THE CIRCLES OF PROPORTION AND THE HORIZONTAL INSTRUMENT.  
The Former showing the manner how to work Proportions both simple and compound: and the ready and easy resolving of Quaestions' both in Arithmetic, Geometry, & Astronomy: And is newly increased with an Additament for Navigation. All which Rules may also be wrought with the pen by Arithmetic, and the Canon of Triangles.  
The Later teaching how to work most Quaestions', which may be performed by the Globe: and to delineat dials upon any kind of Plain.  
Invented, and written in Latin by W. O. Translated into English, and set out for the public benefit, by WILLIAM FORSTER.  
LONDON, Printed by Augustine Mathewes, and are to be sold by Nic: Bourne at the Royal Exchange. 1633.   

TO THE HONOURABLE AND RENOWNED FOR virtue, learning, and true valour, Sir KENELME DIGBYE Knight.  
SIR,  
THE excellent accomplishments wherewith you are adorned both of virtue, and learning, and particularly in the Mathematical Sciences, together with the Honourable respect the Author hereof beareth unto your Worth, and his desire to testify the same, hath made me presume to present unto you, and under the happy auspice of your renowned name, to publish to the world this Treatise: the owning whereof though I may not challenge to myself, yet the birth and production, whereby it hath a being to the benefit of others, is, as unto a second parent, due unto me.  
For being in the time of the long vacation 1630, in the Country, at the house of the Reverend, and my most worthy friend, and Teacher, Mr. William Oughtred (to whose instruction I own both my initiation, and whole progress in these Sciences.) I upon occasion of speech told him of a Ruler of Numbers, Sins, & I angents, which one had be spoken to be made (such as is usually called Mr. Gunter's Ruler) 6 feet long, to be used with a pair of beame-compasses. He answered that was a poor invention, and the performance very troublesome: But, said he, seeing you are taken with such mechanical ways of Instruments, I will show you what devices I have had by me these many years. And first, he brought to me two Rulers of that sort, to be used by applying one to the other, without any compasses: and after that he shown me those lines cast into a circle or Ring, with another movable circle upon it. I seeing the great expeditenesse of both those ways; but especially, of the latter, wherein it fare excelleth any other Instrument which hath been known; told him, I wondered that he could so many years conceal such useful inventions, not only from the world, but from myself, to whom in other parts and mysteries of Art, be had been so liber all. He answered, That the true way of Art is not by Instruments, but by Demonstration: and that it is a preposterous course of vulgar Teachers, to begin with Instruments, and not with the Sciences, and so instead of Artists, to make their Scholars only doers of tricks, and as it were jugglers: to the despite of Art, loss of precious time, and betraying of willing and industrious wits, unto ignorance, and idleness. That the use of Instruments is indeed excellent, if a man be an Artist: but contemptible, being set and opposed to Art. And lastly, that he meant to commend to me, the skill of Instruments, but first he would have me well instructed in the Sciences. He also shown me many notes, and Rules for the use of those circles, and of his horizontal Instrument, (which he had projected about 30 years before) the most part written in Latin. All which I obtained of him leave to translate into English, and make public, for the use, and benefit of such as were studious, & lovers of these excellent Sciences.  
Which thing while I with mature, and diligent care (as my occasions would give me leave) went about to do: another to whom the Author in a loving confidence discovered this intent, using more haste then good speed, went about to preocupate; of which untimely birth, and preventing (if not circumventing) forwardness, I say no more: but advice the studious Reader, only so fare to trust, as he shall be sure doth agree to truth & Art.  
And thus most noble Sir, without any braving flourishes, or needless multiplying of tautologized and erroneous precepts, in naked truth, and in the modest simplicity, of the Author himself (whose known skill in the whole Systeme of Mathematical learning, will easily free him from the suspicion of having the way made for him, and the subject unuailed, to help his sight) I have notwithstanding under the protection of your courteous favour, and learned judgement, persisted in my long conceived purpose, of presenting this tractate to the public view, and light. Wishing withal unto you increase of deserved honour, and happiness.  
May the 1. 1632. By the honourer and admirer of your Worthiness, WILLIAM FORSTER.   


Thos that desire farther instructions in the use of th●s Instruments or other parts of the Mathematics may repair to W Forster at the Red bull over against St Clement's Church yard with out Temple bar   diagram   

THE FIRST PART OF THIS BOOK, Showing the use of the First side of the Instrument, for the working of Proportions both simple and compounded, and for the ready and easy resolving of questions both in Arithmetic, Geometry, and Astronomy, by Calculation.  

CHAP. I. Of the Description, and use of the Circles in this First side.  
1 THere are two sides of this Instrument. On the one side, as it were in the plain of the Horizon, is delineated the projection of the Sphere. On the other side there are diverse kinds of Circles, divided after many several ways; together with an Index to be opened after the manner of a pair of Compasses. And of this side we will speak in the first place.  
2 The First, or outermost circle is of Sines, from 5 degrees 45 minutes almost, until 90. Every degree till 30 is divided into 12 parts, each part being 5 min: from thence until 50 deg. into six parts which are 10 min: a piece: from thence until 75 degrees into two parts which are 30 minutes a piece. After that unto 85 deg. they are not divided.  
3 The Second circle is of Tangents, from 5 degrees 45 min: almost, until 45 degrees. Every degree being divided into 12 parts which are 5 min: a piece.  
4 The Third circle is of Tangents, from 45 degrees until 84 degrees 15 minutes. Each degree being divided into 12 parts, which are 5 min: a piece.  
5 The sixth circle is of Tangents from 84 degrees till about 89 degrees 25 minutes.  
The Seventh circle is of Tangents from about 35. min: till 6 degrees.  
The Eight circle is of Sines, from about 35 minutes till 6 degrees.  
6 The Fourth circle is of Unequal Numbers, which are noted with the Figures 2, 3, 4, 5, 6, 7, 8, 9, 1. Whether you understand them to be single Numbers, or Tenns, or Hundreds, or Thousands, etc. And every space of the numbers till 5, is divided into 100 parts, but after 5 till 1, into 50 parts.  
The Fourth circle also showeth the true or natural Sins, and Tangents. For if the Index be applied to any Sine or Tangent, it will cut the true Sine or Tangent in the fourth circle. And we are to know that if the Sine or Tangent be in the First, or Second circled, the figures of the Fourth circle do signify so many thousands. But if the Sine or Tangent be in the Seventh or Eight circle, the figures in the Fourth circle signify so many hundreds. And if the Tangent be in the sixth circle, the figures of the Fourth circle, signify so many times ten thousand, or whole Radij.  
And by this means the Sine of 23°, 30′ will be found 3987: and the Sine of its compliment 9171. And the Tangent of 23°, 30′ will be found 4348: and the Tangent of its compliment, 22998. And the Radius is 10000, that is the figure 1 with four cyphers, or circles. And hereby you may find out both the sum, and also the difference of Sines, and Tangents.  
7 The Fift circle is of Equal numbers, which are noted with the figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 0; and every space is divided into 100 equal parts.  
This Fift circle is scarce of any use, but only that by help thereof the given distance of numbers may be multiplied, or divided, as need shall require.  
As for example, if the space between 1 ⌊ 00 and 1 ⌊ 0833+ be to be septupled. Apply the Index unto 1 ⌊ 0833+ in the Fourth circle, and it will cut in the Fift circle 03476+; which multiplied by 7 makes 24333: then again, apply the Index unto this number 24333 in the Fift circle, and it will cut in the Fouth circled 1 ⌊ 7512+. And this is the space between 1 ⌊ 00 and 1 ⌊ 0833+ septupled, or the Ratio between 100, and 108⅓ seven times multiplied into itself.  
And contrarily, if 1 ⌊ 7512 bee to be divided by 7: Apply the Index unto 1 ⌊ 7568 in the fourth circle, and it will cut in the fit circle 24333: which divided by 7 giveth 03476+. Then again unto this Number in the Fift circle apply the Index, and in the Fourth circle it will cut upon 1 ⌊ 0833+ for the Septupartion sought for.  
The reason of which Operation is, because this Fift circle doth show the Logarithmes of Numbers. For if the Index be applied unto any number in the Fourth circle, it will in the Fift circle cut upon the Logarithme of the same number, so that to the Logarithme found you prefix a Caracteristicall (as Master Brigs' terms it) one less than is the number of the places of the integers proposed (which you may rather call the Gradual Number). So the Logarithme of the number 2 will be found 0.30103. And the Logarithme of the Number 43 ⌊ 6 will be found 1.63949.  
Numbers are multiplied by Addition of their Logarithmes : and they are Divided by Substraction of their Logarithmes.  
8 In the midst among the Circles, is a double Nocturnal instrument, to show the hour of the night.  
9 The right line passing through the Centre, through 90, and 45 I call the Line of Unity, or of the Radius.  
10 That Arm of the Index which in every Operation is placed at the Antecedent, or first term ', I call the Antecedent arm: and that which is placed at the consequent term, I call the Consequent Arme.   

CHAP. II. Of the Operation of the Rule of Proportion: and also of Multiplication, and Division.  
1. Theorem. IF of three numbers given, the first divide the second and the quotient multiply the third, the product shall be the fourth proportional, to the three numbers given.  
Theorem. If of three numbers given, the second divide the first, and the quotient divide the third; this later quotient shall be the fourth proportional, to the three numbers given.  
Neither is it material whether of the two numbers after the first be second, or third.  
2 And note that in Reciprocal proportion, that term by which the question is made; But in Direct proportion the term that is homogeneal thereto, is the first term, or the Antecedent of the first ratio.  
3 And therefore out of these foundations thus laid, (if you rightly conceive the nature of the Logarithmes) doth follow the finding out of the fourth proportional by this Instrument: whereof this is the Rule.  
Open the Arms of the Instrument to the distance of the first, and second number: then bring the Antecedent arm, or that which stood upon the first number unto the third, and so the consequent arm, keeping the same opening, will show the fourth number sought for.  
In which operation these four things are diligently to be considered.  
First, in constituting the places of each number in the fourth circle; whether the figures written in the spaces do signify Unites, Tenns, or Hundreds, etc.  
Secondly, if that arm which showeth the fourth proportional, do reach beyond the line of the Radius; that then you do account the fourth in a new circle or degree.  
Thirdly, whether the fourth number sought, aught to be greater, or lesser than the third. For if a fourth number be offered greater than the third, when it should be less, or less than the third when it should be greater; it is a sign that that number doth appertain to a circle of another degree.  
Fourthly, that look what true distance was between the first and second, that the same be supposed between the third and the fourth, and also on the same part.  
4 And for because Multiplication and Division, have a certain implicit proportion: we will speak of them in the first place.  
5 In Multiplication, As an Unite is to one of the factores (or numbers to be multiplied:) so is the other of the factores, to the product.  
And the product of two numbers shall have so many places as there be in both the factores, if the lesser of them exceed so many of the first figures of the product: But if it do not exceed, it will have one less.  
6 And in Division, As the Divisor is to an Unite; so is the Dividend, to the Quotient.  
And the Quotient shall have so many places, as the Dividend hath more than the Divisor, if the Divisor exceed so many of the first figures of the Dividend: but if it do not exceed, it shall have one place more.  
7 Wherefore let this rule be still carefully kept in mind: that In Multiplication the first term of the implicit proportion is evermore 1: And in Division, the first term is the Divisor.  
And thus much concerning the operation of Proportion, Multiplication, and Division, I thought meet to admonish, lest hereafter in Multiplying, or Dividing, or seeking out a fourth proportional, we be constrained to repeat the same things many times over.  
8 An example of Multiplication. How many pence are there in 47li. 9 sh? For because 1 shilling contains 12 pence, and 1 pound contains 20 shillings, that is 240 pence: you shall multiply 47 by 240, and 9 by 12, and then add together the products.  
In the first Multiplication. 1 · 47 ∷ 240 · 11280 ·  
For set the Arms of the Index at 1 and 47 in the fourth circle; and then bring the Antecedent arm (which stood at 1) unto 240, and the Consequent arm will show 11280.  
Again in the second Multiplication. 1 · 9 ∷ 12 · 108 ·  
For set the two Arms of the Index at 1 and 9 in the fourth circle; and bring the Antecedent arm unto 12, and the Consequent arm will show 108. Lastly, add together 11280 and 108 and the sum 11388 will be the number of pence contained in the said sum of 47 li, 9sh.  
9 An example of Division. How many pounds, and shillings are in 11388 pence? Divide 11388 by 240: the division is thus. 240 · 1 ∷ 11388 · 47 ⌊ 5—  
For set the two Arms of the Index at 240, and 1 in the fourth circle: and then bring the Antecedent arm (which stood at 240) unto 11388; and the Consequent arm will show 47 and almost an half.  
But how many shillings that excess doth contain will appear, if first you find by Multiplication that 11280 pence are contained in 47 li: which subducted from 11388 there will remain for the excess 108 pence. Afterwards by division you may seek how many shillings are in 108 pence: the division is thus. 12 · 1 ∷ 108 · 9  
For set the Arms of the Index at 12 and 1: then bring the Antecedent arm (which stood at 12) unto 108; and the Consequent arm will show 9  
9 Any Fraction given may be reduced into Decimal parts, thus.  
Set the Antecedent arm of the Index at the Denominator of the Fraction given, in the fourth circle, and the Consequent arm at the Numerator, then keeping the same distance, bring the Antecedent arm unto 1, and the consequent arm will show the decimal parts.  
So ●40/●●● is 0 ⌊ 75. And 19/48 is 0 ⌊ 396—   

CHAP. III. Now follow certain examples of Proportion.  
Example I. IF 54 Elnes of Holland be sold for 96 shillings, for how many shillings was 9 elnes sold? The terms given are 54 · 96 ∷ 9 ·  
According to the 2 Chap. 3 Sect. Set one of the arms of the Index at the Antecedent term 54 in the fourth circle, and the other arm at the consequent term 96: then keeping that distance, bring the Antecedent arm unto 9; and the consequent arm beyond the line of the Radius will show 16 for the fourth proportional, according to the considerations in the 2 Chap. Sect. 3.  
Therefore 54 · 96 ∷ 9 · 16 · are proportionals. And 16 shillings is the price of 9 Elnes.  
Example II. If 108 bushels of corn be sufficient for a company of Soldiers keeping a Fort, for 36 days, How many days will 12 bushels suffice that same number of Soldiers? The terms given are 108 · 36 ∷ 12 ·  
Set one Arm of the Index at the Antecedent term 108 in the fourth circle, and the other Arm at the consequent term 12, (being mindful of the considerations in the 2 Chap. 3 Sect.) then keeping that same distance, bring the Antecedent arm unto 36; and the consequent arm will show 4.  
Therefore 108 · 36 ∷ 12 · 4 · shall be proportionals. And 4 is the number of days sought for.  
Example III. There is laid up in a Fort so much corn as will suffice for 756 Soldiers which keep that Fort, for 196 days: how many Soldier's will that same corn suffice for 364 days?  
The Proportion is reciprocal, therefore the terms given are 364 ∶ 756 ∷ 196 ·  
Set one Arm of the Index at the Antecedent term 364, and the other Arm at the consequent 196: and keeping the same distance, bring the Antecedent arm unto 756; and the consequent arm will show 407+: And therefore for so many Soldiers will the corn laid up suffice for 364 days, or 13 months.  
Example FOUR There is a Tower whose height I would measure with a Quadrant.  
I take two Stations in the same right line from the Tower: and at either Station having observed the height through the sights, I find that the perpendicular cutteth in the nearer Station 28 degrees 7 minutes almost: and in the further Station 21 degr. 58 min. almost: and between both the Stations, the distance was 76 feet.  
The Rule of measuring heights by two Stations is contained in these Theorems.  
Theor. As the difference of the Tangents of the arches cut in either station, is to the distance between the stations; so is the Tangent of the lesser arch, to the nearer distance from the Tower. Again  
Theor. As the Radius is to the Tangent of the greater arch; so is the nearer distance found, to the height.  
And therefore because by the 1 Chap. 6 Sect. the Tangents of the arches 28°, 7′—, and 21° 58′— are 5342, and 4032 whose difference is 1310; the proportions will be  
First, 1310 · 76 ∷ tang. 21°, 58′— · 234 ·  
Wherefore 234 feet is the nearer distance.  
Second Radius · tang. 28°, 7′— ∷ 234 · 125 ·  
Wherefore 125 feet is the height sought for.  
Example V To find the Declination of the Sun the 9th day of May.  
The place of the Sun for every day, may be had ne'er enough out of this Table, by Adding unto the place of the Sun in the beginning of that month so many degrees, as there are days passed in that month: But if the number of degrees exceed 30, the excess is to be accounted in the Sign next following.  
Wherefore the 9th of May the place of the Sun is ♉ 20+9, that is ♉ 29: which is 59 degr. distant from the next Equinoctial point.  

The place of the Sun, in the beginning of every Month.  january ♑ 21 February ♒ 22 March ♓ 20 April ♈ 21 May ♉ 20 june ♊ 19 july ♋ 18 August ♌ 13 Septemb. ♍ 18 October ♎ 17 Novemb ♏ 18 Decemb ♐ 19  
These things being known, the Rule is delivered in this Theorem.  
Theor. As the Radius is to the sine of the sun's distance from the next Equinoctial point; so is the sine of the sun's greatest declination, to the sine of the declination sought for.  
The proportion will be Radius · sine 59° ∷ sine 2●°, 30′ · sine 19° ●9′·  
And so much is the Declination sought for.  
Example VI To find the Right ascension of the Sun, the 9th day of May.  
Seek the place of the Sun for the day proposed in the former Table; and the Sun's distance from the next Aequinoctial point, as in the former example.  
These things being known, the Rule is by one of these two Theorems.  
Theor. As the Radius, is to the sine of the compliment of the sun's greatest declination; so is the the tangent of the sun's distance from the next Equinoctial point, to the tangent of the distance of the right ascension of the sun, from the same Equinoctial point. Or  
Theor. As the tangent of the greatest declination of the Sun, is to the Radius; so is the tangent of the declination of the sun for the time proposed, unto the sine of the right ascension of the sun from the next Aequinoctial point.  
The proportions will be either Radius · sine 66°, 30′ ∷ tang. 59° · tang 56° 46′ ·  
Or, tang. 23°, 30′ · Radius ∷ tang. 19°, ●9′ · sine ●6° 4●′▪   

Of Continual proportion, Or of Progression Geometrical.  
1 THE Ratio of a Progression is the quotient of the consequent term divided by his antecedent. And therefore in the Instrument it is the distance taken between the terms in the fourth circle, by the opening of the Index.  
2 To Double, Triple, or Multiply how often soever any Ratio given, is nothing else but so often to put together the said space or distance between the terms, as is showed in Chap 1. Sect. 7.  
As for example, if the Ratio 60, to 65 be proposed to be septupled.  
Set the Arms of the Index at 60, and 65: and then with the same opening, bring the Antecedent arm which was at 60, unto 1, and the consequent arm will cut 1 ⌊ 08384 in the fourth circle, and 03476 ● in the fift circle: this latter number being multiplied by 7 maketh 23444; unto which number in the fift circle applying the Index, it will in the fourth circle cut 1 ⌊ ●●●2, which is the multiplied number sought for.  
But because in a little Instrument, the arch cut in the fift circle, cannot be estimated exactly: and a small error in the beginning often repeated, by multiplying is made great: it is the most safe way, to take the Logarithmes of the terms of the Ratio out of the Canon, and to multiply them by the number given: As I have done in these examples.  

