CHAP. I. Of the Description, and vse of the Circles in this First side.

1 THere are two sides of this Instrument. On the one side, as it were in the plaine of the Horizon, is delineated the proiection of the Sphare. On the other side there are di­vers kindes of Circles, divided after many severall waies; [Page 2] together with an Index to be opened after the manner of a paire of Compasses. And of this side we will speake in the first place.

2 The First, or outermost circle is of Sines, from 5 de­grees 45 minuts almost, vntill 90. Every degree till 30 is divided into 12 parts, each part being 5 min: from thence vntill 50 deg: into sixe parts which are 10 min: a peece: from thence vntill 75 degrees into two parts which are 30 minutes a peece. After that vnto 85 deg: they are not divided.

3 The Second circle is of Tangents, from 5 degrees 45 min: almost, untill 45 degrees. Every degree being di­vided into 12 parts which are 5 min: a peece.

4 The Third circle is of Tangents, from 45 degrees untill 84 degrees 15 minutes. Each degree being divi­ded into 12 parts, which are 5 min: a peece.

5 The Sixt 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 til 6 degrees.

6 The Fourth circle is of Vnaequall Numbers, which are noted with the Figures 2, 3, 4, 5, 6, 7, 8, 9, 1. Whe­ther you vnderstand them to bee single Numbers, or Tenns, or Hundreds, or Thousands, &c. 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 sheweth the true or naturall Sines, and Tangents. For if the Index bee applyed to any Sine or Tangent, it will cut the true Sine or Tangent in the fourth circle. And wee are to knowe that if the Sine or Tangent be in the First, or Second circle, the figures of the Fourth circle doe signifie so many thousands. But if the Sine or Tangent be in the Seventh or Eight circle, the fi­gures in the Fourth circle signifie so many hundreds. And [Page 3] if the Tangent bee in the Sixt circle, the figures of the Fourth circle, signifie so many times tenne thousand, or whole Radij.

And by this meanes the Sine of 23°, 30′ will bee found 3987: and the Sine of it's complement 9171. And the Tangent of 23°, 30′ will be found 4348: and the Tangent of it's complement, 22998. And the Radius is 10000, that is the figure 1 with foure cyphers, or cir­cles. And hereby you may finde out both the summe, and also the difference of Sines, and Tangents.

7 The Fift circle is of Aequall numbers, which are noted with the figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 0; and e­very space is divided into 100 aequall parts.

This Fift circle is scarse of any use, but onely that by helpe thereof the given distance of numbers may be mul­tiplied, or divided, as neede shall require.

As for example, if the space betweene 1⌊00 and 1⌊0833+ bee to bee septupled. Apply the Index vnto 1⌊0833+ in the Fourth circle, and it will cut in the Fift circle 03476+; which multiplyed by 7 makes 24333: then againe, apply the Index vnto this number 24333 in the Fift circle, and it will cut in the Fouth circle 1⌊7512+. And this is the space betweene 1⌊00 and 1⌊0833+ septu­pled, or the Ratio betweene 100, and 108⅓ seven times multiplied into it selfe.

And contrarily, if 1⌊7512 bee to bee divided by 7: Apply the Index vnto 1⌊7568 in the fourth circle, and it will cut in the fit circle 24333: which divided by 7 gi­veth 03476+. Then againe vnto this Number in the Fift circle apply the Index, and in the Fourth circle it wil cut vpon 1⌊0833+ for the Septupartion sought for.

The reason of which Operation is, because this Fift circle doth shew the Logarithmes of Numbers. For if the Index be applyed unto any number in the Fourth circle, it will in the Fift circle cut vpon the Logarithme of the same num­ber, so that to the Logarithme found you praefixe a Caracte­risticall [Page 4] (as Master Brigs termes it) one lesse then is the number of the places of the integers proposed (which you may rather call the Graduall Number). So the Loga­rithme of the number 2 will bee found 0.30103. And the Logarithme of the Number 43⌊6 will bee found 1.63949.

Numbers are multiplied by Addition of their Loga­rithmes: and they are Divided by Substraction of their Logarithmes.

8 In the middest among the Circles, is a double No­cturnall instrument, to shew the hower of the night.

9 The right line passing through the Center, through 90, and 45 I call the Line of Vnitie, or of the Radius.

10 That Arme of the Index which in euery Operati­on is placed at the Antecedent, or first terme', I call the Antecedent arme: and that which is placed at the conse­quent terme, I call the Consequent Arme.

CHAP. II. Of the Operation of the Rule of Proportion: and also of Multiplication, and Division.

1. Theoreme. IF of three numbers given, the first di­vide the second and the quotient mul­tiply the third, the product shall be the fourth proportionall, to the three numbers given.

Theoreme. If of three numbers given, the second divide the first, and the quotient divide the third; this later quotient shall be the fourth proportionall, to the three numbers given.

Neither is it materiall whether of the two numbers after the first be second, or third.

2 And note that in Reciprocal proportion, that terme by which the question is made; But in Direct proportion the terme that is homogeneal thereto, is the first terme, or the Antecedent of the first ratio.

3 And therefore out of these foundations thus layd, (if you rightly conceive the nature of the Logarithmes) doth follow the finding out of the fourth proportionall by this Instrument: whereof this is the Rule.

