have the same common ratio. This Theorem holds as well in Superficies if you can get the ratio of the Ordinates' to any other rank of Quantities o● Numbers.   
FINIS.   

GAUGING EPITOMISED: OR, An Abbreviation of Solid Geometry, concerning the Business of CASK-GAUGING, taking a Cask in any of the Four Notions following. By Michael Dary.  
PREMONITION.  
IF you put D = the Diameter of the Bouldge, d = the Diameter of either of the Heads, and y = D ● d; 〈◊〉 = the length of the Axis of the Vessel, and C = the Content thereof, the dimension being taken in Inches.  
Prop. 1. If a Cask be taken as the middle Frustum of a Spheroid, intercepted between two Planes parallel cutting the Axis at right angles: Then, 3, 82) ∶ 2DD+dd ∶ ×l. (= C.  
Prop. 2. If a Cask be taken as the middle Frustum of a parabolic Spindle, intercepted between two Planes parallel, cutting the Axis at right angles: Then, 3, 82) ∶ 2DD+dd − 0, 4yy ∶ ●● (= C.  
Prop. 3. If a Cask be taken as the middle Frustum of two parabolic Conods abutting upon one Common Base, intercepted, etc. Then, 3, 82) ∶ DD+dd ∶ ×1 ½●. (= C.  
Prop. 4. If a Cask be taken as the middle Frustum of two Cones abutting upon one Common Base, intercepted, etc. Then, 3, 82) ∶ DD+dd − ⅓yy ∶ ×1½L (= C.  
Now if you would have the Content in Beer Gallons, you must multiply 3, 82 by 282, makes your divisor = 1077 fere: If you would have Wine Gallons, 3, 82 × 231 makes your divisor = 883 fere. And from these two divisors, you may Calculate Tables for Wine or Beer Measure: For the square of any diameter in Inches divided by 883, is the Construction of the Wine Table; or by 1077, is the Construction of the Beer Table: And either of these Table have their second differences equal, therefore they will be made by an easy Collection.  
¶ If a Cask be not full (the Axis being posited parallel to the Horizon) ●●●nd the quantity of Liquor contained in it.  
To do this, you ought to have a Table of the Segments of a Circle, who● Area is unity, the Diameter being divided into 10000 equal parts, and then this Approximation is the readiest hithe to used, which requireth this Data; the whole Content of the Cask, the Diameter at the Bouldge, and the wet Po●●on thereof; and the Proportion runs thus:  
As the whole Diameter, is to its wet Portion; so is the Diameter in the ●able, (i.e. 10000,) to its like Portion: Which being sought in the Table of Segments, abutts you to a Segm●●, by which if you multiply the whole Content of the Cask, the Product is the Content of the remaining Liquor in the Cask.   

Here followeth an Account of some Addenda to, and Faults ●scaped in, Dary's Miscellanies.  
PAge 4. might have been added, or T) RZ − RR (= T of ½ A. p. 11. l. 10. for right anguled r. rectangular. l. 14. for rectanguled r. rectangular. p. 14. pr. 7. should run thus, The 3 Angles of any Spherical Triangle being given, there are likewise 3 sides of another Spherical Triangle given, whose Angles are equal to the sides of the first Triangle, if you take the Compliment to 180 deg. of one of them, but most conveniently of the greatest. p. 19 l. 1. for fact of the sides r. fact of the sins of the sides. p. 20. after the orthographical Projection should have followed, By this means all Circles of the Sphere perpendicular to the Plane, are projected into Right Lines, those parallel to it into Circles of the same bigness, and all other into Ellipses. So after the Steriographical Projection, By this means, all great Circles of the Sphere, passing through the Eye and Touch-point, are Projected into Right Lines, and all other into Circles, and the Angles in this Projection are equal to their Correspondent Angles on the Sphere. So after the Gnomonical Projection, By this means, all great Circles on the Sphere perpendicular to the Plane, are Projected into Right Lines, all those parallel to it into Circles, and all others into Hyperbola's, Parabola's or Ellipses, accordingly as they cut or touch that Great Circle of the Sphere which is parallel to the Plane, or do neither. P. 23. l. 15. for Legs r. Sides. P. 25. l. 2. & 4. for 12, 5637 r. 12, 5664. P. 27 l. 1. for and Chord r. and a Chord; l. 4. for − 15B r. − 16B. P. 28. after Prop. 15 should have followed, The Area of a Parabola, is 〈◊〉 of the circumscribed Parallelogram of the same Base and Height; and the distance of its Centre of Gravity from its Vertex towards its Base, is ⅗ of its Altitude: Moreover in a Semiparab●la the Centre of Gravity, is in a Right Line parallel to the Base. at ⅔ of the Height; and in a Right Line parallel to the Axis at 〈◊〉 of the Base. P. 19 Prop. 18. shoul● run thus, If a Sphere be inscribed in a Cylinder, and that Cylinder be Cut with Planes at Right Angles to its Axis, than the intercepted Surfaces of the Cylinder are equal to the intercepted Surfaces of the respective Segments of the Sphere. P. 31. l. 1. for the begetting r. the Area 〈◊〉 the begetting; l. 13. for 8B+7b r. 8B+● √ Bh● − 3b. P. 32. l. ult. and P. 35. l. 14. & 17. for 8DD+7dd r. 8DD+4Dd+3dd. P. 35. Prop. 12. should run thus, By the word Cylindroid is meant a Solid contained under three Surfaces (i.e.) two Planes parallel and a Peripetasma, whereof the two Planes parallel are the Bases, and are both Circles or both Ellipses, or else one a Circle and the other an Ellipsis, and a Right Line extended from the Centre of one Base to the Centre of the other, may be called the Axis of the Solid; and in the Peripetasma a Right Line may be any where applied from Base to Base, being in the same Plane with the Axis of the Solid.  
But if such a Cylindroid be Cut with two Planes meeting in the Centres of both Bases, Cutting (or rather Inserting) rectangular Conjugate Diameters (or Axes) in both Bases, than etc. P. 37. l. 2. for divers r. many. P. 39 l. 3. for Zone r. Peripetasma; l. 6. for to the foresaid Conjugates r. between themselves, than it is a Prism or Prismoid. P. 42. l. ult. for quantity r. quantity, and their sides proportional.  

This Table of Wine Measure should have come in Pag. 35. of the Miscellanies.  The Names of Wine Vessels. Cubical Inches. Pints. Quarts. Gallons. Runlets. Hogsheads. Tertions. Pipes or Butts. Tuns. 1 Tun 58212 2016 1008 252 14 4 3 2 1 1 Pipe or Butt 29106 1008 0504 12● 07 2 1½ 1  1 Tertion 19404 0672 0336 084 04⅔ 1⅓ 1   1 Hogshead 14553 0504 0252 063 03½ 1    1 Runlet 04158 0144 0072 018 01     1 Gallon 00231 0008 0004 001      1 Quartfield 00057¾ 0002 0001       1 Pint 00028⅞ 0001        1 Cubical Inch 00001