Logarithmes taken out of the Canon.  100 2.0000000 104 2.0170333 105 2.0211893 106 2.0253059 106⅔ 2.0280287 107 2.0293838 107½ 2.0314085— 108 2.0334238— 108⅓ 2.0347621 108½ 2. 0354297+ 108¼ 2.0364293— 109 2.0374265 110 2.0313927  
Or if the Canon be wanting; you may come nearer the mark, if that first single opening of the Index being kept, and the Antecedent arm set at 1; you transfer the Antecedent Arm, unto that place which the consequent arm doth cut; and the consequent arm will cut the same space duplicated. Then holding the consequent arm in that same place, open the Antecedent arm unto 1. and afterward with that duplicated opening, bring the Antecedent arm to the duplicated space, and the consequent arm will cut the space quadrupled. Thirdly, bring the Anticedent arm unto the quadrupled space and the consequent arm, keeping that duplicated opening, will cut the space sextupled. Lastly, having again taken a single opening, bring the Antecedent arm unto the number, or space sextupled; and the consequent arm will show the Ratio sought for septupled.  
And this manner of working may be observed for as many Multiplications as you please of any Ratio given.  
3. The Ratio, and first term being given, to continue the same unto any number of terms.  
Open the Arms of the Index, the one unto the Antecedent of the ratio given, and the other unto the consequent: then the same opening being kept, bring the Antecedent Arm unto the first term given, and the consequent arm will show the second term: again bring the Antecedent arm unto the second term found, and the consequent arm will show the third. After that bring the Antecedent arm unto the third term found, and the consequent arm will show the fourth. And in this manner you may proceed as fare as you please.  
As for example, If a Progression in the ratio 2 unto 5, beginning at 8, Or if a Progression in the ratio 100 unto 108, beginning at 5, is to be instituted; the terms in either progression will be as followeth.  
1 2 · 5 ∷ 8 · 20 2 20 3 50 4 125 5 312 ⌊ 5 6 781 ⌊ 25 7 1953 ⌊ 125 1 100 · 108 ∷ 5 · 5 ⌊ 4 2 5 ⌊ 4 3 5 ⌊ 832 4 6 ⌊ 29856 5 6 ⌊ 8024448 6 7 ⌊ 346640384 7 7 ⌊ 93437161472  
4 Theor; The ratio of any former term, in a row of concinuall proportionals, unto any of the terms following, is equal to the ratio of the first term unto the second, multiplied into itself according to the distance of that latter term from the former.  
As for example, The Ratio 5 ⌊ 4 unto 6 ⌊ ●024448 which is the third term from it, is equal to the ratio of 100 unto 108 triplicated, Or as the Cube of 100 unto the Cube of 108. Wherefore  
5 The Ratio, and the first term of the Progression being given, to find out any other term required.  
First, multiply the ratio given into itself, according to the distance of the term sought from the first term, by the 2 sect: then say  
As 1 is to that multiple found; so is the first term, to the term sought for.  
Example, What will be the amount of 26th, in 7 years by Interest upon Interest, after the rate of 20 pence in the pound?  
Because 1 pound which is 20 shillings containeth 60 groats, and 20 pence contain 5 groats, the rate of the Interest will be 60 unto 65, Or 100 unto 108 ⌊ ●● × ● But the first term is given 26, unto which there are to be acquired 7 other terms in continual proportion.  
First, by the 4 Chap: and 2 Sect. Let the ratio given be septuplicated, that is multiplied sevenfold into itself, which will be 1 ⌊ 552. Then set the consequent arm of the Index unto the septuplicate number 1 ⌊ 552, and open the Antecedent arm unto 1; and keeping the same opening, bring the Antecedent arm unto 26; and the consequent arm will show 45 ⌊ 5312 li the amount sought for.  
6 The Ratio, and any other term of the Progression being given, to find the first term.  
First multiply the Ratio given into itself according to the difference of the term given from the first term. Then say  
As that multuple is unto 1; so is the term given, unto the first term.  
Example, what sum in 7 years did amount unto 45 ⌊ 5312 li by Interest upon Interest after the rate of 100 unto 108 ⌊ 33+?  
First the ratio being septuplicat, by the 4 Chap. 2 Sect. will be 1 ⌊ 7512. Then setting the Antecedent arm of the Index to that septuple 1 ⌊ 7512, open the consequent arm unto 1: and keeping the same opening, bring the Arm unto 45 ⌊ 5312 li: and the consequent arm will show 26, which was the stock, or sum of money, from which that amount did arise.  
7 The Ratio, the First term, and any other term of a Progression being given, to find how many places the term given is from the first term.  
First say, As the first term is unto 1; so is the other term given, unto the ratio multiplied into itself according to the distance of that term from the first.  
Wherefore according to 1 Chap. 7 Sect. and 3 Chap. 2 Sect. by help of the fift circle, see how often the ratio given, is contained in that multuple found.  
Example, In how many years did 26 li, by Interest upon Interest after the rate of 100 unto 108 ⌊ 33+ + increase unto 45 ⌊ 5312 li?  
First set the Antecedent arm of the Index at 26, and the consequent arm at 1: and keeping the same opening bring the Antecedent arm unto 45 ⌊ 5312, and the consequent arm will show 1 ⌊ 7●12: to which in the fift circle answereth 24333. Then because unto the consequent term of the ratio 108 ⌊ 33+ + there agrees in the fift circle 03476, divide 24333 by 03476, and the quotient will be 7, the number of years sought for.  
8 The First term, and any other term of the Progression being given to find the ratio of the Progression.  
First say, As the first term is unto 1 : so is the other term given, unto the ratio multiplied into itself according to the distance of that term from the first.  
Wherefore according to Chap. 1. Sect. 7, by help of the fift circle, Let the multuple found be divided by the distance of the term from the first.  
Example, 26 li by Interest upon Interest in 7 years amounted unto 45 ⌊ 5312: what was the ratio of the Interest compared unto 100?  
First set the Antecedent arm of the Index at 26, and the consequent arm at 1: and then keeping the same opening, bring the Antecedent arm unto 45 ⌊ 5312: and the consequent arm will show 1 ⌊ 7●12: unto which in the fift circle answereth 24333 and this number being divided by 7 will give 03476+: unto which agreeth in the fourth circle 108 ⌊ 33+ + the consequent term of the Interest sought for.  
9 Two numbers being given to find as many Middle proportionals between them as you will.  
Divide the distance of the greater number given from the lesser in the fourth circle justly taken, according to Chap. 1. Sect. 7, by help of the fift circle into equal segments, one more than are the number of Middle proportionals sought for. All these segments added orderly to the first term, do distinguish the terms of the Progression which you seek.  
Example, Let there be four Middle proportionals, sought out between 8 and 19 ⌊ 90656.  
Apply the Index unto 8 in the fourth circle, and it will cut in the first circle 9039: also set the Index unto 19 ⌊ 91— in the fourth circle, and it will cut in the fift circle 2989; which number because it reaches beyond the Unite ●●ne, is indeed 12989, according to Chap. 1. Sect. 7, and so is the distance justly taken. Then subducting 9039 from 12989, there will remain 3950: which divided by 4 + 1, the quotient will give 790. wherefore 9039 + 790, scil. 9829 in the fift circle doth agree with 9 ⌊ 6 in the fourth circle, which is the first middle proportional. And 9039 + twice 790, scil. 10619 in the fift circle doth agree with 11 ⌊ 52 in the fourth circle, which is the second middle proportional. And in this manner 13 ⌊ 824, and 16 ⌊ 5888 will be found the third and fourth middle proportionals.  
10 Theor. If from the Ratio given, being multiplied in itself according to the number of terms, you subduct 1, and multiply the remainder, by the antecedent of the ratio. It will be  
As the difference of the terms of the ratio, is unto the product; so is the first term, to the sum of the Progression.  
As for example if the Ratio of the Progression be R to S: and the difference of R taken out of S be D. and the first term of the Progression α: and the whole sum of the terms Z it shall be, D · rat. multa— 1 in R ∷ α · Z.  
Which is the very Theorem itself expressed in Symbols of words: that it may more easily be fixed in the fantasy. Which proportion also we must consider, doth hold both alternly, and conversly.  
This Theorem may otherways be expressed, by the equality which the product of the two middle terms hath to the product of the two extremes, thus Ratio multiplicata— 1 in R in α = Z in D.  
This manner of setting down Theorems, whether they be Proportions, or Equations, by Symbols or notes of words, is most excellent, artificial, and doctrinal. Wherefore I earnestly exhort every one, that desireth though but to look into these noble Sciences Mathematical, to accustom themselves unto it: and indeed it is easy, being most agreeable to reason, yea even to sense. And out of this working may many singular consectaries be drawn: which without this would, it may be, for ever lie hid. As in this present proportion: because it is D · rat. multa— 1 in ● ∷ α · Z · wherefore Rat. multa— 1 in R · D ∷ Z · α · And α · Z ∷ D · rat. multa— 1 in R  
These are exceeding easy: but this following is more difficult, and requireth attention.  
In the former equation it was Rat. multa— 1 in R in α = Z in D  
Now because Rat. multa— 1 in R in α, and Rat. multa●● in R in α— R in α, and Rat. multa in R— R in α, and Rat. multa in α— α in R, are equal one to another, and also to Z in D, these aequations shall also be consectarious. and and and  
And beside these many more. The practice whereof I leave to the industry of the studious Reader, especially having delivered the whole Art of such operations in my Clavis Mathematica.  
Some of these I have occasion to use in the sections following.  
11. Therefore the Ratio of a Progression, and the first term, and the number of terms being given, to find the sum of the whole progression.  
For this operation the rule, or Theorem last before serveth: for by it  
Example. If an Annuity of 5 li, be detained 7 years, what will be the amount thereof by interest upon interest after the rate of 100 unto 108?  
Now because the Amount is the sum of the Progression, whereof the first term is the annuity, Multiply the ratio into itself according to the number of years, by Chap. 4. Sect. 2, and it will be 1 ⌊ 71382: from which if you subduct an unite, there remains 0 ⌊ 71382, which multiplied by 100 maketh 71 ⌊ 382: then set the Arms of the Index at 1 and 71 ⌊ 382; and bring the Antecedent arm which stood at 1, unto 5 the first term: and the consequent arm will cut 356 ⌊ 915 (that is Rat. multa— 1 in R in α) and last this number being divided by 108— 100, scil. by 8, the quotient will give 44 ⌊ 6●40, for the sum of the whole progression: and so much is the amount sought for.  
12. The Ratio, the Number of Terms, and the Sum of a Progression being given, to find the first term.  
By the converse of the foregoing Theorem, it is manifest, that Rat ∶ multa— 1 in R · D ∷ Z · α ·  
The declaration in words was in the 10 Sect.  
Example. If an Annuity detained 7 years by Interest upon Interest, after the rate of 100 unto 108, did increase unto 44 ⌊ 6140li. how much was that Annuity?  
Multiply the Ratio into itself according to the number of years (per cap. 4. sect. 2.) and the product will be 1 ⌊ 7●●82: from which if you subduct an Unite, there remains 0 ⌊ 7●●82: this being multipled by 100 doth make 71 ⌊ ●82. Then say 71 ⌊ 382 · 8 ∷ 44 ⌊ 6140 · 5 · for the First term: which was the Annuity sought for.  
13 The Ratio, the First term, and the Sum of the Progression being given, to find the Number of terms.  
By the Theorem in the 10. Sect. it was α · Z ∷ D · rat. multa— 1 in R ·  
Wherefore set the Antecedent arm of the Index unto the Antecedent term of the ratio, and the consequent arm unto the Sum of the Progression: and with that same opening, bring the Antecedent arm unto the difference of the terms of the ratio; and the consequent arm will show a number (that is Rat. mul.— 1 in R;) which if you divide by the Antecedent term of the ratio, and unto the quotient add an Unite, you shall have the ratio multiplied into itself according to the number of terms. Therefore taking the distance between the terms of the ratio, with the arms of the Index, measure by help of the fift circle (per Cap. 4. Sect. 7.) how often that distance may be found in the multiplied ratio: for so many are the terms of the progression.  
Example. If an Annuity of 5 li detained by Interest upon interest after the rate of 100 to 108, increased unto 44 ⌊ 6140li. How many years was the Annuity detained?  
Set the Antecedent arm of the Index at 5, and the consequent arm at 44 ⌊ 6140, and with the same opening bring the Antecedent arm unto 8, and the consequent arm will show 71 ⌊ 382 (that is Rat. multa— 1 in R:) this number being divided by 100, will be 0 ⌊ 71382: and if unto the quotient you add 1, you shall have 1 ⌊ 71●82 (the ratio multiplied into itself according to the number of years:) unto which in the fift circle answereth 2338: but unto 108 in the fift circle there answereth 0334. divide therefore 2338 by 0334; and the quotient will be 7, for the number of years sought for.  
Or such questions may be more easily performed by this Theorem, which the industrious Reader may by himself practise.  
14 Theor. If the sum of the whole progression be divided by the ratio multiplied into itself according to the number of terms, the quotient will be the first term; and that sum given will be the last term, of another progression, having the same ratio but one term more.  
15 And because a sum of money the amount whereof in any number of years given, by Interest upon interest, doth equal an Annuity so long detained, is equivalent to the same Annuity; and the amount of an Annuity is the sum of a Progression continued from that Annuity. If therefore an Annuity for any number of years be divided by the ratio multiplied into itself according to the number of years; the quotient will be the just price of an Annuity to endure for so long. And because by the 10 and 11 Sect. it hath been showed that .  
Therefore by the 14 Sect. will be equal to the price of an Annuity in ready money, which shall be the Rule for the operation following.  
Wherefore also Rat. multa— 1 in R in α = Rat. multa in D in Pret. which proportion is thus enuntiated in words.  
Theor. If from the Ratio multiplied into itself according to the Number of years you subduct an Unite, and the remainder be multiplied continually by the Antecedent of the Ratio, and the Annuity itself: And again, If the Ratio multiplied into itself, according to the number of years be multiplied continually by the difference of the terms of the Ratio, and by the Price: both those products will be equal.  
Example. An Annuity of 5 li, to endure for 7 years, is to be sold: what is it worth in ready money, after the Rate of 100 unto 108?  
The Ratio multiplied into itself according to the number of years (per Cap. 4 Sect. 2) is 1 ⌊ 7138: subduct 1, and there will remain 0 ⌊ 71382: which multiplied by 100 maketh 71 ⌊ 382. Then set the Antecedent arm of the Index at 1, and the consequent arm 71 ⌊ 382: and keeping that same opening, bring the Antecedent arm unto 5; and the consequent arm will show 356 ⌊ 915; keep this (for it is . After that set the Antecedent arm of the Index at 1, and the consequent arm unto the multuple ratio 1 ⌊ 7138; and with the same opening bring the Antecedent arm unto 8; and the consequent arm will cut 13 ⌊ 7106, keep this number also (for it is Rat. mul. in α). Lastly, place the Antecedent arm of the Index at 13 ⌊ 7106, and the consequent arm at 1; and with the same opening bring the Antecedent arm unto 356 ⌊ 015: and the consequent arm will show 26 ⌊ 032li. which is the just Price of an Annuity of 5 li in ready money.  
16 And by the last praecedent Theorem or Rule, also which Theorem may be enuntiated in words as was there showed.  
Example. An Annuity for 7 years is bought for 26 ⌊ 032 li. after the rate of 100 unto 108, by Interest upon interest: how much was that Annuity?  
The Ratio multiplied into itself for the number of 7 years (per Cap.: 4, Sect. 2) is 1 ⌊ 7138: which multiplied continually by 8, and by the price, doth make 356 ⌊ 915. Divide this number found, by 71 ⌊ 382, (which is the multuple ratio) itself less by an Unite, and multiplied by 100: and the quotient will be 5 li, the Annuity sought for.  
17. Also by Ratiocination from that praecedent rule will follow this proposition which is thus enuntiated in words.  
Theor. If the product of the Antecedent of the ratio multiplied by the Annuity be divided by itself, being diminished by the product of the difference of the terms of the ratio multiplied by the Price: the quotient will be equal to the ratio multiplied into if self according to the number of years. As  
If the ratio be 100 unto 108; the Annuity 5 li: and the price thereof 26 ⌊ 032 li. Set the Antecedent arm of the Index at 1, and the consequent arm at 8, the difference of the terms of the ratio: and with the same opening bring the Antecedent arm unto 26 ⌊ 032—: and the consequent arm will show 208 ⌊ 256: which subducted from 500, there will remain 291 ⌊ 744, for the divisor. Set therefore the Antecedent arm of the Index at the divisor 291 ⌊ 744, and the consequent arm at 1: and with the same opening, bring the Antecedent arm unto the dividend 500: and the consequent arm will show 1 ⌊ 71●8, which is the ratio multiplied into itself according to the number of years. And by this number so found it will be easy (by help of the fift circle) the ratio of the Interest being given, to find the continuance of the Annuity.  
Example. An Annuity of 5 i'll, was bought for 26 ⌊ 032 li—, after the rate of 100 unto 108: how many years is it to last?  
First seek out the ratio multiplied into itself according to the number of years, which will be 1 ⌊ 71●8, according as was even now showed in this Sect. to this in the fift circle there answereth 2338: but unto 108 there answereth in the fift circle 0334. Divide therefore 2338 by 0334; and the quotient will be 7 for the number of years sought for.   

CHAP. V Of the Quadrating, and Cubing of Numbers, Sides or Roots: and of the Extraction of the Quadrate, and Cubic side, or root, out of Numbers, or Powers given.  
1 IF a number, side, or root be multiplied into itself; the product will be a Quadrat. And if a quadrate be multiplied into his own side, or root, the product will be a Cube. Wherefore 1 · N ∷ N · Q · and 1 · N ∷ Q · C ·  
2 If therefore a number be given to be Quadrated. Set the Antecedent arm of the Index at 1, and the consequent arm at the number given: then with the some opening bring the Antecedent arm to the number given; and the consequent arm will show the Quadrat thereof.  
And the number of figures in a Quadrat of a single root (or which doth not exceed 9) is easily found out by those Rules, that have been before delivered concerning Multiplication. But if a side or root consist of more figures than one; for each figure after the first it acquireth two more places of figures. And if any of the figures of the root given be decimal parts, cut off from the Quadrat found, twice so many of the last figures for decimals.  
Example 1. The Quadrat of the side 7 is required. Say 1 · 7 ∷ 7 · 49 Set therefore the Antecedent arm of the Index at 1, and the consequent arm at 7; and with that opening bring the Antecedent arm unto 7; and the consequent arm will show 49 which is the Quadrat sought for.  
Example 11. The Quadrat of the side 57 is required.  
Set the Antecedent arm of the Index at 1; and the consequent arm at 57: then with that same opening bring the antecedent arm unto 57; and the consequent arm will show 3249, which is the Quadrat sought for, consisting of four places.  
Example 111. The Quadrat of the side, or root 570 is required.  
Having found as before, 3249 for the quadrat of the side 57: put thereunto two circles: and it will be 324900, the quadrat sought for.  
Example 1111. The quadrat of the side 574 is required.  
Set the Antecedent arm of the Index at 1, and the corsequent arm at 574; and with the same opening bring the Antecedent arm unto 574; and the consequent arm will show 329476, the quadrat sought for consisting of six figures: but the two last figures cannot at all be discerned by the Instrument.  
3 If a number be given to be Cubed.  
Set the Antecedent arm of the Index at 1, and the consequent arm at the number given; and with that same opening, bring the Antecedent arm unto the number given; and the consequent arm will show the Quadrat; then bring the Antecedent arm unto the Quadrat, and the consequent arm with that same opening will show the Cube of that side given.  
The number figures in a Cube of a single side, or root which doth not exceed 9, is easily found by that which hath been before delivered concerning Multiplication: But if the side, or root consist of more figures than one; for each figure after the first it obtaineth three more places of figures. And if any of the figures of the root given be decimal parts, cut off from the Cube found thrice so many of the last figures for decimal parts.  
Example. The Cube of the side, or root 7 is required. Say, 1 · 7 ∷ 7 · 49 · again 1 · 7 ∷ 49 · 343 ·  
Set therefore the Antecedent arm of the Index at 1, and the consequent arm at 7; and with that opening bring the Antecedent arm unto 7, and the consequent arm will show 49 the quadrat thereof: Then set the Antecedent arm at 49; and the consequent arm (with that first opening) will show 343, which is the desired Cube of the side proposed.  
Another example, The Cube of the side, or root 57 is required.  
Set the Antecedent arm of the Index at 1, and the consequent arm at 57; and with that same opening bring the Antecedent arm unto 57; and the consequent arm will show the quadrat 3249. Then set the Antecedent arm at 3249; and the consequent arm keeping the former opening will show 185193, which is the required Cube of that side proposed, consisting of six places: but the two last figures cannot be known by the Instrument.  
Example III. The Cube of the side 570 is required.  
Having found as before the Cube of the side, or root 57 to be 185193: put thereunto three circles; and it will be 185 193000 the Cube sought for.  
Examples of greater Cubes, it will be needless to set down.  
4 The Extraction of the Square, or Quadrat root, or side, is done by help of the fift circle, after this manner.  
Set the Index at the Quadrat proposed; and of that number which it cuts in the fift circle, take half: then set the Index at that half; and it will show in the fourth circle, the side, or root sought for.  
But you must know that if the number which is the Quadrat proposed, have only two places of Integers, the side, or root consisteth of one figure. But if it have more places of Integers, dividing them by 2, the quotient will give the true number of figures in the root, if it measure it exactly; or one less than the true number if any thing remain.  
Example I. The side, or root of the Quadrat 49 is required.  
Set the Index at 49 in the fourth circle, and it will cut in the fift circle 6902; indeed 1. 6902 having the gradual number 1 prefixed, because 1 in the fourth circle signifieth 10, one circuition thereof being finished: the half whereof is 0. 8451. Then set the Index at 0. 8451, in the fift circle; and it will cut in the fourth circle 7, the side, or root sought for.  
Example II. The side, or root of the quadrat 3249 is required.  
Set the Index at 3249 in the fourth circle, and it will cut in the fift circle 5118; indeed 3. 5118 prefixing the gradual number 3, because 1 in the fourth circle signifieth 1000, three circuitions thereof being finished: the half whereof is 1. 7559. Then set the Index at 7559, omitting the prefixed gradual number 1; and it will show in the fourth circle 57 the side sought for, consisting of two figures, because 1 in the fourth circle signifieth 10.  
Example III. The side of the quadrat 329476 is required.  
Set the Index at 329476 in the fourth circle, and it will cut in the fift circle 5178; indeed 5. 5178 prefixing the gradual number 5, because 1 in the fourth circle signifieth 100000, five circuitions thereof being passed over; the half whereof is 2. 7589; Then set the Index at 7589, omitting the gradual number 2 prefixed thereto; and it will show in the fourth circle 574 the side, or root sought for, consisting of three figures, because 1 in the fourth circle doth signify 100  
5 The Extraction of the Cubic root, or side is done by help of the fift circle after this manner.  
Set the Index at the Cube proposed; and that number which it cuts in the fift circle divide by 3. Then set the Index at that third part, and it will show in the fourth circle the side, or root sought for.  
And you must know that if the Cube proposed have only three places of Integers, the side, or root thereof consisteth of one figure: But if it have more places of Integers; divide them by 3, the quotient will give the true number of the figures of the Root, if it measure it exactly; or one less than the true number if any thing remain.  
Example I. The side, or root of the Cube 343 is required.  
Set the Index at 343 in the fourth circle, and it will cut in the fift circle 5353; indeed 2. 5353 prefixing the gradual number 2, because 1 in the fourth doth signify 100, two circuitions thereof being passed over: the third part whereof is 8451. Then set the Index at 8451, and it will show in the fourth circle 7, the side sought for.  
Example II. The side, or root of the Cube 185193 is required.  
Set the Index at 185193 in the fourth circle; and it will cut in the fift circle 2677; indeed 5. 2677 prefixing the gradual number 5, because 1 in the fourth circle doth signify 100000, five circuitions thereof being passed over: the third part whereof is 1. 7599. Then set the Index at 7599, omitting the prefixed gradual number 1; and in the fourth circle it will show 57, the side, or root sought for, consisting of two figures, because 1 in the fourth circle doth signify 10.  
Examples of greater Cubes it will be needless to set down.   

CHAP. VI Of Duplicated, and Triplicated proportion. And first of Duplicated proportion.  
Theorem. LIke Plains are in a Duplicated ratio, that is, As the Quadrats of their homologal sides. And therefore questions in the which the sides of like planes are compared, do appertain unto this place.  
And it is to be noted, that if three numbers be given, in which As the quadrat of the first, is unto the quadrat of the second; so ought the third to be unto a number sought for. Let it be thus done, As the first number, is to the second; so is the third to afourth; And again As the first number, is to the second; so is the fourth now found, to the number sought for.  
Example I. There are two like rectangle Areae, or plains, the length of the greater, is 40 feet, the length of the lesser 24 feet: each of them paved with paving tiles; the greater hath 1200 tiles: how many shall the lesser have?  
The Areae, or plains are one to the other, as the quadrats of the longitudes given. And the proportion is direct. Say therefore 1600 (Q ∶ 40) · 576 (Q ∶ 24) ∷ 1200 · 432 · which is the number of tiles contained in the pavement of the lesser Areae, or plain.  
Example II. How many Acres of woodland measured with a Perch, of 18 feet, are there in 73 Acres of champaign land, measured with a Perch of 16½ feet?  
The measures given (18, 16½) being reduced into their jest terms, are as 12 unto 11: and the quadrats of these numbers, are, 144, and 121. And the Proportion is Reciprocal. Say therefore 144 (Q ∶ 12) · 121 (Q ∶ 11) ∷ 73 · and so many are the Acres of Wood land.  

Of Triplicated proportion.  
3 Theor. Like Solids are in a Triplicated ratio, that is, As the Cubes of their homologal sides. And therefore questions in which the sides of like solids are compared, do appertain unto this place.  
4 If three numbers be given in the which, As the Cube of the first is to the Cube of the second; so is the third number to a number sought for. Let it be thus done, As the first number is to the second; so is the third to a fourth: Again, As the first is to the second; so is the fourth now found unto a fift. And thirdly, As the first is to the second; so is that fift to the number sought for.  
Example. If 4●/100lib. of gunpowder, suffice to charge a Gun, whereof the concave diameter is inch 1½. How many pounds of powder will suffice to charge a Gun, whose concave diameter is 7 inches?  
The capacities are one to another, as the Cubes of the diameters. And the proportion is direct. Say therefore 3 ⌊ 375 (C ∶ 1 ⌊ 5) · 343 (C ∶ 7) ∷ 0 ⌊ 43 · 4●● + wherefore 43 ⌊ ●lib. of Gunpowder, will be needful to be had.  
Another example, 43 ⌊ 7lib. of Gunpowder are sufficient to charge a Gun, whose diameter in the concave is 7 inches: now there is another sort of Gunpowder, much more strong and forcible, that is in strength unto the former, as 5 unto 2: How much of this stronger powder, will suffice to charge a Gun of 4 inches diameter?  
Here are two operations: the first seeks out, how much of that stronger powder sufficeth to charge a Gun of 7 inches diameter: and the proportion is reciprocal, that is 5 · 2 ∷ 43 ⌊ 7 · 15 ⌊ 48  
The second operation is like that in the former example. 343 (C ∶ 7) 64 (C ∶ 4) ∷ 15 ⌊ 48 · 3 ⌊ ●●●●    

CHAP. VII. Concerning the Measuring of Circles, Cones, Cylinders, and Spheres.  
1 ARchimedes in a peculiar Treatise found the proportion of the Diameter of a circle to the Circumference to be a very small deal greater than o● 7 unto 22: And of late Ludolph Van Ceulen insisting in the same steps of Archimedes, hath more precisely found it to be of 1 unto 3 ⌊ 141592653589793 but for our Instrument it will be sufficient to take the ratio of 1 unto 3 ⌊ 1416, Or of 0 ⌊ 3183+ unto 1: leaving the diligent practizer, to more exactness, if he please to use his Pen.  
And note that the Rules following are set down in proportions, to be wrought as hath been taught in Chap. 2, Sect. 3. Wherein  

D, or Diana. signifieth the Diameter.  
Dq, or Q. Diana. the Quadrat of the Diameter.  
Dc, or C. Diana. the Cube of the Diameter.  
R, or Rad. the Radius, or Semidiameter.  
P, or Perif. the Periphery, or Circumference.  
Pq, the Quadrat of the Periphery.  
Long. the length.  
L the side, or latus  
Alt. the altitude.  
●, showeth that the two magnitudes between which it is set, are to be multiplied together.   

In a Circle.  
2. The Diameter of a circle being given, to find the Periferia. Say, 7 · 22, Or 1 · 3 ⌊ 1416 ∷ Diana · Perif ·  
Example. A circle is given, the Diameter whereof is 12, I would know the circumference, or Periferia of it. Say, 1 · 3 ⌊ 1416 ∷ 12 · 37 ⌊ 6992 ·  
3 The Periferia of a circle being given, to find the Diameter. Say, 22 · 7, Or 1 · 0 ⌊ 3183 + ∷ Perif · Diana ·  
4 The Diameter of a circle being given, to find the Area. Say, 7 × 4 · 22, Or 1 · 0 ⌊ 7854 ∷ Q ∶ Diana · Area · Or else 1 · 3 ⌊ 1416 ∷ Q ∶ Rad · Area ·  
Example. A circle is given, the diameter whereof is 12, I would know the content, or Area of it. Say, 1 · 0 ⌊ 7854 ∷ 144 (Q ∶ 12) · 113 ⌊ ●●76 · Or 1 · 3 ⌊ 1416 ∷ 36 (Q ∶ 6) · 113 ⌊ 0176  
5 The Area of a circle being given, to find the Diameter. Say, 22 · 7 × 4, Or 1 · 1 ⌊ 27324 ∷ Area · Q ∶ Diana ·  
Example. A circle is given, the content whereof is 113 ⌊ 0976, I would know the Diameter of it. Say, 1 · 1 ⌊ 27324 ∷ 113 ⌊ 0976 · 144 ·  
6 The Periferia of a circle being given, to find the Area. Say, 22 × 4 · 7, Or 1 · 0 ⌊ 0795775 ∷ Q ∶ Perif · Area ·  
7 The Area of a circle being given, to find the Periferia. Say, 7 · 22 × 4, Or 1 · 12 ⌊ ●6637 ∷ Area · Q ∶ Perif ·   

In a Cone.  
8 The side of a right Cone, and the Diameter of the base being given, to find the Superficies. Say, 7 · 22, Or 1 · 3 ⌊ 1416 ∷ ½ D × L · Superf ·  
Example. A Cone is given, whereof the side is 18, and the diameter of the base 12, I would know the superficies of it. Say, 1 · 3 ⌊ 1416 ∷ 108 (½ D × L) · 33 ⌊ ●2928 ·  
9 The Axis, or height of a right Cone, and the Diameter of the base, being given, to find the Solidity. Say, 7 × 4 · 22, Or 1 · 0 ⌊ 7854 ∷ Dq in 1/● Axis · Solidity ·  
Example. A Cone is given, whereof the Axis is 18, and the Diameter of the base 12, I would know the Solidity. Say, 1 · 0 ⌊ 7854 ∷ 864 (Dq in 1/● axis) · 678 ⌊ 2856 ·   

In a Cylinder.  
10 The side of a right Cylender and the Diameter being given, to find the Superficies. say, 7 · 22, Or 1 · 3 ⌊ 1416 ∷ Diana ∶ × axem · Superfic.  
11 The Side of a right Cilinder, and the Diameter being given, to find the Solidity. Say, 7 × 4 · 22, Or 1 · 0 ⌊ 7854 ∷ Dq × L · Solidity ·  
12 The Side of a right Cylinder, and the Circumference P, being given, to find the Solidity. Say, 22 × 4 · 7, Or 1 · 0 ⌊ 0795775 ∷ Pq × L · solidit ·   

In a Sphere.  
13 The Axis, or Diameter of a Sphere being given, to find the Superficies. Say, 7 · 22, Or 1 · 3 ⌊ 1416 ∷ Dq · Superficies ·  
14 The Superficies of a Sphere being given, to find the Axis. Say, 22 · 7, Or 1 · 0 ⌊ 31831 ∷ Superfic · Dq ·  
15 The Segment of a Sphere being given, to find the Superficies. Say, 7 · 22, Or 1 · 3 ⌊ 1416 ∷ Q ∶ chord of ½ Segm. · Superfic ·  
16 The Axis, or Diameter of a Sphere being given, to find the Solidity. Say, 7 × 6 · 22, Or 1 · 0 ⌊ 5236 ∷ Dc · Solidity ·  
Example. A Sphere is given, whereof the Axis is 12, I would know the solidity of it. Say, 1 · 0 ⌊ 5236 ∷ 1728 (Dc) · 59● ⌊ 6208 ·  
17 The Solidity of a Sphere being given, to find the Axis. Say, 22 · 7 × 6, Or 1 · 1 ⌊ 90986 ∷ Solidity · Dc ·  
Example. A Sphere is given, the Solidity whereof is 590 ⌊ 6208, I would know the Axis thereof. Say, 1 · 1 ⌊ 90986 ∷ 590 ⌊ 6208 · 1728 (Dc) ·  
18 A Segment of a Sphere being given to find the Solidity. Say,  
First. As the altitude of the other Segment, is to the altitude of the Segment given: so is that altitude of the other Segment increased by half the Axis, unto a fourth. Then again say, As 7×3 is to 22, Or as 1 is to 10472: so is the product of the quadrat of half the chord of the Periferia of that Segment, multiplied by that fourth, to the Solidity. Viz. 7 × 3 · 22, Or 1 · 1 ⌊ 0472 ∷ Q ∶ chord in quartam · Solidit ·  
19 For note that a Sphere, is equal to two Cones, having their height and the diamerer of their base, the same with the Axis of the Sphere. Or which is all one, A Sphere is two third parts, of a Cylinder, having the height and the diameter of the base the same with the Axis of the Sphere.    

CHAP. VIII. Concerning Plain, and  Measures.  
1 THe dividing of the Carpenter's ruler into Inches, and half, and quarters, and half quarters of Inches, that is of every Inch into eight parts, is most inartificial, and unfit for measuring, by reason of the manifold denominations, which must be brought into one, and is hard to be done of them that are unskilful. I would wish therefore that every Inch were divided into 10 parts, or rather that the foot were divided into 100 parts, which is best of all: for then there will need no reduction. And all other divisions must be reduced unto this, by these Rules following.  
2. If the measures be taken upon a Ruler divided into Inches and tenth parts of an Inch, first take out all the whole feet, and then divide the Inches remaining, with their decimal parts if there be any by 12.  
Example. How many feet and decimal parts of a foot, are in Inches 17 ⌊ 3?  
First take out the whole foot which is 12 Inches, and there will remain Inches 5 ⌊ 3: which being divided by 12, you shall have 442 thousand parts almost: wherefore Inches 17 ⌊ 3, is feet 1 ⌊ 442—. And contrariwise, feet 1 ⌊ 442—, shall be reduced into Inches 17 ⌊ ●, being multiplied by 12.  
3. If the measure be taken upon a Ruler divided into inches and half quarters, that is each inch into 8 parts, First you must reduce the eight parts into decimals of Inches, by dividing the number of parts given by 8 the Denominator thereof: and afterward by the former Rule, reduce the Inches, and decimal parts, into decimal parts of a foot.  
Example. How many feet, and decimal parts of a foot, are in 7 inches, and 5 eight parts?  
First divide the 5 eight parts by 8, and you shall have 625 thousand parts: which being put to 7 inches, will make inches 7 ⌊ 625: Again divide these by 12, as was showed in the former rule: and the whole measure will be feet 0 ⌊ 635. And contrariwise feet 0 ⌊ 635, will be reduced into inches 7 ⌊ 625, being multiplied by 12.  
4. I must advice all those that have occasion, to measure Plains, or Solids, to make themselves very perfect in this kind of Reduction (because most Rulers they shall ordinarily meet withal, are divided into inches, and half quarters) which will be very easy to them, if they do but remember, that In division the first term of the proportion employed, is the Divisor itself: but in Multiplication, the first term is evermore 1. as hath been showed in Chap: 2, Sect: 7. And therefore, presuming on the diligence of the Practiser herein, I shall not need in this kind of measuring, to speak any more of inches, but of feet and decimal parts of feet, as if the Ruler were so divided.  

Of Plain measures.  
5. A Parallelogram, or four sided rectangle Superficies, being proposed, to find the length of a Superficial foot.  
Take with your Ruler the breadth thereof in feet, and decimals of a foot: and by the breadth so taken divide 1. the quotient shall be the length of a superficial foot.  
Example. A board is feet 1 ⌊ 17 broad, how much thereof will make a foot?  
Divide 1 by 1 ⌊ 17, the quotient will be 0 ⌊ 855 almost: so much shall the length of a foot be, which multiplying the parts by 12, will give inches 10 ⌊ 26. And again those parts multiplied by 8, will give 2 eight parts of an inch.  
Example, II. In tileing, or healing they use to reckon by the Square, which is 10 foot every way, in all 100 ●eet. There is roof, feet 16 ⌊ 2● broad, how much thereof maketh a Square?  
Divide 100 by 16 ⌊ 25 the quotient will be 6 ⌊ ●●4 almost: so much shall be the length of one square; which multiplying the parts by 12, will be 6 feet, and inches 1 ⌊ 8●●— almost. And again those parts multiplied by 8, will give somewhat more than 6 eight parts of an inch.  
Example, III: In paving they use to reckon by the yard, which is 3 feet every way, in all 9 feet. There is a room to be paved, which is feet 17 ⌊ 35 broad; how much thereof maketh a yard?  
Divide 9 by 17 ⌊ 35, the quotient will be 0 ⌊ 519 almost, the length of one yard.  
6: A four sided rectangle Superficies, with all the opposite sides parallel being proposed, to find the content.  
Take with your Ruler both the breadth and length of and multiply the one number into the other.  
Example, A board is feet 1 ⌊ 17 broad, and feet 16 ⌊ 32 long: how many feet doth it contain in all?  
Multiply 16 ⌊ 3● by 1 ⌊ 17 the product will be feet 19 ⌊ ● almost: the whole quantity of that board.  
Example. II. A certain barn tiled, hath the breadth of the roof feet, 16 ⌊ 2●, and the length of the barn is feet 47, how many squares of tiling hath it?  
Double the length (that you may have both sides of the roof) and it will be 94, which being multiplied by 16 ⌊ 25, will give feet 1527 ⌊ ●. Again divide those feet by 100, so shall you have squares 15 ⌊ 275.  
Example. III. A certain hall paved hath the breadth feet 17 ⌊ 35, and the length feet 30 ⌊ 5, how many yards doth it contain?  
Multiply 17 ⌊ 35 by 30 ⌊ 5, the product will be feet 229 ⌊ 175. Divide these by 9 and the quotient will be yards 58 ⌊ 797.  
7. A four sided Superficies with the two sides of length only parallel being proposed, to find the content thereof.  
Take with your Ruler the length of the two parallel sides thereof: add both those numbers together, multiply half that sum, by the breadth of the Superficies taken the nearest way over, and the product will be the content thereof.  
Example. A Trapezium, or four sided figure is proposed, having two sides thereof parallel, the length of the longer parallel side is feet 18 ⌊ 75 and the length of the shorter side is feet 14 ⌊ 4●, the breadth thereof being taken the nearest way over is feet 12 ⌊ 5, I would know how many feet are contained in the whole Superficies?  
The length of the parallel sides are feet 18 ⌊ ●5, and 14 ⌊ 45, which added together make 33 ⌊ ●, half whereof is 16 ⌊ ●, which multiplied by the breadth 12 ⌊ ●, the product will be 207 ⌊ 5, so many feet are contained in the superficies of the Trapezium proposed.  
8. A four sided Superficies which hath none of the sides parallel, as also every plain figure of more sides the four being proposed, must with Diagoniall lines be divided into triangles. And note that every such figure containeth so many triangles as it hath sides, abating two out of the number. Then those triangles are to be measured severally as followeth.  
9 To find the content, or Area of a Triangle.  
Take the perpendicular height, or nearest distance between the base or known side, and the angle opposite: and by that height multiply half the base, or multiply the whole base by half that perpendicular height: and the product shall be the content, of that Triangle.  
But if it be an Aequilater triangle: say, 1000 · 433 ⌊ 0127 ∷ the side of the triangle · Area ·  
10. To find the content of a segment of a Circle, whereof the Periferia is given in degrees and decimal parts.  
First say, As 100000, is to 1745 ⌊ 32025: so is the Arch in degrees, to the Arch in the divisions of the Radius. keep this number found.  
Again by the 6 Sect: 1 chap: find out the true sinus of the Arch given. Then take the difference of these two numbers found, by subducting the Sinus out of the Arch. And lastly multiply half that difference by the Radius 100000, the product shall be the content of that segment.  
11 The chord of any arch, together with the Radius, or semidiameter of the whole circle being given, to find out the Arch itself. Say,  
At the Radius given, is to half the chord (reckoned in the fourth circle) so is 100000, to the sinus of half the arch (to be reckoned in the first, or eighth circle). Wherefore double the arch found, and so have you the arch of the chord proposed.  
12 To find a Quadrat, or Square equal to a superficies given.  
First, seek out (as hath been taught) the content of that superficies: then take the quadrat root thereof by Chap. 5, Sect. 4.   

Of Solid measures.  
13 In a Column, or Cylinder, having the base, to find how much of it maketh a foot solid.  
By a Column I mean a solid body arising from a plain base, the angular lines whereof are parallel, and equal: and if the angular lines make right angles with the base, it is a right Column, and the length is the height thereof: but if they make obliqne angles, it is an Obliqne Column, and the length is not the height, but the height is a perpendicular line let down from the top of the Column unto the base, extended if need be: as in the Diagrame, the Solid ABCDEFGH, is an Obliqne Column, because the angular line EBB, standeth obliquely upon the side of the base BASILIUS, and indeed upon the base itself. Wherefore the height of it shall be equal to the line FP, let fall from the top unto the base extended.   diagram 
And after this manner also a Cylinder, and a Pyramid, and a Cone is esteemed either right, or obliqne, and the height taken accordingly.  
First therefore the base is to be found, of what fashion soever it is, as hath even now been showed, either in this Chapter, or in the last before: and then divide 1 by that same base: the quotient shall be, the height of a Section thereof, which is equal to one foot solid.  
Example. A Column, or piece of timber, whose sides are all parallel, hath the breadth feet 1 ⌊ 75, and the thickness thereof is feet 1 ⌊ 25: which multiplied together the product will be 2 ⌊ 1875. Divide therefore 1, by 2 ⌊ 1875: and the quotient shall be 0 ⌊ 457143 almost. And so much is the height of a solid foot, of that piece of timber.  
14 Having the Base, and the height of a Column, or Cylinder, to find the whole content.  
Multiply the base into the height, and the product shall be the content.  
Example. A Column hath the base feet 2 ⌊ 1875, and the height thereof is feet 17 ⌊ 34, how many feet are contained in the whole?  
Multiply the base 2 ⌊ 1875, by 17 ⌊ 34 and the product will be 36 ⌊ 086875, so many solid feet are contained in that Column.  
And in this very manner may you find the content of a Cylinder, having either the diameter, or circumference given, together with the height.  
15 To measure tapering timber, the base, or bases thereof, together with the height being given.  
A Tapering piece of timber, according as the base thereof is right lined, or circular, is either a Pyramid or a Cone, or else a segment of one of these two: If it be a complete Pyramid, or Cone, it hath but one base, Multiply that base by ⅓ of the height, and the product shall be the content.  
But if it be the segment of a Pyramid, or Cone, First find out the bases at both the ends, and multiply the one by the other, and out of the product thereof extract the Quadrat root: then add together both the bases, and that quadrat root, and multiply the Aggregate thereof by ⅓ of the height, the product shall be the content.  
Example. There is a tapering piece of timber, the height whereof is feet 12 ⌊ 6, and the breadth of the base at the greater end is feet 1 ⌊ 75, and the thickness is feet 1 ⌊ 32, which multiplied together, the product will be feet 2 ⌊ 3● for the greater base: the breadth of the base at the lesser end is feet 1 ⌊ 2, and the thickness there, is feet 0 ⌊ 91, which multiplied together, the product will be feet 1 ⌊ 052 for the lesser base. Multiply the bases together, the product will be 2 ⌊ 52252, the quadrat root whereof is 1 ⌊ 588 almost, to which if you add the sums of both the bases, the aggregate will be 4 ⌊ 99, which being multiplied by ⅓ of the height, scil. 4 ⌊ 2, the product will be feet 20 ⌊ 958, the content of the piece of timber.  
16 Wherefore that vulgar manner which Carpenters use in measuring of tapering timber, is not true: for if a piece of timber be tapering, they measure it in the very middle, and take the base, or Section there, multiplying it by the whole length. Which their manner of working, Isay, is erroneous.  
For first by practise a content will be given, ever less than the true content found according to the former Sect: which way of working is infallibly true, as is Analitically demonstrated, in my Clavis mathematica, Cham 20, Sect: 15.  
And secondly Isay, that the product of that middle base, or Section, multiplied by the length, shall be less than the true content, by four Pyramids, having for their bases, a rectangle under half the difference of the thickness at the ends, and a quarter of the difference of the breadths: and are as long as half the piece of timber: Or which is all one, by a Parallelepipedon, under half the differences of the breadthes, and thicknesses, at both ends, and a third part of the whole length.   diagram 
Which that I may show, suppose one quarter of a tapering piece of timber given, sawed in ●unner, at half the breadth, and half the thickness be ABCDGFEH: the middle Section is IK●M. Measure upon the greater base BN, and CO, equal to OF, and BY, and NZ, equal to FE. Divide CT, and DO to the midst, in the points X, and Q, and draw the lines PQ, NO parallel to BC, and YZ, SX parallel to BG. Measure also FER, equal to GS, or KL. And so the Parallelegram BPTX, shall be equal to the middle Section IKLM. Lastly, draw the lines LO, LV, LZ, and KP, KN, and MX, MY.  
I say that in this one quarter of the piece of timber, A solid having the bases equal to IKLM, and the length AB (which is the usual measuring of Carpenters) is less than the true content, by the Pyramid DSVOL, in which DO, or SV, is half the difference of thickness, and DS, or VO, one quarter of the difference of breadths in both ends: and the height of it equal to half the length.  
For the two solides AFEHIKLM, BNZYIKLM, are apparently equal, again the two  wedges GSTPKL, PTUNKL are equal, and also the two  wedges OCXULM, VXYZLM are equal. Now if you turn over the wedge GSTPKL, unto the lesser part AFEHIKLM, it will overreach in thickness, at the lesser end the quantity of ER, and if you also turn over the wedge OCXULM, unto the lesser part, you shall find it to fill up the former overreaching, and to make an exact Parallelepipedon, the bases or ends whereof are equal to the middle Section IKLM. but over and above those two wedges turned over, you shall have left the Pyramid DSVOL, which was to be proved.  
And in like manner if a round tapering , or piece of timber, be measured by the middle Section, or circular base, I say, That the product thereof multiplied by the length, shall be less than the true content, by a Cylinder, the diameter of whose base, is equal to half the difference of the diameters of the two bases: and the length is one third part of the whole length.    

CHAP. IX. Concerning the Measuring, or Gauging of Vessels.  
AWine, or Beer vessel, whether Pipe, Hogshead, barrel, Kilderkin, or Pirkin, and such like, is in form of a Sphaeroides, having the two ends equally cut off: and accordingly may be measured thus.  
Measure the two diameters of the Vessel, in inches, or else in tenth parts of a foot, the one at the bung hole, the other at the head, and also the length within. And by the diameters found, find out the circles; then add together two third parts, of the greater circle, and one third part of the less: Lastly, multiply the aggregate by the length: so shall you have the content of the Vessel, either in Cubic inches, or cubic tenth parts of a foot.  
Example. Suppose a vessel, having the Diameter at the bung 32 inches, and at the head 18 inches, and the length 40 inches. The quadrat of 32 is 1024. and the quadrat of 18 is 324: Say then evermore, 1 · 0 ⌊ ●236 ∷ 1024 · 536 ⌊ 166, ●/3 the circle at the bung · 1 · 0 ⌊ ●618 ∷ 324 · 84 ⌊ 823, ⅓ circle at the head. Or else by Chap. 7, Sect. 4.  
The aggregate of those two circles is 620 ⌊ 989, which being multiplied by 40, the length giveth 24839 ⌊ 56 cubic inches for the whole content of that vessel.  
2 Mr. Edm: Gunter in his second book of the Cross Staff, Chap. 4. pretending to show the manner of gauging Wine vessels, beginneth with these words.  
The Vessels which are here measured, are supposed to be Cylinders, or reduced into Cylinders by taking the mean, between the Diameter at the head, and the Diameter at the boungue, after the usual manner.  
And according to this supposition, teacheth to find a Gage point, For a gallon of wine, in that his imagined Cylindriacall vessel.  
Because his words are cautelous, and subterfugious, we must a little examine them; for if his way be true, my Rule before set down, though grounded upon demonstration, cannot stand.  
Well then, that reduction of a wine Vessel into a Cylinder, is either true, or false; if it be true what need those ambiguities of Vessels which are here measured: and are supposed to be, etc. and after the usual manner? if false why is it not noted, but delivered as a Rule to confirm an error. And what meaneth, the mean between the Diameter at the head, and the Diameter at the boungue? is it the mean Geometrical, or Arithmetical, that is the mean proportional, or that which equally differeth from both? such shifting is unworthy an Artist.  
First therefore, Let it be the mean in respect of difference, which is equal to half the sum of the two Diameters: I say that the Vessel cannot truly be reduced to a Cylinder by such a mean Diameter. For seeing it is most apparent that such a Vessel, is greater in the middle then at the ends, the boards, or sides thereof, shall from the middle to the ends, go either straight, and so the Vessel shall be as it were, two equal segments of Cones, set base to base: Or else arching, and so the Vessel shall (as before I said, and is commonly taken for a truth) be a Sphaeroides, having the two ends equally cut of.  
If it be considered as two segments of Cones: the measure by that mean Diameter, or middle Section is quite false, as hath been demonstrated in the former Chapter, Sect: 16, 17. and will be given less than the true content, although the sides go strait: much more then, if the sides go arching; for that convexnesse, must needs yield a greater capacity. And therefore in neither can that manner of Gauging be true.  
Again, if the mean Diameter be understood to be the mean proportional between the two Diameters, it is much more false, for between any two numbers, the mean Geometrical is less than the mean Arithmetical.  
Thus much I have thought good in this place ingenuously to signify to the inexpert Learner, that he might not beguile himself with a prejudged opinion.  
3. I have showed the measuring of vessels by the cubic inch: but our usual reckoning is by the Gallon, and parts thereof. We must therefore do the best we can, to inquire the true quantity of a Gallon in inch measure, which will be difficult to do exactly, both because the Standards usually are not straight sided, but a little arching, neither do they agree perfectly one with another: but what partly by experience, both mine own, and others, which hath come to my sight; and partly by reasoning shall seem to me most probable, I will not refuse to set down.  
4 Our English Gallon is understood to be either in Ale measure, or Wine measure: and these two measures not a little differing. And first we will inquire about our Ale measure.  
I myself have measured Bushels, and P●cks, which have exactly been fitted to the Standards, and have still in my account found a Gallon to contain better than 270 cubic inches, indeed much about 272, or 273 as precisely as I could measure in a vessel not truly regular. Also my worthy friend Master William Twine, who hath undergone great pains and charge, in finding out the true content of our English measures, gave unto me two several measures of an Ale-gallon, and those in due consideration but little differing. The one was found out by a brassen vessel made in manner of a Parallelepipedon, the base whereof was exactly six inches square, and the sides divided into inches and twentieth parts: into which vessel he pouring out a standard Gallon of Queen Elizabeth, filled with water, found it therein to ari●e unto 7 inches, and 6 tenth parts: which being computed maketh cubic inches 273 ⌊ 6. The other was found by taking the dimensions of that standard Gallon, which was made in form of a segment of a Cone, but that the sides were a little arching: the dimensions were thus; the Diameter of the top was inches 6 ⌊ 5: the Diameter of the bottom was inches 5 ⌊ ●: and the height of it was inches 9 ⌊ 8: which being cast up by Chap: 8, Sect: 15, will be found to contain cubic inches 268 ⌊ 85: differing from the former, only cubic inches 4¾: which difference might well arise through the curuitie of the sides. These measures he did not only take himself, but to give me satisfaction, shown me the experience in the said Vessel and Standard: but the truth is, I observed the Standard, besides the arching of the sides, to be not exactly circular within, nor the brim of an even height, nor the bottom plain: and in taking the height of the water in the Vessel, our sight was not able to estimate the ascent thereof so precisely, that a spoonful of water, more or less, could breed any sensible difference.  
What therefore shall we do in this difficulty? indeed look to the first ground, and principle of our English measuring, from Barley corns. For the length of 3 Barley corns taken out of the middle of the ear is an Inch, or Vncia, that is a twelfth part of a foot. 3 feet make a Yard: and 16 feet and an half, that is 5 yards and an half, a Perch, with which we measure our land; for 40 perches is a Furlong, and 8 furlongs an English mile: and again 40 square perches is a Roodland, 4 roodland make an Acre. So then a Perch which is feet 16½, or yards 5½ is as it were the beginning of all land measure in length: and a square Perch which is feet 272¼, is as it were the beginning of all land measure in the superficial content.  
Now therefore seeing in Vessels a gallon is as it were the beginning of Vessel measure (for a pottle is but a diminative of it, and a quart, the quarter) it is not unlikely that our wise Ancestors had such a consideration also in  measures, that as a square Perch (the beginning of Superficial land measure) did contain 372 square feet and a quarter; so a Gallon (the beginning of vessel measure) should contain 272 Cubic inches and a quarter. And the rather seeing that the ancient Geographers, divide a foot into 4 Palms, a palm being 3 of our inches, as 3 feet are a yard. So that as the side of a square Perch consisteth of yard's 5½, a Gallon also should consist, of a number of Cubic inches the square side whereof is palms 5½.  
Wherefore saving the exact truth when it shall appear, and in the mean time the more probable reasons of other men, I make bold to tender this my conjecture, to the censures of more diligent Inquirers, That the measure of an English Ale gallon should be a square Vessel of inch 16½, or Palms 5½ every way, and 1 inch deep: that is 272 ⌊ 2● Cubic inches.  
5. And this my opinion may peradventure receive some confirmation by the inquiry of an English wive Gallon.  
M. Henry Briggs that learned Geometrician, and my very loving friend, made an experiment, with a cubical vessel, which was 12 inches every way, which having filled with water carefully measured, found it to contain 7 gallons and an half wanting a moment, as he himself long since, being then of Gresham College, signified to me. Now if it had contained exactly 7 gallons, and an half, a wine gallon should have been 230 ⌊ 4 cubic inches, but because it wanted a little, the gallon must be somewhat bigger, for which moment therefore if you will put 6 hundred parts of an inch, the wine gallon shall contain 231 cubic inches.  
Again M. Gunter in the place before mentioned showeth, that the common opinion is that at London, a Cylindriacall vessel, whose diameter is 38 inches, and length 66 inches, doth contain 324 gallons: wherefore by this account a gallon should be 231 cubic inches almost exactly, which in both so nearly agreeing, we may well conclude, That an English wine gallon doth contain 231 cubic inches.  
It is also commonly received, that the reason of the greatness of an Ale gallon above the Wine gallon is, that because of the frothing of the Ale or Beer, the quantity becometh less, and therefore such liquors that did not so yield froth, as Wine, Oil, and the like, should in reason have a lesser measure. If then we compare these two gallons together, we shall find that 272 ⌊ 25 · 231 ∷ 16 ⌊ 5 · 14 · which abatement might to our Ancestors, in apportioning those measures, seem to be reasonable.  
6 To find how many Cubic tenth parts of a foot are in a gallon, both of beer, and Wine: or also in any number of Cubic inches.  
Because there be in a Cubic foot, 1000 Cubic tenth parts, and 1728 Cubic inches, say, 1728 (C ∶ 12) · 1000 (C ∶ 10) ∷ 272 ⌊ 25 · 157 ⌊ ●521— cubic tenth parts of a Beer gallon.  
And 1728 (C ∶ 12) · 1000 (C ∶ 10) ∷ 231 · 133 ⌊ 6803— cubic tenth parts in a Wine gallon.  
Also 1728 · 1000 ∷ 24839 ⌊ 56 · 14374 ⌊ 746, cubic tenth parts, are in the vessel measured in Sect. 1.  
7 And according to these measures so found, you may easily find the content of a pint, or quart, or peck, or bushel, which two last are to be reckoned in Ale measure.  
8 The Content of a Vessel, being given in cubic inches, or in cubic tenth parts of a foot, to find how many gallons it containeth.  
This is easily done if you divide the content given in inches, by 272 ⌊ ●5 for Ale measure: and by 231, for Wine measure. But if the content be given in decimal parts of a foot divide it by 157 ⌊ 5521— for Ale measure; and by 133 ⌊ 6803— for Wine measure.  
Example. How many wine gallons are in a vessel containing 24839 ⌊ 56 cubic inches, or 14374 ⌊ 746 cubic tenth parts of a foot. Divide 24839 ⌊ 56 by 231, or divide 14374 ⌊ 746 by 133 ⌊ 68, and the quotient shall be 107 ⌊ ●3 wine gallons.   

CHAP. X. Concerning the Comparison of sundry Metals, in quantity and weight.  
1 IF four pieces of Metals, whereof the third is of the same kind with the first, and the fourth of the same kind with the second, are proportional, their gravities also, or weights, shall be proportional.  
2. If there be four pieces of metal, whereof the third is of the same kind with the first, and the fourth of the same kind with the second: and the first and second be of equal greatness, and the third and fourth of equal weight; the weights of the first and second, shall be reciprocal to the magnitudes of the third and fourth.  
3. Two Spheres of the same matter are in weight, as the cubes of their Diameters, are in magnitude. Et contra.  
4. Pieces of Metal if they be of equal magnitude, have their weights in direct proportion, as is here set down: but if they be of equal weight, they have their magnitudes in proportion reciprocal: According to the experiments of Marinus Ghetaldi, in his tractate called Archimedes promotus.  
Gold. 3990 Brass. 1890 Arg. Viu. 2850 Iron. 1680 Lead. 2415 Tin. 1554 Silver. 2030    
5. To find the Weight of a Sphere of Tin, having the Diameter 1 Inch, or else ● tenth part of a foot.  
Take a piece of Tin, and turn it exactly in a Lathe, into a Cylinder, having both the Diameters of its base, and also the length equal, to the Diameter of the Sphere given. Then weight that Cylinder, that you may have the weight thereof in grains: And lastly, take two third parts of the whole number of grains: for the weight of the Sphere.  
After this manner Marinus Ghetaldi found a Cylinder of 1 inch, or twelfth part of a foot thick, and long to weigh 1824 grains: whereof ● is 1216, the weight of a Sphere of that thickness.  
Again if you say, 1000 (C ∶ 10) · 1728 (C ∶ 12) ∷ 1216 · 2101 ⌊ 248  
You shall have the weight of a Sphere whose Diameter is one tenth part of a foot.  
Wherefore also a Cubed inch of Tin weigheth 2322 ⌊ 4—, and a Cubed tenth part of a foot weigheth 4013 ⌊ 1—  
And note that Mar: Ghetaldi useth the ancient Roman foot, which by the measure set down in his book seemeth to be very little less, then our-usuall English foot, if not exactly the same.  
Note also that he divideth one pound, into 12 ounces, and every ounce into 24 Scruples, and every Scruple into 24 grains: So that an ounce with him weigheth 576 grains: and a pound 6912. Whereas our English pound of Iroy weight by Assize, or Gold smith's weight, is but 5760 grains, and our ounce 480. But whether the Raguzean grain, be the same with our English, I leave to be tried by the diligent Practizer.  
6 To find the Weight of a Sphere of Tin, at any other Diameter assigned.  
Multiply the Cube of the Diameter given by 1216, if it be in inch measure; or by 2101 ⌊ 248, if the measure be by decimal parts of a foot: and the product will be the weight of that Sphere.  
And contrariwise to find the Diameter of a Sphere of Tin, by the weight given in grains. Divide the weight given in grains by 1216 if you would have inch measure: or by 2101 ⌊ 248 if you would measure by decimal parts of a foot, and the quotient shall be the Cube of the Diameter.  
7. To find the Weight of a Sphere of any Mettle, at any Diameter given, either in Inch measure, or in decimal parts of a foot.  
First by Sect. 6, seek the weight of a Sphere of Tin, at that Diameter: then by Sect. 4, say, As the proportional number of Tin, is to the proportional number of that other Metal: so is the weight of the Sphere of Tin now found, to the weight of the Sphere proposed.  
Example. Suppose a Sphere of Iron, whose Diameter is 3 inches; what shall be the weight thereof?  
First, the weight of a Sphere of Tin, of 3 inches Diameter, will be found to be 32832 grains. Then say, 1554 · 1680 ∷ 32832 · 35494 ⌊ 054 grains the weight of the Sphere proposed.  
8 To find the Diameter of a Sphere of any Metal in inch measure, or decimal parts of a foot, the weight thereof being given.  
First by the contrary of Sect. 6, seek the Cube of the Diameter of a Sphere of Tin of that weight. Then by Sect. 4, say reciprocally. As the proportional number of that other Metal, is to the proportional number of Tin: so is the Cube of the Diameter now found, to the Cube of the Diameter of the Sphere proposed.  
Example. A Sphere of Iron weigheth 35494 ⌊ 0●4 grains: how many inches is the Diameter thereof?  
First, the Cube of the Diameter of a Sphere of Tin of 35494 ⌊ 054 grains weight, will be 29 ⌊ 18910695, then say, reciprocally, 1680 · 1554 ∷ 29 ⌊ 18910691 · 27—  
The Cubic root whereof is 3, the Diameter of a Sphere of Iron of that weight proposed.   

CHAP. XI. Concerning the Ordering of Soldiers, in any kind of rectangular form of battle.  
1 Battles are considered either in respect of the number of men, or in respect of the form of ground. As asquare battle of men is that which hath an equal number of men, both in Rank and File, though the ground on which they stand, be longer on the File, then on the Rank. And a square battle of ground is that which hath the Rank as long as the F●le, though the men in Rank be more than in File.  
2. In respect of the number of men, it is called either asquare battle, or a double battle, or a battle of the grand front, which is quadruple, or a battle of any proportion, of the number in Rank, to the number in File.  
3. If it be asquare battle of men: Extract the quadrat root out of the whole number of men, and the same shall be the number of Soldiers, to be set in a Rank.  
Example. 576 Soldiers are to be martialled in a square battle, that so many may be in Rank, as in File.  
Take the quadrat root of 576, which is 24: the same shall be the number to be placed in a Rank.  
4. If it be a double battle of man: Extract the quadrat root out of half the number of men, and the same doubled shall be the number of Soldiers to be set in a Rank.  
Example. 1458 Soldiers, are to be placed in a double battle; so that twice so many may be in Rank as in File.  
Take half the given number 1458, which is 729, the quadrat root whereof is 27: double it and you shall have 54 men to be placed in a rank.  
5. If it be a quadruple battle, which is called of the great front: Extract the quadrat root out of one quarter of the number of men, and the same quadrupled shall be the number of Soldiers to be set in a rank.  
Example. 1024 Soldiers are to be martialled into a battle of the grand front, so that four times so many be in rank as in file.  
Take one quarter of 1024 the number given, which is 256, the quadrat root whereof is 16: quadruple it, and you shall have 64 men to be placed in a rank.  
6. If a battle be required of any other form, that is, if a Ratio be given, according to which the number of men in Rank, shall be to the number in file. Multiply the two terms of the Ratio given: Then say As the product is to the quadrat of the term which is for the rank, or As the term which is for the file, is to the term which is for the rank: so is the whole number of Soldiers, to the quadrat of the number of men to be placed in a rank.  
Example. 1944 Soldiers, are to be martialled so, that the number of the rank, be to the number of the file, as 8 unto 3, that is for 8 men in rank, 3 are to be set in file.  
First multiply the two terms of the Ratio 8 and 3, the product whereof is 24, also quadrat 8, the term of the rank, which will be 64. Then say, 3 · 8 ∷ 1944 · 5184 · out of which extract the quadrat root 72, and it will give you the true number of the rank.  
7. In respect of the form of ground, the battle is either a square of ground, or longer one way then the other. For the distance, or order of Soldiers martialled in array, is distinguished either into Open order, or Order.  
Open order is when the very centres of their places, are distant 7 feet asunder, both in rank and file.  
Order is when the centres of their places, are distant 3 feet and a half in rank, and so much in file. Or else 3 feet and a half in rank, and 7 feet in file: which last order, and whatsoever order else there is, in which the distance of the ranks one from another is greater, than the distance of files, causeth that a square of men, maketh not a square of ground, but the ground is longer on the file then on the rank.  
8 If it be a square battle of ground, the centres of the distances being feet 3½ in rank, and 7 feet in file. Because 3½ is half of 7, the ratio of the distances, is as 1 unto 2. And seeing the number in rank, to the number in file, is reciprocal to the distances, the ratio of the number of men in rank, to the number of men in file, shall be as 2 unto 1. And so the Rule shall be the same with that in Sect. 6, namely, As the term of the file, is to the term of the rank; so is the whole number of Soldiers, to the true number of the rank.  
Example, 1352 Soldiers, are to be set in a square of ground, that their distances may be feet 3½ in rank, and 7 feet in file.  
The Ratio of the rank to the file, shall reciprocally be, as 7 to 3½, that is as 2 to 1. Say therefore 1 · 2 ∷ 1352 · 2704, the quadrat root whereof 52 is the number of men to be set in a rank.  
9 If a battle wherein the distance in rank is unequal to that in file, be longer one way then the other, according to any Ratio given: there is to be considered a double ratio, one reciprocal in respect of the distances, the other according to the form of the ground. Wherefore to find the Ratio of men in rank, to the men in file, Multiply the two terms of the rank, for the rank, and the two terms of the file, for the file. And then the Rule shall be the same with that in Sect. 6, namely, As the term of the file, is to the term of the rank: so is the number of Soldiers, to the quadrat of the true number of the rank.  
Example. 10290 soldiers, are to be set in a battle, so that they may stand only 3 feet asunder in rank, and 7 feet in file, and the length of the ground for the rank, to the length of the ground for the file, shall have the ratio of 5 unto 2.  
First in respect of the distances, the Ratio of Rank to file, reciprocally is as 7 unto 3. Secondly, in respect of the ground, the ratio of rank, to file, is as 5 to 2. Wherefore by multiplication of like terms, the true ratio of rank to file shall be 7 × 5 to 3 × 2, that is as 35 to 6. Say therefore 6 · 35 ∷ 10290 · 60025 the quadrat root whereof is 245, the number of men to be set in rank.  
10 If 1000 Soldiers, may be lodged in a square, of 300 feet, how many feet must the side of a square be, which will serve to lodge 5000? Say, 1000 · 5000 ∷ 300 × 300 · 450000 · the quadrat root whereof 671— is the square side sought for.  
And this is the order for resolution of all other questions of this sort.   

CHAP. XII. A collection of the most necessary Astronomical operations,  
1 BEfore we deliver the Rules of such operations, it will not be inconvenient, to set down certain Reductions, whereof we may have frequent use.  
To reduce sexagesime parts into decimals. Divide the sexagesimes given by 60.  
Example. How many decimals are 34′, 12″?  
Here are required two reductions, first of the seconds into decimals of minutes: then of the minutes with their decimals, into decimals of degrees, Thus 60 · 1 ∷ 12″ · 0 ⌊ 2 Again 60 · 1 ∷ 34 ⌊ ′ · 0 ⌊ 57° Wherefore 34′, 12″ are equal to 0 ⌊ 57 of a degree.  
And contrariwise to reduce decimal parts of degrees in sexagesimes. Multiply the decimal part given by 60.  
Example. How many sexagesime parts are 0 ⌊ 57°? 1 · 60 ∷ 0 ⌊ 57° · 34 ⌊ ●′ Again 1 · 60 ∷ 0 ⌊ 2 · 12″  
To reduce hours into degrees. Multiply the hours with their decimal parts by 15.  
Example. How many degrees are 8 Ho, 34′, 12″; that is by the former reduction 8 ⌊ 57 Ho? thus 1 · 15 ∷ 8 ⌊ 57 · 128 ⌊ 55  
Wherefore Hours 8, 34′, 12″ do contain 128 ⌊ 55 degrees.  
And contrariwise to reduce degrees into hours. Divide the degrees with their decimal parts by 15.  
Example. How many hours are in degrees 128 ⌊ 55? 15 · 1 ∷ 128 ⌊ 55 · 8 ⌊ 57  
2 It is to be understood, that if four numbers are proportional, their Order may be so transposed, that each of those terms, may be the last in proportion. In this manner,  

I. As the first is to the second; so is the third to the fourth.  
II. As the third is to the fourth; so is the first to the second.  
III. As the second is to the first; so is the fourth to the third.  
FOUR As the fourth is to the third; so is the second to the first.   
Wherefore every proportion doth implicitly contain four Orders, two descending, and two ascending, as may be seen by their combinations: By one of which orders, if of four proportional numbers, any three be given, that other which is unknown, may be found out.  
Example. To find out any of these,  
terms 1 As the Sine of the compliment of the sun's declination, 2 is to the Sine of the compl. of his altitude; 3 So is the Sine of the Sun's Azumith from the meridian, 4 to the Sine of the horary distance from the meridian.  
If the first, second, and third terms be given, the fourth shall be found out by the I order.  
If the first, third, and fourth terms be given, the second shall be found out by the TWO order.  
If the first, second, and fourth terms be given, the third shall be found out by the III order.  
If the second, third, and fourth terms be given, the first shall be found out by the IIII order.  
3. To find out any one of these.  
terms 1 As the Radius, or total Sine 2 is to the Sine of the distance, or longitude of the Sun in the Ecliptic, from the next Aequinoctial point: 3 So is the Sine of the Sun's greatest declination (which is the angle of the Ecliptic with the Aequinoctial), 4 To the Sine of the Sun's declination in that longitude.  
4. To find out any one of these.  
terms 1 As the Radius, 2 is to the Sine of the Sun's right ascension, from the next equinoctial point: 3 So is the tangent of the Sun's greatest declination, 4 to the tangent of the Sun's declination in that place.  
5. To find out any one of these.  
terms 1 As the Radius 2 is to the Sine of the compl. of the Sun's greatest declination: 3 So is the tangent of the longitude of the Sun from the next equinoctial point. 4 to the tangent of the Right ascension of the Sun, from the same aequinoctial point.  
6. To find out any one of these.  
terms 1 As the Radius 2 is to the Sine of the compl. of the longitude of the Sun from the next equinoctial point; 3 So is the tangent of the Sun's greatest declination, 4 to the tangent of the compl. of the angle of the Ecliptic with the Meridian.  
7. To find out any one of these.  
terms 1 As the Radius, 2 is to the Sine of the Sun's greatest declination: 3 So is the Sine of the compl. of the Sun's right ascension from the next aequinoctial point, 4 to the Sine of the compl. of the Angle of the Ecliptic with the Meridian.  
8. To find out any one of these.  
terms 1 As the Sine of the compl. of the Poles height, 2 is to the Radius; 3 So is the Sine of the Sun's declination, 4 to the Sine of the Sun's Amplitude ortive, that is the arch of the horizon from the place of the Sun's rising or setting to the true East, or West point.  
9 To find out any one of these.  
terms 1 As the Radius, 2 is to the Sine of the Sun's greatest amplitude ortive, which is in the Tropic: 3 So is the Sine of the longitude of the Sun from the next aequinoctial point, 4 to the Sine of the Sun's amplitude ortive.  
Or also of these.  
terms 1 As the Sine of the compl. of the Poles height, 2 is to the Sine of the compl. of the Sun's greatest declination: 3 So is the Sine of the Sun's longitude from the next aequinoctial point, 4 to the Sine of the Sun's amplitude ortive.  
10. To find out any one of these.  
terms 1 As the Radius, 2 is to the tangent of the height of the Pole. 3 So is the tangent of the Sun's declination, 4 to the Sine of the Sun's Ascensional difference.  
11. To find out any one of these.  
terms 1 As the Radius 2 is to the Sine of the height of the Pole: 3 So is the tangent of the Sun's amplitude ortive, 4 to the tangent of the Sun's ascensional difference,  
12. To find out any one of these.  
terms 1 As the Sine of the compl. of the Sun's declination, 2 is to the Radius: 3 So is the Sine of the compl. of the Sun's amplitude ortive, 4 to the Sine of the compl. of the Sun's ascensional difference.  
13. To find out any one of these.  
terms 1 As the tangent of the height of the pole, 2 is to the Radius: 3 So is the tangent of the Sun's declination, 4 to the Sine of the Sun's horary distance from the Meridian, being due East or West.  
14. To find out any one of these.  
terms 1 As the Sine of the height of the pole 2 is to the Radius: 3 So is the Sine of the Sun's declination, 4 to the Sine of the Sun's altitude being due East, or West.  
15. To find out any one of these.  
terms 1. As the Radius, 2 is to the Sine of the height of the Pole: 3 So is the Sine of the Sun's declination, 4 to the Sine of the Sun's altitude above the Horizon at six of the Clock.  
16. To find out any one of these.  
terms 1 As the Radius, 2 is to the Sine of the compliment of the Poles height. 3 So is the tangent of the Sun's declination. 4 to the tangent of the Sun's Azumith from the North meridian, at 6 of the Clock.  
17 The hour of the Sun's Rising, and setting is found out by the ascensional difference. For if you reduce the degrees of the ascensional difference, into hours, it will show you how much the Sun riseth, or setteth before, or after 6 a Clock.  
18. The Obliqne ascension also of the Sun is found out by the ascensional difference. For if you subduct the Sun's ascensional difference, out of the right ascension of the Sun, from the beginning of Aries, for the six Northern signs which are ♈, ♉, ♊, ♋, ♌, ♍, or if you add it thereto, for the six Southern Signs, which are ♎, ♏, ♐, ♑, ♒, ♓, you shall have the Sun's obliqne ascension.  
19 The declination of the Sun, and his Altitude above the Horizon at any time, together with the height of the Pole being given, to find the hour of the day. Say,  
As the Radius, is to the Sine of the compliment of the Sun's declination.  
So is the Sine of the compl. of the Poles height, to a fourth number. Keep it,  
Then out of the Sun's distance from the North pole, subduct the compliment of the Pole; and of that remains, and the compliment of the Sun's altitude, take both the Sum, and also the difference. And say again,  
As the fourth before kept, is to the Sine of half that sum: so is the Sine of half the difference, unto a number which being multiplied by the Radius, is equal to the quadrat, of the Sine of half the Angle of the Sun's horary distance from the Meridian.  
20. The declination of the Sun, and his altitude above the Horizon at any time together with the height of the Pole being given, to find the Sun's Azumith. Say,  
As the Radius is to the Sine of the compl. of the Sun's altitude. So the Sine of the compl. of the Poles height, is to a fourth, Keep it,  
Then out of the compliment of the Sun's altitude, subduct the compliment of the Pole; and of that remains, and the distance of the Sun from the North Pole, take both the Sun, and also the difference: and say again,  
As the fourth before kept, is to the Sine of half the sum: So is the Sine of half the difference, unto a number which being multiplied by the Radius, is equal to the quadrat of the Sine of half the angle of the Sun's horizontal distance from the Meridian.  
21. To find the length of the Crepusculum, or Twilight.  
Between the light of the day, and the darkness of the night, the Twilight is set by the wise Creator; that we here upon the earth might not in an instant, pass from one extreme into another, but by successive degrees. The Twilight is nothing else but the refraction of the Sun's beams, in the density of the air. And Pet-Nonnius to find the length of the Twilight, watched the time after Sun set, when the twilight in the West was shut in, so that no more light appeared there, then in any other part of the sky near the Horizon: then by one of the known fixed Stars, having taken the true hour of the night, found by many observations, that at the time of shutting in the Twilight, the Sun was under the Horizon 18 degrees, and until the Sun was gone so low, the Twilight continued. Say therefore  
As the Radius is to the Sine of the compl. of the Sun's declination:  
So the Sine of the compl. of the height of the Pole, is to a fourth, Keep it.  
Then out of the Sun's distance from the South Pole, subduct the compliment of the Pole; and of that remains, and degrees 62, take both the Sum and also the difference; and say again,  
As the fourth kept, is to the Sine of half the sum: so is the Sine of half the difference, to a number which being multiplied by the Radius is equal to the quadrat of the Sine of half the angle of the Sun's distance at the ending of the Twilight, from the high Noon next to it.  
Wherefore if out of the whole angle converted into hours, you subduct half the diurnal arch, or the hour of the Sun's setting, you shall have the true length of the Crepusculum, or Twilight.  
22. To find the length of the least Crepusculum in the year.  
The Sun being in the winter Tropic maketh the longest Crepusculum, of the whole winter half year, and from thence, as the day's increase, the Crepuscula do decrease until they come to be shortest, which is in a certain Parallel, between that Tropic, and the Equinoctial: the declination whereof is thus found out.  
As the tangent of the compliment of the Pole, is to the Sine thereof: So is the tangent of 9 degrees, to the Sine of the declination of the Parallel, in which the Sun maketh the shortest Crepusculum of the whole year.  
23 But before the Crepusculum come to be shortest, there is another Parallel, in which the Crepusculum is equal to that in the Equinoctial: the declination whereof is thus found out.  
As the Radius is to the Sine of the altitude of the Pole: So is the Sine of 18 degrees to the Sine of declination of the Parallel in which the Sun maketh the Twilight equal to that in the Equinoctial.  
24. If an Arch of the Ecliptic, be equal to his Right ascension, one end thereof being known, to find out the other end. Say,  
As the Sine of the Compl. of the declination of the arch given. is to the Radius: so is the Sine of the compl. of the greatest declination, to the Sine of the compl. of the other and.  
25. To find the point of one quadrant of the Ecliptic, wherein the difference of longitudes cease to be greater, than the differences of the right ascensions.  
Multiply the Sine of the compliment of the greatest declination, by the Radius, and out of the product extract the quadrat root: the same shall be the Sine of the compliment of the declination sought for.  
26. To find the quantity of the angles, which the circles of the 12 Houses make with the Meridian. Say  
As the Radius is to the Tang. of 60 degr. for the 11th, 9th, 5th, and third hours, or to the Tang. of 30 deg. for the 12th, 8th, 6th, and second houses; so is the Sine of the compliment of the Pole, to the tang. of the compl. of any house with the Meridian.  
And note that on the Eastern part of the upper hemisphere, there are three circles of Houses, the Horoscope, which is also the Horizon, and next to that is the circle of the 12th House, than the circle of the 11th House. On the Western part also, are three circles of Houses, the circle of the 7th House, which also is the Horizon, and next thereto the circle of the 8th House, than the circle of the 9th House. But the circle of the 10th House, is the very upper Meridian it self. Contrary Houses are 1 and 7, 2, and 8: 3 and 9: 4 and 10: 5 and 11: 6 and 12.  
27. Resolve the whole time from the Noon last passed into degrees (by multiplying the hours with their decimal parts by 15, according to Sect: 1) which add unto the right Ascension of the Sun: and you shall have the right ascension of the point of the Aequator in the upper Meridian, which is called the Right ascension of Medium coeli.  
28. Add 99 degrees to the Right ascension of Med. Caeli: and it shall be the degree of the Aequator then rising upon the East Horizon.  
29. If the first quadrant of the Aequator do arise, the beginning of γ is distant from the meridian Eastward, so much as is the distance of the Right ascension of Med. coeli, from 360. But if the second quadrant of the Aequator do arise, the beginning of γ, is distant from the Meridian Westward, so much as is the distance of ☉, from the Right ascension of Med. coeli.  
And in both of them the lower angles of the Ecliptic with the Meridian, on the East side is obtuse, and on the West side acute: and the 90th degree of the Ecliptic, commonly called non agesimus gradus, is on the East part.  
30. If the third quadrant of the Aequator do arise, the beginning of ♎ is distant from the Meridian Eastward, so much as is the distance of the Right ascension of Med. coeli from 180. But if the fourth quadrant of the Aequator do arise, the beginning of ♎, is distant from the Meridian Westward, so much as is the distance of 180, from the Right ascension of Med. coeli.  
And in both of them the lower angle of the Ecliptic with the Meridian, on the East side is acute, and on the West side obtuse: and on the 90th degree is on the West part.  
31. The point of the Ecliptic culminant in the Meridian, which is called Medium coeli, or Cor coeli, and is the cuspis of the 10th house, may be found by Sect. 5.  
32. The declination of the said culminant point, may be found by Sect. 3. Wherhfore also by adding or subducting that declination, to, or from the elevation of the Aequator, (which is the compliment of the Pole) the Altitude of Med. coeli may be had.  
33. The Angle of the Ecliptic with the Maridian, may be found by Sect. 7.  
34. To find the Altitude of the 90 degr. Or the Angle of the Ecliptic with the horizon.  
As the Radius is to the Sine of the compl. of the altitude of Med: coeli. So is the Sine of the angle of the Ecliptic with the Meridian, to the Sine of the compl. of the angle sought for.  
35. To find the Azumith of 90 degr. which is also the Amplitude ortive of the Ascendent, or Horoscopus.  
As the Radius, is to the Sine of the Altitude of Med. coeli. So is the tang. of the Angle of the Ecliptic with the Meridian, to the tang. of the compl. of the distance of that Azumith from the Meridian.  
36. To find the Horoscopus, or Ascendent degree of the Ecliptic, Or the Cuspis of the first house.  
The Distance of the Azumith of 90 degrees from the Meridian, is equal to the Amplitude ortive of the Ascendent degree. Wherefore the Ascendent degree or the Ecliptic, may thence be found, by Sect: 8, or 9 Or else thus.  
As the Radius is to the Sine of the compliment of the angle of the Ecliptic with the Meridian: So is the tang. of the compliment of the altitude of Med. Coeli, to the tangent of the distance of Med. coeli from the Ascendent degree.  
37. To find the parts of the angle of the Ecliptic with the Meridian, cut with an arch perpendicular to the Circle of any of the Houses. Say  
As the Radius is to the Sine of the compl. of the altitude of Med. coeli: so is the tangent of the circle of any House with the Meridian, to the tang: of the compl: of the part of that angle, which is next the Meridian  
Then subduct that part found out of the whole Angle for the remaining or latter part.  
38. To find the Distance of the cuspis of any house, from Med: coeli. Say  
As the Sine of the compl. of the later part of the angle of the Ecliptic with the Meridian, is to the Sine of the compl: of the former part of that angle: So is the tang▪ of the altitude of Med: coeli, to the tang: of the distance of the cuspis of that House sought for.  
39 To find the Altitude of the Pole above any of the circles of the Houses.  
First find out the Angle which the circle of the House proposed maketh with the Meridian, by Sect: 23: And then say.  
As the Radius is to the Sine of the angle of the circle of the House with the Meridian: So is the Sine of the height of the Pole above the Horizon of the place, to the Sine of the height of the Pole above that circle of position.  
40. The longitude, and latitude of any fixed Star being given, to find out the Right ascension, and Declination thereof.  
The angle which the Circle of the Sun's longitude maketh with the Meridian, at the Pole of the Ecliptic, I call the Angle of longitude.  
And the angle which the Circle of the Sun's Right ascension, maketh with the Meridian at the Pole of the world, I call the Angle of right Ascension. The condition and quantity of which two angles, is thus found out.  

In Stars of the Northern latitude  
If the longitude be in the I quadrant of the Ecliptic: subduct it out of 90: the remains will be the angle of longitude, acute. And the Angle of Right ascension, being found, must be added unto 270.  
If the longitude be in the TWO quadrant: subduct 90 out of it: the remains will be the angle of longitude, acute. And the Angle of right ascension being found must be taken out of 270.  
If the longitude be in the III quadrant: subduct 90 out of it, the remains will be the angle of longitude, obtuse. And the Angle of right ascension being found, must be taken out of 270.  
If the longitude be in the IIII quadrant, subduct it out of 90+360: the remains will be the angle of longitude, obtuse. And the Angle of right ascension being found, must be added unto 270.   

In Stars of the Southern latitude  
If the longitude be in the I quadrant, subduct 270 out of it + 360: the remains will be the angle of longitude, obtuse. And the Angle of right ascension being found, must be taken out of 90.  
If the longitude be in the TWO Quadrant, subduct it out of 270: the remains will be the angle of longitude, obtuse. And the Angle of right ascension being found, must be added to 90.  
If the longitude be in the III quadrant: subduct it out of 270; the remains will be the angle of longitude, acute. And the Angle of right ascension being found, must be added to 90.  
If the longitude be in the IIII quadrant: Subduct 270 out of it: the remains will be the angle of longitude, acute. And the Angle of right ascension being found, must be subducted out of 90× 360.  
Then say,  
As the Radius, or total Sine, is to the Sine of the compliment, or excess of the angle of longitude:  
So is the tang. of the compl. of the latitude, to the tang. of the first base.  
If the angle of longitude be obtuse; unto the first base found, add the greatest declination deg. 23½: and the sum shall be the second base; and the angle of right ascension shall be acute.  
But if the angle of longitude be acute; out of the first base subduct the greatest declination: and the remains shall be the second base. And the angle of right ascension shall be obtuse.  
Or else out of the greatest declination of the Sun, subduct the first base; and the remains shall be the second base: and the angle of right ascension shall be acute.  
Say again,  
As the Sine of the second base, is to the Sine of the first base:  
So is the tang. of the angle of longitude, to the tang. of the angle of right ascension.  
Whence by adding or subducting as was before delivered in the conditions of those angles, shall be given the Right ascension of that Star sought for.  
Lastly say  
As the tang. of the second base, is to the Radius:  
So is the Sine of the compl. or excess of the angle of right ascension to the tang. of Declination.  
Where note that if the second base exceed 90 degr: the declination found, shall not be of the same kind, that the latitude is, but in the contrary Hemisphere.  

A Table of the Right Ascensions, and Declinations, of 40 of the chiefest fixed Stars, Calculated for the year of our Lord. 1650.  Names of Stars Right Ascension Declination mag. The Polar Star 7° 4′7 80° 2′7 N 2 Andromedaes' girdle 12 32 33 48 N 2 the former horn of ♈ 23 38 17 37 N 4 the bright star in the head of ♈ 26 56 21 48 N 3 the jaw of the Whale 41 03 2 42 N 2 Medusa's head 41 27 39 35 N 3 the eye of the Bull 64 00 15 46 N 1 the Goat star 72 44 45 35 N 1 the former shoulder of Orion 76 38 4 59 N 2 the latter shoulder of Orion 84 07 7 18 N 2 the great dog star 97 27 16 13 S 1 the higher head of ♊ 108 01 32 35 N 2 the lesser Dog star 110 17 6 06 N 2 the lower head of ♊ 111 00 28 49 N 2 the Cribb, or Manger 125 4 20 52 N Neb. the heart of Hydra 137 39 7 10 S 2 the heart of the Lion 147 27 13 39 N 1 In the loins of the Lion 163 54 22 26 N 2 In the tail of the Lion 172 49 16 32 N 1 In the girdle of Virgo 189 32 5 20 N 3 Names of Stars R. Ascen. Declination mag. Aliot 189° 36 57 53 N 2 Vindemiatrix 191 15 15 51 N 3 Spica Virgins 196 44 9 17 S 1 Arcturus 209 56 21 34 N 1 the Southern balance 217 56 14 32 S 2 the Northern balance 224 31 8 2 S 2 the bright star in the serpent's neck 231 49 7 35 N 3 the heart of Scorpius 242 4 25 34 S 1 the head of Hercules 254 40 14 51 N 3 the head of Ophiucus 259 41 12 52 N 3 bright star in the Harp 276 17 38 30 N 1 bright star in the Vulture 293 27 8 1 N 2 upper horn of ♑ 299 30 13 32 S 3 the left hand of ♒ 307 10 10 43 S 4 the left shoulder of ♒ 318 18 7 2 S 3 the mouth of Pegasus 321 49 8 18 N 3 the right shoulder of ♒ 326 59 1 58 S 3 Fomahant 339 29 31 23 S 1 In the upper wing of Pegasus 341 53 13 21 N 2 In the tip of the wing of Pegasus 358 52 13 15 N 2  
41. The longitude, and latitude of any two Stars being given, to find their distances.  
If the Stars have both the same longitude, differing only in latitude; the difference of latitude, is the distance of the stars.  
And if they differ only in longitude having the same latitude; Say  
As the Radius, is to the Sine of half the difference of longitude:  
So is the Sine of the compl. of the latitude given, to the Sine of half the distance of the Stars.  
But if they differ both in longitude, and latitude, whether the latitudes be both of the same kind, or one Northern, and the other Southern. Take the difference of both the stars, from the pole of the Ecliptic, toward which the star having the greater latitude is. And say,  
As the tang. of the compl. of the lesser distance from the pole, is to the Radius:  
So is the Sine of the compl. or excess of the difference of longitudes, to the tang. of the first base.  
Take this first base out of the greater distance from the pole, and the remains shall be the second base, Then say,  
As the Sine of the compl. of the first base, is to the Sine of the compl. of the second base:  
So is the Sine of the compl. of the lesser distance from the pole, to the Sine of the compl. of the distance of the two Stars.  
If any man will take pains to calculate (by this last Rule) the distances of some noted stars of the first, second, and third magnitudes, round about the heavens, which are not above 5, or 6 degrees, at the most, one from the other: and shall keep them written in his book: they may serve as a Rule, or Instrument, whereby he may reasonably estimate with his eye, the distance of any Planet, or Comet, or other apparition from a known fixed star, not very fare remote: by comparing the distance which he would know with some of those known distances which he shall find, either to be equal, or else to have some proportion thereto.  
42. The longitude, and latitude of any two Cities being given, to find their distance.  
The manner of the operation is the very same with the former, unto which therefore I refer the Reader: only will note, that in the heavens, the longitude and latitude is taken in respect of the Ecliptic, which being the way of the Sun, all the stars in their proper motion, have reference unto it, as unto their measure and rule. But in the Earth the principal Circle is the Equinoctial, dividing it into the Northern, and Southern he misphaeres. And therefore in the earth, the longitude, and latitude is reckoned by the Equinoctial.  
The distance of two places upon the Earth, being found in degrees, may be converted into English miles, by taking 60 miles for every degree, and one mile for every minute.  
43. To find at what hour a fixed star cometh into the Meridian any day.  
Seek the Right ascension of the Sun, for that day, by Sect 〈◊〉; and subduct it out of the Right ascension of the Star. And reduct the degrees remaining into hours, by Sect: 1. The same shall show how long time from the Noon before, the same star shall come into the Meridian.  
Wherefore if at any time of the night, a Star whose Right ascension is known, be in the Meridian, the hour of the night is easily found.  
44. The height of any known Star above the Horizon, being by any means given, to find the hour of the night.  
First seek out the hour of that stars coming into the Meridian the same day, by Sect. 43. Again seek out the horary distance of that star from the Meridian, according to Sect: 19 And then if the star be on the East side, not yet come to the Meridian, take the difference of those two numbers; but if the star be past the Meridian, take the Sum of them, for the hour of the night.  
45. The height of the Pole being given to find the coming of any fixed Star, in the due East, or West. Say  
As the Radius is to the tang: of the stars declination: So is the tang: of the compl: of the Pole, to the Sine of the compl. of the Stars horary distance from the Meridian.  
46. The height of the Pole being given, to find the Altitude of any fixed star above the Horizon, being due East or West. Say,  
As the Sine of the height of the Pole, is to the Radius: so is the Sine of the Stars declination, to the Sine of the Altitude, at due East or West.  
47. By the Altitudes of any two known fixed stars taken when they are both in the same Azumith, to find the height of the Pole.  
First say,  
As the Sine of the difference of the stars altitudes, is to the Sine of the difference of their Right ascensions: so is the sine of the nearer stars distance from the apparent Pole, to the Sine of an angle to be kept.  
Again compare the furthest stars distance from the Pole with the distance from the Zenith, and say  
As the Radius is to the Sine of the compl. of the Angle kept: so is the tang: of the lesser of the compared arches, to the tang: of the first base.  
Subduct the first base out of the greater of the two compared arches; and the remains shall be the second base.  
Then lastly say,  
As the Sine of the compliment of the first base, is to the Sine of the compl. of the second base: so is the Sine of the compl. of the lesser of the two compared arches, to the Sine of the height of the Pole.  
48. To find out the horizontal Parallax of the Moon.  
First the distance of the Moon from the Centre of the earth must be known in Semidiameters of the earth: which unto them that are acquainted with the Theory of the Planets, is not very difficult. And whereof peradventure, I may hereafter teach the practice, by most easy and exact instruments, which I have long since framed.  
Say,  
As the distance of the Moon, from the centre of the earth, is to the Semidiameter of the earth: So is the Radius, to the Sine of the Moons horizontal parallax in that distance.  
49. The horizontal Parallax of the Moon being given, to find her Parallax in any apparent altitude.  
As the Radius, is to the Sine of the altitude of the Moon: so is the Sine of the horizontal Parallax, to the Sine of the Parallax in that altitude.  
50. The place of the Moon in the Ecliptic having little or no latitude (as in the Eclipse of the Sun) together with her Parallax of altitude being given, to find the Parallaxes of her longitude, and latitude.  
If the Moon be in the 90th degree of the Ecliptic: she hath no Parallax of longitude, and the Parallax of latitude, is the very Parallax in that altitude.  
But if the Moon be not in the 90th deg. say,  
As the Radius is to the tang. of the angle of the Ecliptic with the horizon: So is the Sine of the compl. of the distance of the Moon, from the Ascendent, or descendent degree of the Ecliptic, to the tang. of the compl. of the angle of the Ecliptic with the Azumith of the Moon.  
Again say,  
As the Radius is to the Sine of that angle: So is the Parallax of the Moon's altitude, to the Parallax of her latitude.  
Lastly say,  
As the Radius is to the Sine of the compl. of that former angle: So is the Parallax of the Moon's altitude, to the Parallax of her longitude. which is to be added to the true motion of the Moon, if she be on the East part of the 90th degree of the Ecliptic: or to be subducted out of it, if she be on the West part.  
Many other Astronomical and Geographical problems might be added. But because it is impossible to set down all, which may be of use, at some time or other: I have in the next Chapter delivered briefly the doctrine of triangles fitted unto practice: with all the several cases belonging thereto.    

CHAP. XIII. Of Trigonometria, or the manner of calculating both Plain, and Spherical triangles. And first concerning certain general notions, and rules necessary thereto.  
IN every triangle both Plain, and Spherical, the greater side subtendeth the greater angle. And the greater angle hath the greater side opposite unto it. Also the greater angle lieth to the lesser side, and the greater side hath the lesser angle lying unto it.  
In every plain triangle, any two angles being given, the third is also given: and one of the angles being given, the sum of both the other two is given. For all the three angles together, are equal to two right angles, that is to 180 degrees.  
In a plain rectangle triangle, one of the acure angles is the compliment of the other. Where note that when the compliment is named without any other addition, it is meant of the arch, which is wanting of a quadrant of that circle, or 90 degrees. In like manner the excess is meant of the arch, which is above a quadrant. But when it is said the compliment to a semicircle, it is understood of so many degrees as will make up 180.  
But in a Spherical rectangle triangle, one of the obliqne angles is always greater than the compl: of the other.  
If two arches together make up a Semicircle, the excess of the greater arch, is equal to the compliment of the lesser.  
The same Right Sine, and the same Tangent, and Secant, doth belong both to the arch itself, and also to the compliment of it to a Semicircle. But their versed Sins differ: For the versed Sine of an arch less than a quadrant, is equal to the difference of the Radius, and the Sine of the compliment of that arch: and the versed Sine of an arch greater than a quadrant, is equal to the sum of them. And the versed Sine is thus found out, As the Radius is to the Sine of half the arch; so is the Sine of that half arch, to half the versed Sine of the whole arch.  
In a right angled triangle both Plain, and Spherical, one of the sides containing the right angles, is called the Base, and the other the Cathetus: and the fide subtending the right angle is the Hypotenusa. And know that every rectangled triangle, is most fitly noted with the letters ABC; so that BASILIUS may be the Base, and CA the Cathetus, and BC the Hypotenusa: and B the angle at the base, and C the angle at the Cathetus, and A the right angle. Likewise every oblique-angled triangle with the letters BCD; so that out of the angle C a perpendicular CA, being let down, it may in the base BD distinguish the two cases BASILIUS and DA, which are the bases of the two particular triangles into which it is cut. And in noting the triangles with letters, observe diligently, that if any angle be given together with one of the sides including it, the same angle be noted with B; and the side with BC.  
If both the angles at the Base BD be acute, the perpendicular CA shall fall within the triangle: And BD = BA + DA that is BD is equal to the sum of BASILIUS and DA. And if the angle B be obtuse, the perpendicular CA shall fall without the triangle, beyond the obtuse angle B: And BD = DA − BASILIUS, that is BD is equal to the excess of DA above BASILIUS or if the angle B. be obtuse, the perpendicular CA: shall fall without the triangle beyond the obtuse angle D: And BD = BA − DA, that is BD is equal to the excess of BASILIUS above DA. The lesser case being still taken from the greater angle.  
And note that this sign +, or pl (that is plus) showeth that the magnitude before which it is set, is affirmed and positive in nature; and therefore to be added. And that this sign— or mi (that is minus) showeth the magnitude before which it is placed, is denied, and privative in nature; and therefore to be substracted, as you may see in those former examples.  
Again, some magnitudes are taken severally and apart; as s BA, that is the Sine of the Base; s co BC, that is the Sine of the compliment of the Hypotenusa; t B, that is the tangent of the angle ABC at the base; t co C, or t co ABC, the tangent of the compliment of the angle at the Cathetus: So also VqZ, that is the quadrat side of the plain Z. And some magnitudes are taken universally, and then they are included in pricks: as : that is the Sine of half the arch, which is composed of the sum of the two arches DC, and BD, abating thereout the arch BC. So also Vq ∶ Z × X ∶ that is the quadrat side of the two plains Z and X put together: also Vq ∶ Q in R ∶ Or Vq ∶ Q × R ∶ that is the quadrat side of a rectangular plain, the two sides whereof are the lines Q and R, or some fourth proportional already found, and the Radius, or Semidiameter, which is the total Sine. For by the sign in, or ×, I use to express multiplication.  
When any triangle is given to be resolved by trigonometry, note the parts thereof (either-sides or angles) which are given and known, with a little line drawn cross each such part: and note the unknown part which is sought for with a little circle.  
And if a triangle Spherical (be it right angled or oblique-angled) proposed hath two sides each of them severally greater than a quadrant: you shall in resolving thereof, keep the least side with the least angle opposed to it: and for the two other both sides, and angles, take the compliments of them to a semicircle.  
Lastly, if a triangle with all the three angles given, be required to be converted into a triangle having the three sides given. You shall for the greatest angle of the triangle proposed, & for the greatest side subtending it, take the compliments to a semicircle; keeping the other two lesser angles, with their subtendent sides as they are.  

THE CALCULATION OF PLAIN right-angled-triangles.  
I.   diagram 
BC · BA ∷ R · 〈…〉 B (●C) ·  
II.   diagram 
BA · CA ∷ R · ●B (●●● C) ∶ Of plain Oblique-angled triangles.  
III.   diagram 
●B · DC ∷ ●D · BC ∷ ●C · BD · and here it is necessary to be known, whether the angle sought for be greater, or less than a right angle, or 90 deg.  
FOUR   diagram 
First seek the angle D, by the III; then both the angles Band D being subducted out of 180, you shall have 180 − B − D = C ·  
V   diagram 
First seek the angle D, by the III; then both the angles B and D being subducted out of 180, Say ●B · DC ∷ s ∶ 180 − B − D · BD ·  
VI   diagram 
Let the side BD be greater than the side BC: First, then for the other two angles: — the greater — the lesser.  
VII.   diagram 
Let the side BD be greater than the side BC:  
First, the angles C and D are to be sought, by the VI and the side DC, by the III.  
VIII.   diagram 
Take the greatest side BD for the base: and let the side BC, be greater than the side DC. First say, BD · BC + DC ∷ BC − DC · Q (viz BD − 2DA) · then  
Nextly seek the angles B and D, by the III. Lastly 180 − B − D = C.   

THE CALCULATION OF Spherical rightangled, and quadrantall triangles.  
I.   diagram 
R · ●B ∷ sBC · sCA ·  
II.   diagram 
R · sB ∷ ●●●BA ∶ s ●●CA ·  
III.   diagram 
R · ●●●BA ∷ ●●●CA ∶ ●●●BC ∶  
FOUR   diagram 
R · sBA ∷ tB · tBA ·  
R · t coB ∷ tCA · sBA ·  
R · sBA ∷ t coCA · t coB ·  
V   diagram 
R · s coB ∷ tBC · tBA ·  
R · t coBC ∷ tBA · ● coB ·  
R · ● coB ∷ t coBA · t coBC ·  
VI   diagram 
R · ● coBC ∷ ●B · ●●●C ·  
VII.   diagram 
If a triangle BCD be quadrantall, having one side BC equal to a quadrant; upon the pole D describe an arch of a great circle CA, cutting the side DB extended in A: and so making a rightangled triangle ABC without the other. This outward rightangled triangle shall be resolved in steed of the quadrantall proposed.   

Of Spherical Oblique-angled triangles.  
VIII.   diagram 
sB ∶ ●DC ∷ ●D · sBC ∷ ●C · sBD · and in these it is necessary to be known whether the term sought for be greater than a quadrant, or not. The same also is to be known in the ten rules next following, if the sides BC and DC are both given.  
IX.   diagram 
First, R · s coB ∷ tBC · ●BA · then, s coBA · s coDA ∷ ● coBC · s coDC ·  
X.   diagram 
First, R · s coB ∷ tBC · tBA · then ●DA · sBA ∷ ●B tD ·  
XI.   diagram 
First, R · s coB ∷ ABC · ●BA then, s coBC · ●●● DC ∷ ●●●BA · s ●●DA ·  
XII.   diagram 
First, R · s coB ∷ tBC · tBA · then, tD · tB ∷ ●BA · sDA ·  
XIII.   diagram 
First, R · ● coBC ∷ ●B · t coBCA · then, s coDCA · s coBCA ∷ tBC · tDC ·  
XIIII.   diagram 
First, R · s coBC ∷ tB · t coBCA · then, sBCA · sDCA ∷ s coB · s coD ·  
XV.   diagram 
First, R · s coBC ∷ tB · t coBCA · then, tDC · tBC ∷ s coBCA · s coDCA ·  
XVI.   diagram 
First, R · s coBC ∷ tB · t coBCA · then, s coB · s coD ∷ ●BCA · sDCA ·  
XVII.   diagram 
First, R · sBD ∷ sBC · QI · then,  
See what QII cutteth in the fift circle, which is of equal divisions: and thereto add the Radius, by setting I before that number. Divide the whole into two equal parts: and reckoning one half in that fift circle, set the Index to it, and it shall in the first circle cut the Sine of half the angle B.  
XVIII.   diagram 
If all the three angles be given: convent the triangle into another having all the three sides given: and resolve the same for the triangle proposed.    

CHAP. XIIII. Of the Nocturnal Dial's.  
THere are in the Instrument, two several Nocturnal Dial's. The innermost of them is fitted to the star in the rump of the great Bear, commonly called Aliot. The other is composed of 12 several stars: whose names you shall find written within, near to the centre.  
The outermost circle of the Nocturnal Dial is divided into twice 12 hours: each hour being subdivided into quarters, and are noted with figures belonging to the hours, as may be seen in the Instrument.  
The middlemost circle of the Nocturnal is divided into 12 months, having their names written: each month being distinguished into tenth days with longer lines; and into fift days with shorter lines. And if the Instrument be large enough, each day of the months throughout the year, may be noted with pricks.  
In the innermost circle are the divisions and names of 12 fixed stars: which are these.  

The bright star in the head of ♈.  
the Bull's eye  
the latter shoulder of Orion  
the little dog  
the heart of the Lion  
the tail of the Lion  
Spica Virgins  
the North balance  
the head of Ophiuchus  
the heart of the Vulture  
the mouth of Pegasus  
the tip of the wing of Pegasus.   

To find out the hour of the night by Aliot.  
Seek the day of the month in the annual circle of the Nocturnal: and apply the Index thereto: mark what hour it cutteth, in the hour circle. Remember this hour for all that day: then at night when you would find out the hour, hold up your Instrument by the handle, and move it up and down, till you see the pole star through the middle hole, and the star called Aliot by the limb: Set the Index or label to Aliot, and mark what hour the label cutteth, for if unto this hour you add the hour kept in mind for that day, the sum of both shall give you the true hour of the night: so that you cast out 12 hours, from the said sum if it shall chance to be more.  
Example. If on the 15th of November you would find the hour of the night by the star Aliot. Apply the Index to the day of the month, and it will cut in the hour circled 8 and an half: then suppose the Index being set to the star Aliot, as above taught, doth cut in the hour circled 10. these two numbers being added together, the sum will be 18½, out of which subduct 12; and the remains 6½, will be the true hour of the night.   

To find out the hour of the night by the Inner Nocturnal Dial.  
To perform this it is necessary that you know the true Meridian of the place wherein you are, and can find it out by night, which you may thus do. Having a Meridian line drawn in some window, or other convenient place (as is showed in the Second part of this book, Use 19) stick up therein a long needle perpendicularly, and watch till the Sun casteth the shadow of the needle, upon the Meridian line. Or else in a true Sun Dial observe when the shadow falls just on 12 a clock, for than is the Sun in the Meridian. Wherefore go instantly into some place about your house where you may see some mark, either a chimney, or the corner of an house, or else some tree, or such like, directly between you and the Sun: then have you the true Meridian.  
Or otherwise you may in a clear night go into some plain place near your house, and setting up a strait pole perpendicularly on the ground, go a good distance from it Southwards; and then move up and down, till you see the top of your pole, directly between your eye, and the North polar star: then set up another pole perpendicularly between your feet, so that both your poles, and the Polar star, may be in one right line. And then going back again to your first pole, look what known star is directly over your last pole, for that star is in the Meridian. You may therefore instantly go to some convenient place, and take a mark whereby you may at all times know the Meridian as is afore taught.  
When therefore at any time of the night you would know what a clock it is, go to that place where you stood, and looking directly over your mark, see if any of the 12 fixed stars, be in the Meridian; or if none of them be therein, observe which two of them are on either side thereof, and what part of that space is in the Meridian. Then go into the light, and take your instrument, and set the Index to that star, or point which you saw in the Meridian: mark what hour it cutteth, for that same hour being added to the hour, which the day of the month showeth, shall give you the true hour of the night: so that you cast out 12 hours, from the said sum, if it shall chance to be more.  
Example. Suppose the fifth of December, that the middle point of the space between the bright star in the head of Aries, and the Bull's eye, be in the Meridian. Set the Index to the middle point of the space between those two stars in the Instrument: and it will cut in the hour circled 2 and an half: then again set the Index to the fifth of December, and in the hour circled it will cut 7: which added unto 2 and an half, giveth 9 and an half, for the true hour of the night.  
Another example. Suppose the 19th of December, that one third part of the space between the Bull's eye, and the right shoulder of Orion, be in the Meridian. Set the Index to one third part of that space in the Instrument, and it will cut in the hour circled 4 and half a quarter almost: again, set the Index to the 19th of December, and in the hour circled it will cut 6, which being added unto 4, and half a quarter almost, giveth 10 and almost half a quarter for the hour of the night.   diagram    

THE SECOND PART OF THIS BOOK.  
Showing the use of the Second side of the Instrument, for the working of most questions, which may be performed by the Globe: And the declination of Dial's, upon any kind of Plain.  
Upon the second side of the Instrument, is delineated the projection of the upper Hemisphere upon the plain of the Horizon: The Horizon itself is understood to be the innermost circle of the limb: and is divided on both sides, from the points of East, and West into degrees, noted with 10, 20, 30, etc. unto 90. And the centre of the Instrument is the Zenith, or Vertical point.  
Within the Horizon, the middle strait line, or Diameter pointing North and South, is the Meridian, or 12 a clock line: and the other shortarching lines, on both sides of it are the hour lines, distinguished accordingly by their figures. These hour lines should indeed be drawn through the whole plain, crossing one another in the Pole of the world: but that the Instrument may be more fair, they are only drawn short.  
And because diverse excellent uses, do require the total delineation of the hour circles, I have in a several paper, inscribed entirely, both the hour lines, and also two other circles between them, containing every one five degrees. (But if the Instrument were large enough to receive them, it were best if every degree had his circle: and so every 15 circle should be an hour line.) And of the parallels, there needs no more, but the Aequinoctial, and both the Tropics.  
For as much as there will be great use of this paper Instrument; I have in the 24 Use showed the manner of making it: so that any that 1● ingenuous, and ready handed may himself delineate one sufficient enough to serve his turn, for any elevation.  
The two arches which cross the hour lines meeting on both sides in the points of intersection of the six a clock lines with the Horizon, are the two Semicircles of the Ecliptic, or Annual circle of the Sun: the upper of which arches serveth for the Summer half year, and the lower for the Winter half year: and are therefore divided in 365 days: which are also distinguished into 12 months with longer lines, having their names set down: and into tenths, and fifthes with shorter lines: and the rest of the days with pricks: as may plainly be seen in the Instrument.  
And this is for the ready finding out of the place of the Sun every day: and also for showing of the Sun's yearly motion: because by this motion the Sun goeth round about the heavens in the compass of a year, making the four parts, or seasons thereof. Namely the Spring, in that quarter of the Ecliptic which beginneth at the intersection on the West side of the Instrument, and is therefore called the Vernal intersection. Then the Summer in that quarter of the Ecliptic which beginneth with the intersection of the Meridian in the highest point next the Zenith. And after that Autumn in that quarter of the Ecliptic, which beginneth at the intersection on the East side of the Instrument, and is therefore called the Autumnal intersection. And lastly, the Winter in that quarter of the Ecliptic, which beginneth at the intersection with the Meridian, in the lowest point next the Horizon.  
But besides this yearly motion, the Sun hath a Diurnal or daily motion, whereby it maketh day and night with all the diversities, and inequalities thereof: which is expressed by those other circle's drawn cross the hour lines: the middlemost whereof being grosser than the rest, meeting with the Ecliptic in the points of the Vernal, and Autumnal intersections, is the Equinoctial: and the rest on both sides of it, are called the Parallels, or Diurnal arches of the Sun: the two outermost whereof are the Tropics, because in them the Sun hath his furthest digression, or Declination from the Equinoctial, which is degrees 23½: and thence beginneth again to return to the Equinoctial. The upper of the two Tropics next the centre (in this our Northern Hemisphere) is the Tropic of Cancer: and the Sun being in it, is highest into the North, making the longest day of Summer. And the lower next the Horizon, is called the Tropic of Capricorn; and the Sun being in it, is lowest into the South, making the shortest day of Winter.  
Between the two Tropics, and the Equinoctial infinite such parallel circles are understood to be contained: for the Sun is what point soever of the Ecliptic it is carried, describeth by his lation, a circle parallel to the Aequinoctial. Yet those parallels which are in the Instrument, though drawn but to every second degree of Declination, may be sufficient to direct the eye, in imagining and tracing out, through every day of the whole year in the Ecliptic, a proper circle, which may be the Diurnal arch of the Sun for that day. For upon the right estimation of that imaginary parallel, doth the manifold use of this Instrument especially rely: because the true place of the Sun, all that day, is in some part, or point of that circle. Wherefore for the better conceiving, and bearing in mind thereof, every fift parallel, is herein made a little grosser than the rest.  
I Use. And thus by the eye, and view only, to behold and comprehend the course of the Sun, both for his Annual and Diurnal motion, may be the first use of this Instrument.  
TWO Use. To take the height of the Sun above the Horizon.  
Set up the pin, (which is therefore made fit for the hole at the centre) perpendicular in the centre: and put the Indices on both sides, down upon the Meridian, that they with their weight, may not sway the Instrument any way as it hangeth: then with a thread put into the hole above in the handle, hang it perpendicularly, bearing the edge toward the Sun, that the pin may cast a shadow, upon the degrees in the limb: for that degree which the shadow of the pin cutteth in the limb, is the height of the Sun above the Horizon, at that present.  
III. Use. To find the Declination of the Sun every day.  
Look the day of the month proposed in the Ecliptic, and mark how many degrees the prick showing that day, is distant from the Equinoctial, either on the Summer, or Winter side, viz North, or South.  
Example. I. What will the Declination of the Sun be, upon the 11th day of August? Look the 11th day of August: and you shall find it in the sixth Circle above the Equinoctial: now because each Parallel, standeth (as hath been said before) for 2 degrees, the Sun shall that day decline North-wards 12 degrees.  
Example. II. What Declination hath the Sun, upon the 24th day of March? Look the 24th day of March, and you shall find it, between the second, and third Northern parallels, as it were an half and one fift part more of that distance from the second: reckon therefore 4 degrees for the two Circles, and one degree for the half space: so shall the Sun's declination be 5 degrees, and about one fift part of a degree Northward, that same day.  
Example. III. What Declination hath the Sun upon the 13th day of November? Look the 13th day of November, and you shall find it below the Equinoctial, ten parallels and about one quarter, which is 20 degrees, and an half Southwards. So much is the Declination. And according to these examples judge of all the rest.  
FOUR Use. To find the Right ascension of the Sun every day.  
Imagine an hour line through the day of the month given, and mark in what point it will cr●sse the Equinoctial: then lay a Ruler, or a straight Scroll of paper, to the Pole of the world (noted in the Instrument with P W) and that same point. For the Ruler shall in the innermost Circle of the limb, of the South side, cut the Right ascension of the Sun for that day, to be reckoned from the West, to the point of intersection, for the first, or upper Semicircle of the Ecliptic: or from the East together with 180, for the second, or lower Semicircle of the Ecliptic.  
V Use. To find the longitude of the Sun, or in what degree of the Sign he is every day.  
The Pole of the first Semicircle of the Ecliptic is noted P I. and the Pole of the second Semicircle is noted P I I. Lay a Ruler, or a straight Scroll of paper, to the day of the month, and the proper Pole of the Semicircle of the Ecliptic, in which it is: for the Ruler shall in the innermost Circle of the limb, on the South side, cut the degree of the Sun's place in the Ecliptic, reckoning it in the same manner as you did in finding the Sun's Right ascension: and the Arch thus found is called the longitude of the Sun. which may be expansed into signs, by reckoning on the limb, from the West to South ♈, ♉, ♊, and from South to East ♋, ♌, ♍: then back again from East to South ♒, ♏, ♐; and lastly from South to West ♑ ♒, ♓, allowing 30 degrees, for each of those twelve signs.  
VI Use. To find the Diurnal Arch, or Circle of the Sun's course every day.  
The Sun every day by his motion (as hath been said) describeth a Circle parallel to the Equinoctial, which is either one of the Circles in the Instrument, or somewhere between two of them. First then seek 〈◊〉 of the month; and if it fall upon one of those 〈◊〉, that is the Circle of the Sun's course that same day: But if it fall between any two of those Parallels, imagine in your mind, and estimate with your eye, another Parallel through that point, between those two Parallels, keeping still the same distance from each of them.  
As in the first of the three former Examples, The circle of the Sun's course, upon the 11th day of August, shall be the very sixth Parallel above the Equinoctial towards the Centre.  
In Example II. The Circle of the Sun's course upon the 24th day of March, shall be an imaginary Circle between the second, and third Parallels, still keeping an half of that space, and one fift part more of the rest from the second.  
In Example III. The Circle of the Sun's course upon the 13th day of November, shall be an imaginary Circle, between the tenth, and eleventh Parallels, below the Equinoctial, still keeping one quarter of that space from the tenth.  
VII. Use. To find the Rising and Setting of the Sun every day.  
Seek out (as was last showed) the imaginary Circle or Parallel of the Sun's course, for that day, and mark the point where it meeteth with the Horizon, both on the East and West sides thereof, for that is the very point of the Sun's rising, and setting that same day, and the hour lines which are on both sides of it, by proportioning the distance reasonably, according to 15 minutes, for the quarter of the hour, will show the hour of the Sun's rising, on the East side, and the Sun's setting on the West side.  
VIII. Use. To know the reason, and manner, of the Increasing, and Decreasing of the days and nights throughout the whole year.  
When the Sun is in the Equinoctial, it riseth, and setteth at 6 a Clock, for in the instrument, the intersection of the Equinoctial, and the Ecliptic with the Horizon, is in the 6 a clock Circle on both sides. But if the Sun be out of the Equinoctial, declining toward the North, the intersections of the Parallel of the Sun with the Horizon, is before 6 in the Morning, and after 6 in the Evening: and the diurnal Arch of the Sun, greater than 12 hours; and so much more great, the greater the Northern Declination is. Again if the Sun be declining toward the South, the intersections of the Parallel of the Sun with the Horizon, is after 6 in the Morning, and before 6 in the Evening; and the diurnal Arch lesser than 12 hours; and by so much lesser, the greater the Southern Declination is.  
And in those places of the Ecliptic in which the Sun most speedily changeth his Declination, the length also of the day is most altered, and where the Ecliptic goeth most parallel to the Equinoctial changing the Declination but little, the length of the day also is but little altered.  
As for example, when the Sun is near unto the Equinoctial, on both sides, the day's increase, and also decrease suddenly and apace: because in those places, the Ecliptic inclineth to the Equinoctial in a manner like a straight line, making sensible declination. Again when the Sun is near his greatest Declination, as in the height of the Summer, and the depth of Winter, the days keep for a good time, as it were at one stay, because in those places the Ecliptic is in a manner parallel to the Equinoctial, scarce altering the declination: and because in those two times of the year, the Sun standeth as it were still, at one declination; they are called the Summer Solstice, and the Winter Solstice. And in the mean spaces, the nearer every place is to the Equinoctial, the greater is the diversity of days.  
Wherefore we may hereby plainly see, that the common received opinion, that in every month, the days do equally increase, is erroneous.  
Also we may see that in Parallels equally distant from the Equinoctial, the day on the one side, is equal to the night on the other side.  
IX. Use. To find the ascensional difference of the Sun every day.  
Seek out the time of the Suns Rising, or Setting that same day (by the VII Use) and see how much it differeth from six a clock, then convert the same difference into degrees (as was taught in 1 Part. Chap. 12. Sect. 1.) by multiplying the hours with their decimal parts, by 15. And so have you the ascensional difference for that day.  
X. Use. To find out the Obliqne ascension of the Sun every day.  
Seek out the Sun's Right ascension (by the IIII Use) and the ascensional difference (by the IX Use:) And if the Sun be in the first Semicircle of the Ecliptic, Subduct the ascensional difference, out of the Right ascension: But if the Sun be in the second Semicircle of the Ecliptic, add the ascensional difference to the Right ascension: and you shall have the obliqne ascension.  
XI. Use. To find how fare the Sun riseth, and setteth, from the true East and West points, which is called the Sun's Amplitude ortive, and occasive.  
Seek out (as was showed in the VI Use) the imaginary Circle, or Parallel of the Sun's course, and the points of that Circle in the Horizon, on the East, and West sides, cutteth the degree of the Amplitude Ortive, and occasive.  
XII. Use. To find the length of every day and night.  
Double the hour of the Sun's setting, and you shall have the length of the day; and double the hour of the Sun's rising, and you shall have the length of the night.  
XIII. Use To find the true place of the Sun, upon the Instrument, which answereth to the point, wherein the Sun is in the heavens: and is the ground of all the questions following.  
Take with your Instrument the height of the Sun, and reckon it on the movable Index, or Label: and then move the said Label, till you find the height of the Sun, exactly to fall upon the Parallel of the Sun for that day, on the East side if it be in the Forenoon, and on the West side, if it be in the Afternoon; the point of intersection, where the Index, or Label crosseth the Parallel, in that point of the Sun's altitude, shall be the true place of the Sun on the Instrument.  
XIIII. Use. To find the Hour of the day.  
The true place of the Sun on the Instrument (found out as was last showed) showeth among the hour lines the true hour of the day.  
XV. Use. To find out the Azumith or vertical Circle in which the Sun is, or the horizontal distance of the Sun from the Meridian.  
The Index or Label fastened at the Centre, is a movable Azumith: apply therefore the edge thereof, unto the true place of the Sun on the instrument (found out as was showed by XIII. Use.) And mark what point of the Horizon, or Limb, the same edge of the Label cutteth; reckon how many degrees of the Horizon, are intercepted between that point, and the Meridian line, or South point, either on the East, or West side: and that Arch shall be the horizontal distance sought for, whereby is showed the Azumith of the Sun at that instant: and consequently the Angle which the vertical Circle, or Azumith of the Sun maketh with the Meridian.  
XVI. Use. The Azumith of the Sun being known, to find out the Altitude of the Sun, and the Hour of the day.  
Set the edge of the Label to the Azumith given, and mark in what point the same edge crosseth the Parallel of the Sun for that day: that point of intersection showeth the height of the Sun above the Horizon, upon the Label: and also it showeth the hour of the day among the hour lines.  
XVII. Use. To find at what hour the Sun cometh to be full East, or West every day in Summer.  
Apply the edge of the Label, unto the East, or West points of the Limb, and mark in what point, the said edge cutteth the Parallel of the Sun for that day, for that same point among the hour lines, shall show the time of the Sun's coming to be full East, or West in that day, and likewise of what altitude the Sun will be above the Horizon, at that time of his being full East, or West.  
XVIII. Use. To find the height of the Sun at high Noon every day, and likewise at every other hour.  
Mark in what point the Parallel of the Sun for that day, cutteth the line of that hour, for which you would know the Sun's altitude: And unto that point of intersection, apply the edge of the movable Label, or Index: and thereon shall you find, the very degree of the Sun's altitude, at that hour.  
By this XVIII Use, and by the XVI, are made the Quadrants, described by Gemma Frizius, Munster, Clavins, Mr. Gunter, and others: and also all manner of Rings, Cylinders, & invumerable other topical Instruments, for the finding out of the hour, and other like conclusions. And likewise the reason, of finding the hour of the day, by a man's shadow, or by the shadow of any Gnomon, set up perpendicular to the Horizon, or else parallel to it.  
XIX. Use. To find out the Meridian line, and the points of the compass without a Magnetical needle, yea more exactly then with a needle.  
Take the height of the Sun, by the shadow of the pin: and apply the same height, reckoned on the Index, or Label to the parallel of the Sun for that day, whereby you have the true place of the Sun, in the instrument, as hath been showed in the XIII. Use. Then keeping both the Label at that point, and the pin upright in the Centre, hold, or set your instrument parallel to the plain of the Horizon, with the pin toward the Sun, and move it gently, till the shadow of the pin shall fall, exactly upon, the fiducial edge of the Label. For then the Meridian line of the instrument, shall be in the true Meridian of the place: and the four quarters of the instrument, shall look into the four cardinal points, of East, West, North, and South. Wherefore if with a bodkin, you make a prick at each end of the Meridian of your instrument where it standeth: and with a Ruler draw a line through them: the same shall be the Meridian of that place.  
This is a most excellent practice, for finding out the Meridian in any place, and is in an instant performed, and that easily. And hereby you may examine the Variation of the Compass. And also exactly place any Sun Dial.  
XX. Use. Considerations for the use of the instrument in the night.  
In such questions as concern the night, or the time before Sun rising, and after Sun setting, the instrument representeth the lower Hemisphere, wherein the Southern Pole is elevated. And therefore the Parallels, which are above the Equinoctial, shall be for the Southern, or Winter Parallels, and those beneath the Equinoctial, for the Northern, or Summer parallels. And the East shall be accounted for West, and the West for East: and the North shall be accounted for South, and the South for North: contrary to that which was before, when the Instrument represented the upper Hemisphere.  
XXI. Use. To find how many degrees the Sun is under the Horizon, at any time of the night.  
Seek the declination of the Sun for the day proposed: and at the same declination, on the contrary side, imagine a Parallel for the Sun that night: and mark what point of it is in the very hour and minute proposed: then set the Index, or Label to that point of the Parallel, and it will show you thereon the degree of the Sun's depression under the Horizon.  
XXII. Use. To find out the length of the Crepusculum, or Twilight every day.  
Because the question concerneth the night time, you must seek out the Sun's Parallel, for the night, on the other side of the Equinoctial, having the same declination with that which the day of the month showeth: then move about the Label, until the said Parallel cutteth the edge thereof in the 18th deg. on the West side for the Morning Twilight, and on the East side for the Evening Twilight, of the same day.  
And note that in the height of Summer, the Twilight in our Horizon, continueth all night long: because the same goeth not under the Horizon, full 18 degrees  
XXIII. Use. To find the Declination of any Wall, or Plain.  
Take a board having one straight edge, and a line drawn perpendicular unto that edge: apply the straight edge unto the Wall, at what time the Sun shineth thereon, holding the board parallel to the plain of the Horizon: and hang up a thread with a plummet, so that the shadow of the thread may fall on the board, crossing that perpendicular line. Then take with your Instrument the height of the Sun, and instantly make two pricks, in the shadow of the thread on the board, a good way distant one from the other: and laying a Ruler to those two pricks, draw a line, which line shall be the Azumith of the Sun, on the board: again with the height of the Sun lastly taken, find out on your instrument, the Azumith of the Sun; or the Angle which the Sun's Azumith maketh with the Meridian, (by the XV. Use.) And on the board taking the intersection of the shadow line with the perpendicular for the Centre, describe a Circle equal to the innermost Circle of the Limb: (which you may easily do, if you set one foot of your compasses upon the East, or West point, and extend the other foot unto 60 degrees, on the same innermost Circle, for this distance is equal to the Radius thereof.) Again with your compasses, take of the Arch between the Azumith of your Instrument, and the Meridian, and set that on the Circle of the board, that way that the true South is: and through the end of that Arch measured on the board, draw a straight line for the Meridian. Lastly take with your compasses, the Arch intercepted between the Meridian on the board, and the perpendicular line, and by applying it to the in most Circle of the limb, from the East, or West points see how many degrees it containeth: for that is the declination of the Wall. Or else you may find the Meridian upon the board, by XIX Use.  
If the Angle of the Meridian with the perpendicular, on the board, be a right Angle, the Wall is direct East, or West.  
But if the Meridian fall upon the perpendicular, or be parallel there to making no Angle with it, the Wall is direct North, or South.  
XXIIII. Use. The Art of dialing. And first how to make the Instrument in paper, promised in the beginning of this second part.  
For the Delincation of this instrument in paper, it will be necessary first to show the manner how the Semidiameter is to be graduated, or divided into degrees: and how the Centres, and Semidiameters, of the several kinds of Arches are to be found.  
Upon half a sheet of strong large Dutch paper, (the larger, the better) draw two straight lines, making a right Angle near one of the corners, the one through the length, and the other through the breadth of the paper; which two lines I therefore call the longer, and the shorter perpendicular.  
Upon the right Angle point, being the Centre, with a Semidiameter equal to that by which you intent to delineate your instrument, describe a quadrant of a Circle: and on the point where it meeteth with the shorter perpendicular, draw a long tangent line parallel to the longer pependicular.  
Divide the Quadrant into 90 degrees, among which from the beginning at the shorter perpendicular, reckon the elevation of the Pole, for which you will make your instrument, and applying a Ruler to the end thereof, and to the Centre, where the Ruler cutteth the tangent line make a prick. And taking with your compasses the distance from the Centre to that prick, measure it upon the shorter perpendicular: this shall be the Semidiameter of the sixth hour Circle. At the end thereof draw another long line parallel also to the longer perpendicular.  
Then out of the Centre unto the second parallel through every degree of the quadrant, draw fine straight lines, cutting also the first Parallel. The intersection of those lines with the first Parallel, shall be The scale of centres of Arches. And their intersection with the second Parallel shall be The scale of centres of hour Circles. And the segments of those lines, intercepted between the Centre, and the first Parallel, shall be the Semidiameters of Arches: and the whole lines between the Centre and the second Parallel, shall be The Semidiameters of hour Circles. And that you may know for what Circle, every Centre, and Semidiameter serveth, you shall note every fift line from the beginning, with the figures 5, 10, 15, 20, etc. Set under the second Parallel, unto 90 which will fall upon the longer perpendicular: that so you may readily find the Centre, and Semidiameter of any Circle required.  
Again divide the first 45 degrees of the Quadrant in the midst: and applying your Ruler to the Centre, and to every one of those half divisions, where in each place the Ruler cutteth the first Parallel, or tangent line, make a prick. So shall you have upon the tangent line between the shorter perpendicular and the middlemost line 45, a third scale, which is, The scale of 90 degrees, for the graduating of the Semidiameter of your instrument on the paper: In which you shall also distinguish every fift degree, with figures set under the tangent line.  
Having thus prepared your paper of scales with lines neatly and exactly drawn, keep it by you to have it still in a readiness for the making, and using of the Instrument in paper. The making whereof is thus.  
Take with your compasses the Semidiameter of the Quadrant in your paper of scales: and therewith upon a piece of strong Dutch paper, Describe the horizontal Circle: which you shall cut into two Semicircles with a Meridian line drawn through the Centre: divide them into Quadrants in the points of East, and West: and each Quadrant into 90 degrees to be marked with figures just as is done in the Instrument.  
Then with your compasses take the elevation of the pole upon the scale of degrees in your paper: & set it upon the Meridian line from the Centre which way you please: that shall be the intersection of the Equinoctial with the Meridian. Also reckon the compliment of the height of the Pole, upon the scale of Centres of Arches, and with your compasses take the distance from the end thereof to the Centre: the same shall be the Semidiameter of the Equinoctial, to be drawn from the East point of the Horizon through the point of intersection with the Meridian unto the West point.  
Again take with your compasses upon the scale of degrees in your paper the compliment of the height of the Pole: and set it upon the Meridian on the other side of the Centre from the Equinoctial: there shall be the Pole of the Equinoctial, or of the World, in which all the hour lines shall cross one another.   diagram 
Moreover with your compasses, take the distance between the Centre and the second Parallel in your paper, which is the Semidiameter of the sixth hour Circle: and set it on the Meridian from the Pole beyond the Equinoctial: that shall be the Centre of the sixth hour Circle: upon which you may draw the same Circle, from the East point of the Horizon through the Pole to the West point. Then through the centre of the sixth hour Circle erect a line perpendicular to the Meridian, extending it infinitely on both sides of the Meridian: and in that line both ways, prick down the Centres of the horary Circles, out of the scale in paper: And lastly opening your compasses from every one of those Centres unto the Pole severally, describe all the horary Circles, or at least every fift of them, and so is your paper instrument perfectly finished.  
The use of this instrument on paper is, that lines, and arches may be designed upon it with a fine pennicell of black lead, and afterward be wiped out again. Wherefore it will be needful for him that will use this instrument, to all the purposes thereof, to get a good pair of large compasses with three points, ●ne sharp, another for ink, a third for black Lead. And I suppose it would do well to fast ●n over your instrument a piece of thin oiled paper, through which the lineaments may be conspicuous: and upon it to trace such lines, and arches as you have occasion to use: that so your instrument may be kept clean, and last longer.  
For as much as in delineating the horary Circles, which are within 30 degrees of the Meridian, the Semidiameters will be too long for your compasses: you may in that straight thus help yourself. First say, As the Radius, is to the Sine of the Elevation of the Pole; So is the tang: of the distance of any Horary Circle from the Meridian, suppose 25, or 20, or 15, or 10, or 5 degrees, to the arch of the Horizon between the Meridian, and that horary Circle.  
Reckon this distance on the limb of your Instrument from both ends of the Meridian, and mark it. Thus do for the 25th, 20th, 15th, 10th, and 5th horary Circle on both sides of each end of the Meridian.  
Then in any piece of clean paper, through the midst of the longer way, draw aline: and toward one end which (I call the upper end) cross the same with a perpendicular line exactly equal to the Diameter of your Instrument, the point of Intersection being the centre.  
Take with your Compasses out of the paper of scales, the semidiameter of 60. degr: (which you may well do for an ordinary instrument): and setting one foot on either end of the Diameter, that point wherein the other foot shall cut the first long line, make your Centre, and thereon draw an Arch through both ends of the Diameter, and cutting the upper part of the first long line: this Arch is equal to that horary Circle, which is distant from the Meridian 30 degrees the compliment of 60.   diagram 
Divide each half of the Diameter into 3 equal parts, with 4 points, and from every of those points unto the Arch draw lines parallel to the first long line. And having divided every one of those five parallel lines intercepted between the Diameter and the Arch into 6 equal parts, for the 6 times 5 degrees which remain to the Meridian, draw through those divisions from the ends of the Diameter (with a smooth and even hand) the Arches 25 20, 15, 10 and 5.  
Those Arches you may transfer from the paper to your instrument in this manner. Rub the back side of the paper against the Arches, with fine powder of black lead: then applying the paper with Arches to your instrument, that the ends of the Diameter may exactly fall upon the two opposite marks, in the limb of your instrument, which serve for the horary Circle that you would draw, either 25, 20, 15, 10, or 5, trace over that Arch with the point of any hard piece of wood sharpened: and the black lead on the back side will upon the instrument leave the print of that Arch.  
XXV. Use. To set an upright Wall or plain upon the instrument: and to find how many hours the Sun shall shine thereon at some time of the year.  
The situation of Walls, or Plains is considered either in respect of the Meridian, or of the Horizon. And unto both it is either perpendicular, or obliqne, or parallel.  
The plain perpendicular to the Meridian, is that which standeth directly North, or South: which if it be also perpendicular to the Horizon, is called North, or South direct upright. But if it stoop from the Zenith forward, it is called North, or South inclining: if backward, it is called North or South reclining. And note that in a stooping Plain that side which is toward the Horizon, is inclining, and that which is toward the Zenith is reclining.  
The Plain obliqne to the Meridian is that which standeth not directly North, or South, but declineth one side into the East, and the other into the West: and is therefore called Declining Eastward, or Westward, according as either side of the Plain looketh: As if an upright Wall being Southern, declineth from the South into the East, it is called South declining Eastwards upright. But if it be not upright, it is called South declining Eastward, and inclining, or reclining.  
The Plain parallel to the Meridian, is that which looketh directly East, or West; and accordingly, hath his denomination, whether it be Upright, Inclining, or Reclining.  
The Plain Parallel to the Horizon, is called horizontal: and is represented by the instrument itself, or at least by the inner most Circle of the limb thereof.  
And note that the Arch of Declination, is reckoned from the next East, or West point. And that the Arch of Inclination, or Reclination is reckoned from the Zenith, or the compliment of it from the Horizon. So that every upright Plain is understood to pass through the Zenith, which in the instrument is the Centre.  
And thus having showed the several affections of Plains, we will now proceed to show the manner how to set them upon the Instrument.  
A Direct North, or South upright Plain, is represented in the instrument by a line drawn through the Centre from the East point to the West, which is also the horizontal intersection of the Pla●ne. And by it you shall see that the Southern side or face of the plain is open to all the hours between six in the morning and six in the evening. And that about London, the Northern side, only in the Summer enjoyeth the Sun from his rising till after seven in the morning: and from before 5 a clock in the afternoon, till his setting.  
A direct East, or West upright plain, is represented in the Instrument by the Meridian, which is also the horizontal intersection of the plain. And in it you shall see that all the forenoon hours are open to the East side: and all the afternoon hours to the West side.  
A Declining Plain is thus set upon the Instrument, reckon on the Horizon the arch of Declination, from the East, or West point: and at the end draw a line through the Centre unto the opposite point of the Horizon: So that each side thereof may be open to that point, either East, or West, into which the Declination is supposed. That line so drawn through the centre is the horizontal intersection of the plain, and representeth the plain itself, if it be upright. For example, there is about London an upright Wall declining Eastwards 35 degrees: which I would set upon the Instrument. Hold the Southern part of the Instrument to you, and reckon from the East backward into the North upon the Horizon 35 degrees: there draw a line through the Centre: this line shall not only upon the South side represent a Southern Plain declining Eastward 35 degrees. But also upon the North side shall represent a Northern Plain declining Westward 35 deg. And moreover it will appear that on the Southern side shall be drawn the hours from almost 4 a clock in the morning, till 3 in the afternoon. And that in the Northern side shall be drawn upon one side 4 a clock in the morning only: and upon the other side all the hours from 3 in the afternoon till Sun set.  
And ●o consequently the declination of an upright wall, or Window being given, it may be found at what hour the Sun upon any day in the year will come to that Wall, or Window, and when it will go from it. As in the former example, There is about London a Northern wall declining Westward 35 deg. I would know at what time of the day the Sun will begin to shine upon it on the 24th day of March. Set the Index at 35 deg. from West toward South: and because that day the Sun's Declination is 6 degrees Northward; Look at what hour the sixth Parallel above the Equinoctial toward the Centre meeteth with the Index so placed: and you shall find it at 3 ● clock in the Afternoon. Wherefore at that time the Sun will begin to shine upon the Wall that same day.  
The Poles of every upright Wall are in the Horizon 90 deg. that is a quarter of a Circle, distant from the line representing the Plain. Wherefore if upon that line in the Centre you erect a perpendicular, the ends thereof in the Horizon shall be the poles of that Plain: and are so fare distant from the North and South points, as the Plain itself is from the East, and West.  
XXVI. Use. To set an Inclining, and Reclining Wall, or Plain upon the Instrument: and to find how many hours the Sun shall shine thereon, at some time of the year.  
When you have an Inclining, or Reclining Plain to be described on the Instrument. First the horizontal intersection is to be set thereon, as if it were upright; together with the line perpendicular thereto, in which are the Poles of the Plain: according as was taught in the XXV Use.  
Then upon the scale of degrees in your paper, reckon the arch of Inclination, or Reclination; and with your compasses take, & set it in your Instrument upon the line perpendicular to the horizontal intersection of your Plain, from the Centre that way into which the Inclination, or Reclination tendeth: the same shall be the uppermost point of your Plain.  
Again, with your Compasses take the Compliment of inclination, or reclination, both upon the scale of degrees, and also upon the scale of centres of arches in your paper: and set both spaces upon the same perpendicular line, but on the other side of the centre (extended if need be): At the shorter of those spaces shall be the pole of your plain: and at the longer of them shall be the Centre of it.  
Lastly, setting one foot of your Compasses in the centre of your Plain, and extending the other foot to the uppermost point, describe in your Instrument an Arch of a Circle: which if you have done well, will exactly fall upon the ends of the Horizontal intersection of your Plain. That Arch shall represent your Plain, inclining upon the lower side, which is toward the Horizon, or Limb: but reclining upon the upperside, which is toward the Zenith, or Centre. And so either side shall show in what hour lines the Sun, at some time of the year, will shine upon it: that in delineating a Dial thereon, it may not be cumbered with unnecessary hour lines. For Example, suppose that the former Plain, which with the South declined Eastward 35 deg. do also incline 41 deg. 30 min. Wherefore also with the North side it shall decline Westward 35 deg. and Recline 41 deg. 30 min. Describe this plain upon your Instrument with an Arch of a Circle, found out as was taught last before. And it will appear that upon the Inclining side shall be drawn all the hour lines from almost 4 in the morning, to 4 in the afternoon: And upon the Reclining side shall be drawn first 4 and 5 in the morning: and then beginning at 0 a clock, all hours to Sun set.  
And note that in all Northern plains, the North Pole is elevated: and in all Southern plains the South Pole is elevated. Except such North inclining, and South reclining Plains, that in the Instrument fall below the North Pole, between it and the Horizon: For in them the contrary Pole is elevated. And also that a direct East and West plain, if it Recline, hath the North Pole elevated: and if it Incline, the South Pole.  
XXVII. Use. The Plain being set upon the Instrument, to find the distances of the hour lines, and Substile from the horizontal Intersection. And also the height of the Style above the Substile.  
Every Dial either hath a Centre in which all the hour lines, together with the Substile, and Style do meet: or else it hath no centre, & so they are all parallel one to another.  
If the Plain being set upon the Instrument, cutteth the Pole of the Equinoctial (that is the point in which all the horary Circles cross the Meridian) the Dyal to be drawn upon that Plain shall have no Centre. But if it cutteth not the Pole, the Dyal shall have a Centre. And of these Dial's with Centres we will first entreat: as being most proper for the use of the Instrument.  
Behold therefore the Pole of your plain heedily what horary Circle it falleth upon: Or if it fall between any two, the distance of each being reasonably apportioned, imagine a horary Circle passing through it. Mark in what point that horary Circle, either real, or imagined, doth cut the Plain, that same point shall be the place of the Substile in your Plain: and the height of the Style above it, is the Arch of that horary Circle intercepted betwixt the Pole of the Aequinoctial, and the point of the Substile noted in the Plain.  
Therefore applying a Ruler to the Pole of your Plain, carry it about unto all the intersections of the Plain with the hour Circles, and the substile severally: and where in every place the Ruler shall cut the innermost Circle of the Limb, there make a visible mark: For the arches of the Limb intercepted between the horizontal points of your Plain, and every one of those marks, shall be the distance intended to be sought.  
But for the horizontal plain, the ends of the hour Circles in the Limb of your instrument, do give the distance without any more ado.  
Concerning the height of the Style above the Substile: It is apparent by the instrument, that in a Horizontal dyal, the 6 a clock line lieth directly East and West: and the Meridian perpendicular to it, directly North and South. And that the Meridian is the Substile. And that the height of the Style above the Substile, is equal to the height of the Pole in that place.  
It is also apparent, that in all direct North and South Dial's, the 6 a clock Line is drawn parallel to the Horizon, and the Meridian perpendicular to it: And that the Meridian is the substile. And that the height of the Style above the substile, if the Plain be upright, is equal to the compliment of the height of the Pole. But if it be North inclining, or South Reclining, it is equal to the difference, of the height of the Pole, and the Arch of Inclination, or Reclination. And if the Plain be North Reclining, or South Inclining, it is equal to the Sum of the compliment of the height of the Pole and of the Arch of Inclination, or Reclination. And if the Plain fall upon the Equinoctial, the style shall stand up perpendicular upon it in the Centre: and the hour lines shall be drawn all at equal Angles, viz. 15 degrees one from another.  
In such Dial's as have not the Meridian for the substile, the height of the Style above the Substile is thus sound by the Instrument. It was showed before that the height of the style above the substile is the Arch of the horary Circle through the Pole of the Plain, intercepted between the pole of the Equinoctial, and the point noted in the Plain for the substile. Therefore from the horizontal points, or intersections of that horary Circle reckon 90 degrees both ways: and thereto through the Centre draw an obscure line: in which line shall be both the Inclination of that horary Circle, which is the distance of the intersection, or uppermost point thereof from the Centre: and also the Pole. Then with your compasses take that distance, or Inclination: and setting it upon the scale of degrees in your paper, see how many degrees it containeth upon that scal●. Again upon the same scale take with your compasses the compliment of that inclination or distance, which being set upon the obscure line on the other side of the Centre, shall show the Pole of that horary Circle. Lastly applying a Ruler to the Pole of that horary Circle, and both to the Pole of the Equinoctial, and to the point of the substile in the Plain severally: mark where in both places the Ruler cutteth the innermost limb of your instrument: For the degrees of the limb intercepted betwixt those marks, shall be the height of the style, above the substile, which was sought for.  
And by this which hath been taught you shall find that in an upright Dial declining, as before 35 deg. from the South into the East, or from the North into the West, the substile shall fall upon that horary Circle, which is about 3 deg. after 9 a clock in the morning: and the style elevated above the substile about 31 deg. And also that in a South Dial declining Eastward 35 deg. and inclining 41 deg. 30 min: Or in a North declining Westward 35 deg. and reclining 41 deg. 30 min: the substile shall fall upon that horary Circle which is about 8 degrees after 6 a clock: and the style elevated above the substile about 63 deg. 30 minutes.  
XXVIII. Use. The making of all manner of plain Dial's with Centres.  
I have already showed how to find out in our Instrument the distances of all the hour lines, and the substile, from the Horizontal intersection. Now the delineating of a Dyal is nothing else but to transfer those distances out of your Instrument into the Dyal plain, every one in his due situation: and then through them, out of the Centre, to draw such hour lines as shall be of use, together with the substile.  
The due situation of those distances upon the Dyal plain, dependeth on the true placing of the Meridian, or 12 a clock line: for that being truly described, all the rest will be easy enough.  
First therefore I will show the manner how the Meridian, or 12 a clock line, is to be described.  
Take in your Dyal some point for the Centre, where you shall think fit: and through it draw a line parallel to the plain of the Horizon. Cross it in the Centre with a perpendicular line. And having opened your compasses to the length of the Semidiameter of your paper Instrument, describe on the Centre a Circle equal to the innermost Limb thereof. In which Circle the line parallel to the Horizon is for the Horizontal intersection: and the other for the line perpendicular to it: and the Circle itself representeth the plain: Mark therein the East and West sides of the Plain with E and W.  
In the Horizontal, and in all North and South direct Plains, both upright, and stooping; and in all upright declining plains, the Meridian is perpendicular to the Line parallel to the Horizon.  
In North ●●clining, and South reclining plains, the Meridian is to be drawn on that side of the Dyal plain either East, or West, into which the declination is: But in North inclining and South reclining, on the contrary side. And if the plain be Northern, the Meridian shall be above the Line parallel to the Horizon: and if the plain be Southern, it shall be under it. And if the contrary Pole be elevated, it shall be drawn through the Centre into the opposite Quadrant of the Circle in your Dyal plain.  
Lastly in a direct East and West plain, both inclining and reclining, the Meridian is the same with the line parallel to the Horizon.  
Wherefore with your compasses take the distance in the limb of your Instrument, from the next horizontal point, unto the mark of the Meridian; and measure it upon the Circle of the Dyal plain, in that part, and on that side, according as in consideration of the elevated Pole, and of the quality of the Plain, was showed to be agreeable. And at the end of that arch, through the Centre, draw a line for the Meridian.  
Again with your compasses take the distances in the limb of your Instrument, between the mark of the Meridian, and the marks of all the hour Lines severally: and setting them upon the Circle of the Dyal plain orderly from the Meridian, the Forenoon hours on the West side of it, and the Afternoon hours on the East side: at the end of every one of those arches draw the hour Lines: and distinguish them with their proper figures accordingly.  
Lastly fasten the style in the Centre, so that it may hang perpendicular unto the plain in the Substile, at the just height. And because the style in every Dyal is understood to be a segment of the Axis of the world which is a line imagined to pass from the North to the South Pole through the Centre of the earth; the style being rightly placed shall still with the end point towards the elevated Pole, that is upward from the Centre, if the North Pole be elevated; or downward from the Centre if the South Pole be elevated.  
XXIX. Use. The making of all manner of plain Dial's not having Centres.  
If the plain represented on the Instrument (as was taught before in the XXV and XXVI Uses) cut the Pole of the Equinoctial, it is an horary Circle, either one of them which are drawn in the Instrument, or falling between some two of them: and the Dial plain itself shall not cross the axis of the world, but lie parallel to it, without any Angle of elevation. And therefore such a Dyal can have no Centre: But the style, the substile, and all the hour lines shall be parallel one to another.  
Every such Plain represented on the Instrument,  
Either, First it is the Meridian of the place, the horizontal intersection whereof is the 12 a clock Line drawn from North to South: and the Dial made thereon, is a direct East, or West upright Dyal: In which the substile is distant from the Line, in the Circle of the Dial plain parallel to the Horizon, with an Arch equal to the elevation of the Pole, and upward toward the Pole. And is also the 6 a clock line in your Dyal.  
The rest of the hour lines are thus described. Draw through the substile, in any point, a long Line at right Angles: that line shall be the Aequinoctial intersection usually called the Contingent line: And taking a convenient distance for the style to hang parallel over the substile (according to the greatness of your Dial plain) measure it upon the substile from the Equinoctial intersection: and upon the end of that measure, describe half a Circle for the Equinoctial itself. Divide each Quadrant thereof from the substile, into 6 equal parts, or hours. Then applying a Ruler to the Centre, and to every one of those divisions severally, where in every place the Ruler shall cut the long line of Equinoctial intersection, make pricks: and through those pricks draw the hour lines, parallel to the Substile, or 6 a clock line: distinguishing so many of them as be needful, with their figures: that is all the Forenoon hours on the East plain, and all the Afternoon hours on the West plain. But in these Dial's there is no 12 a clock line, it being infinitely distant from the Substile. Lastly hang the style directly over the Substile, and parallel to it, at the distance formerly taken. And thus are your East, and West Dial's finished.  
Or Secondly, it is the sixth hour Circle, the Horizontal intersection whereof is the line of East, and West: and the Dyal made thereon is direct North inclining, or South reclining, with an Arch equal to the compliment of the height of the Pole. And the parallel to the Horizon is the Aequinoctial intersection: and the line perpendicular to it is the 12 a clock line, and also the Substile.  
The rest of the hour Lines, from 7 a clock in the morning, to 5 in the evening, are thus described. Take a convenient distance for the Style from the Substile, measuring it upon the Substile from the Aequinoctial intersection: and on the end of that space describe the Semicircle of the Aequinoctial, to be divided on both sides of the Substile into 6 hours: through every one of which out of the Centre, a Ruler being applied; at the points of the several intersections of the Ruler with the Aequinoctial intersection, draw the hour Lines parallel to the Substile, or 12 a clock Line: distinguishing them with their figures, namely 11, 10, 9, 8, 7, on the West side: and 1, 2, 3, 4, 5, on the East side: but in these Dial's there is no six a clock Line, it being infinitely distant from the Substile. Lastly hang the Style directly over the Substile, and parallel to it, at the distance formerly taken.  
Or Thirdly, it is North inclining, or South reclining, and also declining: in which.  
As the tangent of the Elevation of the Pole, is to the Radius; So is the Sine of the compl: of Declination, to the tang: of the compl: of Inclination or Reclination.  
The Plain being set upon the Instrument by the Arches of Declination and stooping thereof (as hath been taught in XXV ● Use) shall cut the pole of the Equinoctial. Apply therefore a Ruler to the Pole of the plain, and to the Pole of the Aequinoctial; and the point in which it cutteth the Limb, mark for the substile: which is to be transferred unto the Circle of the Dyal plain, by taking the distance between that point, and the next Horizontal intersection, and setting it on that Circle from the line parallel to the Horizon, upward if the plain be North: or downward if the plain be South: and on that side which is contrary to the Declination. The substile being thus sound, draw a long line perpendicular to it, for the Aequinoctial intersection. And taking a convenient distance for the style from the substile, measure it upon the substile from the Equinoctial intersection: and on the end of that space describe the Semicircle of the Aequinoctial. Then look in your Instrument how many degrees of the Aequinoctial are intercepted between the Meridian, and the Arch representing your Plain: and reckoning the same number of degrees upon the Aequinoctial of the Dyal plain, from the substile towards the side of Declination, there make a mark for the Meridian point thereof: in which you must begin to divide the Aequinoctial semicircle into hours both ways: And that being divided, apply a ruler to the centre, and to every one of the divisions: and at the points of the several intersections of the ruler with the Aequinoctial intersection, draw the hour lines parallel to the substile. Set 12 at that hour line which was drawn at the intersection through the Meridian point of the Equinoctial: and 11, 10, 9, 8, etc. on the West side: and 1, 2, 3, 4, &c, on the East side. Lastly, hang the style directly over the substile, and parallel to it, at the distance formerly taken.  
XXX Use. How by Sines and tangents to calculate the places of the Meridian, and Substile, and the height of the Style above it: and the distance of the Meridian of the Equinoctial from the Substile; together with the places of hour lines, both by calculation, and also Geometrically.  
I have already taught the making of all manner of plain Dial's most easily by the Instrument, for the same height of the Pole. But if any man either want an Instrument, or else desireth greater exactness, I will also here show how to perform the same by calculation, on the other side of the Instrument.  
In a plain erect Dial declining.  
As the Radius is to the Sine of doclination; So is the tang. of the compl. of the Poles height. to the tang. of the distance of the substile from the Meridian.  
In plain Dial's both declining and also inclining, and reclining. As the Radius is to the Sine of inclination reclination So is the tang: of Declination, to the tang. of the compl. of the distance of the Meridian, from the line parallel to the horizon.  
Again As the Radius is to the Sine of the compl. of Declination; So is the tang. of the compl. of the Poles height, to the tang. of Base I.  
If the Dial be South reclining, or North inclining, the sum of Base I, and of the compliment of Inclination or reclination shall be Base II. But if the Plain be South inclining, or North reclining, the difference of Base I, and of the compliment of inclination, or reclination shall be Base II.  
Then say thirdly, As the Sine of the compl. of Base I, is to the Sine of the compl. or excess of Base TWO; So is the Sine of the height of the Poles to the Sine of the height of the style above the substile.  
Fourthly, As the Sine of the compl. of the height of the style above the substile, is to the Sine of Declination; So is the Sine of the compl. of the height of the Pole, to the Sine of the compl. of the distance of the substile from the line parallel to the horizon.  
Again As the Radius is to the Sine of the compl. of Declination; So is the Sine of the compl. of the Poles height, to the Sine of the height of the style above the substile.  
Thirdly As the Sine of the Poles height is to the Radius; So is the tang. of Declination, to the tang. of the Distance of the Meridian of the Equinoctial from the substile. And this distance is ever less than 90 degrees.  
In a plain East and West Inclining and Reclining Dial. As the Radius is to the Sine of the compl. of inclination reclination So is the tang. of the height of the Pole, to the tang. of the distance of the substile from the Meridian.  
Again As the Radius is to the Sine of inclination reclination So is the Sine of the height of the Pole, to the Sine of the height of the style above the substile.  
Thirdly, As the Sine of the compl. of the Poles height; is to the Radius; So is the tang. of the compl. of inclination reclination to the tang. of the distance of the Meridian of the Equinoctial from the substile. And this distance is ever greater than 90 degrees.  
Fiftly As the Sine of the compl. of the height of the style above the substile, is to the Sine of Declination; So is the Sine of the compl. of inclination reclination to the Sine of the distance of the Meridian of the Equinoctial from the substile.  
And note that in South reclining, and North inclining Plains, if Base TWO be less than a quadrant, the contrary pole is elevated above the Plain: And if Base TWO be equal to a quadrant, the Plain doth cut the Pole of the Equinoctial.  
Now concerning the placing of the substile upon the Dial plain (as I have already in the XXVIII Use showed for the Meridian) We are to know, First that the substile is to be drawn upward from the line parallel to the horizon, if the Plain be Northern; or downward from it, if it be Southern. Except in North reclining, and South inclining Dial's, in which the Base I exceedeth the compliment of inclination, and reclination: for in them it is quite contrary. And secondly that the substile is to be drawn in the contrary side from the Declination. But in North inclining, and South reclining Dial's, in which the contrary Pole is elevated, the substile must be drawn through the centre into the opposite quadrant of your Dial circle.  
Lastly, the hour lines in all manner of plain Dial's, are thus to be found.  
If the substile and hour be both on the same side of the Meridian: the arch of the Equinoctial between the substile, and the hour line, shall be equal to the difference of the two distances, namely of the hour line from noon, and of the Meridian of the Equinoctial from the substile. But if the substile be upon one side of the Meridian, and the hour on the other side: it shall be equal to the sum thereof. Then say As the Radius is to the Sine of the height of the style above the substile; So is the tang. of the arch of the equinoctial, between the substile and the hour line, to the tang. of the arch of the circle of your Dial plain, between the substile and that lower line.  
Or else you may without calculation Geometrically inscribe the hour lines in Dial's having centres (for how to do it in Dial's not having centres, I have already showed in the XIX Use) thus.  
Describe in your Dial plain a line for the style, at the same height or distance from the substile, that the true style ought to have. Take also in the substile (as in reason you shall see fit) a point, and through it draw at right angles a long line, for the contingent, or Equinoctial intersection. Again from the same point let fall a perpendicular unto the style: the length of this perpendicular is the nearest distance between that point and the style: and it is also the distance of the centre of the Equinoctial from that point: measure it therefore upon the substile, the contrary way from the centre of the Dial: and having thus the centre of the Equinoctial, describe thereupon toward the contingent line one half of the Equinoctial circle: which if the substile be the Meridian, or 12 a clock line of your Dial, you must begin to divide into hours at the substile: But if the substile and Meridian of your Dial be several lines, apply a ruler to the centre of the Aequinoctial, and to the intersection of the 12 a clock line with the contingent, and there draw a line: this line shall be the Meridian of the Aequinoctial: at which you must begin to divide the Equinoctial circle into hours, both ways. Then applying a ruler unto the centre of the Aequinoctial & every one of those divisions, where the ruler in every place shall cut the contingent line, there make a mark: and lastly, through every one of those marks from the centre of the Dial, draw the hour lines themselves.  
And if in any hour line it shall happen, that the ruler so applied will not reach to intersecat the contingent line: you may thus help yourself. Which rule also may serve you to find the Meridian of the Equinoctial, as often as the intersection of the Meridian of the Dial with the contingent, falleth without your paper or plain.  
Draw the hour line as fare as it will go. And take with your Compasses the distance of the intersection point of the contingent with the substile, both from the centre of the Dyal, & from the centre of the Aequinoctial. And taking at all adventure a point in the contingent line, on that side in which the hour line is, measure from that point on the contingent, both those distances: and at the ends of them both draw two lines parallel to the substile, crossing the contingent. Then applying a ruler to the point, which you took at all adventure, and to the intersection of the parallel, which hath the distance of that centre, whence the hour line given proceedeth, with that hour line: where the ruler shall cut the other parallel, make a prick: and measure the distance between that prick and the contingent, upon the former parallel, on the other side of the contingent. Lastly, out of the proper centre through the end of that measure, draw a line: which shall be that you desire.  
An example of this Geometrical way of delineating th● hour lines you shall find in the description of a South upright Dial declining 35 degr. and reclining degr. 41 min. 30. by considering whereof these rules will be found exceeding plainly set down: As also all the other rules and observations here delivered, to one that is any whit pregnant and ingenious, will need no other exemplification, than the inspection of the instrument itself, and of these several Dial's following  
FINIS.   


A North Dial declining Eastwards 35 deg. reclining 41 deg. 30 min. Latitude 51 deg. 30 min.   diagram  

A horizontal Dial.   diagram  

South direct upright.   diagram  

South direct inclining 24 deg.   diagram  

North direct reclining 24 deg.   diagram  

South direct reclining equal to the compliment of the poles height.   diagram  

West direct upright.   diagram  

East direct reclining 32 deg.   diagram  

South upright declining Eastward 35 deg.   diagram  

South declining Westward 76 degr. reclining. 48 deg.   diagram  

South declining Westward 61 degr. Reclining 21 deg. ●½ min.   diagram   

The Translator to the Reader.  
Gentle Reader, by reason of my absence, whilst this Book was in the Press, it is no marvel though some faults have escaped, which you will be pleased to amend thus.  
Pag. 14 lin. 14, 2.0413927. Pag. 15. lin. 1, the first term of a progression Pag. 17, lin 17, the Antecedent arm Pag. 18 lin. 19, term given from Pag. 19 lin. 11, in the fift circle Pag. 20 lin. 19, lie hid. As in this lin. 20 D. Rat. multa— 1 in R. lin. 28, and Rat multa in R in α— R in α,  
Also in the aequations pag. 21, 24, 26 which have a magnitude equal to a fraction: the same magnitude together with the note of equality, aught to be set right against the line that is between the Numerator and Denominator of the fraction, as in these, . And . And so of the rest.  
Pag. 25 lin. 20, arm at 71 ⌊ 382: Pag. 26 lin. 16, Ratiocination. Pag. 29 lin. 29. number of figures Pag. 35 lin. 5, 61 49/144 Pag. 36 lin. 11, 17 ⌊ 48 lin. 14, 17 ⌊ 48 · 3 ⌊ 26 +. Pag. 44 lin. 8, is a roof line 26, thereof Pag. 94 lin. 21, a circle, or 90 degrees. Pag. 100 lin. 3, and then the side DC Pag. 113 lin. 7, delineation Pag. 127 lin. 1, the Sun goeth not Pag. 131 lin. 19, in the paper Pag. 132 lin. 10, to the tangent of the arch of Pag. 143 lin. 3, in North reclining, and South inclining Pag. 152 lin. 30 North Dial declining Eastward 35 deg.