Open the Armes of the Instrument to the distance of the first, and second number: then bring the Antecedent arme, or that which stood vpon the first number vnto the third, and so the consequent arme, keeping the same opening, will shew the fourth number sought for.

In which operation these foure things are diligently to be considered.

[Page 6] First, in constituting the places of each number in the fourth circle; whether the figures written in the spaces doe signifie Vnites, Tenns, or Hundreds, &c.

Secondly, if that arme which sheweth the fourth proportionall, doe reach beyond the line of the Radius; that then you doe account the fourth in a new circle or degree.

Thirdly, whether the fourth number sought, ought to be greater, or lesser then the third. For if a fourth number bee offered greater then the third, when it should be lesse, or lesse then the third when it should be greater; it is a signe that that number doth appertaine to a circle of ano­ther degree.

Fourthly, that looke what true distance was betweene the first and second, that the same bee supposed betweene the third and the fourth, and also on the same part.

4 And for because Multiplication and Division, have a certaine implicite proportion: we will speake of them in the first place.

5 In Multiplication, As an Vnite 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 [Page 7] exceede so many of the first figures of the product: But if it doe not exceede, it will have one lesse.

6 And in Division, As the Divisor is to an Vnite; so is the Dividend, to the Quotient.

And the Quotient shall have so many places, as the Dividend hath more then the Divisor, if the Divisor ex­ceede so many of the first figures of the Dividend: but if it doe not exceede, it shall have one place more.

7 Wherefore let this rule bee still carefully kept in minde: that In Multiplication the first terme of the impli­cite proportion is evermore 1: And in Division, the first terme is the Divisor.

And thus much concerning the operation of Propor­tion, Multiplication, and Division, I thought meete to ad­monish, least hereafter in Multiplying, or Dividing, or seeking out a fourth proportional, wee be constrained to repeate the same things many times over.

8 An example of Multiplication. How many pence are there in 47^li. 9^sh? For because 1 shilling containes 12 pence, and 1 pound containes 20 shillings, that is 240 pence: you shall multiply 47 by 240, and 9 by 12, and then adde together the products.

In the first Multiplication. 1 · 47 ∷ 240 · 11280 ·

For set the Armes of the Index at 1 and 47 in the fourth circle; and then bring the Antecedent arme (which stood at 1) vnto 240, and the Consequent arme will shew 11280.

Againe in the second Multiplication. 1 · 9 ∷ 12 · 108 ·

For set the two Armes of the Index at 1 and 9 in the fourth circle; and bring the Antecedent arme unto 12, and the Consequent arme will shew 108. Lastly, adde together 11280 and 108 and the summe 11388 will be the number of pence contained in the said summe of 47^li, 9^sh.

[Page 8] 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 Armes of the Index at 240, and 1 in the fourth circle: and then bring the Antecedent arme (which stood at 240) unto 11388; and the Consequent arme will shew 47 and almost an halfe.

But how many shillings that excesse doth containe wil appeare, if first you finde by Multiplication that 11280 pence are contained in 47^li: which subducted from 11388 there will remaine for the excesse 108 pence. After­wards by division you may seeke how many shillings are in 108 pence: the diuision is thus. 12 · 1 ∷ 108 · 9.

For set the Armes of the Index at 12 and 1: then bring the Antecedent arme (which stood at 12) unto 108; and the Consequent arme will shew 9.

9 Any Fraction given may bee reduced into Decimal parts, thus.

Set the Antecedent arme of the Index at the Denomi­nator of the Fraction given, in the fourth circle, and the Consequent arme at the Numerator, then keeping the same distance, bring the Antecedent arme unto 1, and the consequent arme will shew the decimal parts.

So [...]40/ [...] is 0⌊75. And 19/48 is 0⌊396—

CHAP. III. Now follow certaine examples of Proportion.

Example I. IF 54 Elnes of Holland bee solde for 96 shil­lings, for how many shillings was 9 elnes sold? The termes given are 54 · 96 ∷ 9 ·

According to the 2 Chap. 3 Sect. Set one of the armes of the Index at the Antecedent terme 54 in the fourth circle, and the other arme at the consequent terme 96: then keeping that distance, bring the Antecedent arme vnto 9; and the consequent arme beyond the line of the Radius will shew 16 for the fourth proportionall, 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 corne be sufficient for a company of Souldiers keeping a Fort, for 36 dayes, How many dayes will 12 bushels suffice that same num­ber of Souldiers? The termes giuen are 108 · 36 ∷ 12 ·

Set one Arme of the Index at the Antecedent terme 108 in the fourth circle, and the other Arme at the con­sequent terme 12, (being mindfull of the considerations in the 2 Chap. 3 Sect.) then keeping that same distance, [Page 10] bring the Antecedent arme vnto 36; and the consequent arme will shew 4.

Therefore 108 · 36 ∷ 12 · 4 · shall bee proportionals. And 4 is the number of dayes sought for.

Example III. There is layd vp in a Fort so much corne as will suffice for 756 Souldiers which keepe that Fort, for 196 dayes: how many Souldiers will that same corne suffice for 364 dayes?

The Proportion is reciprocall, therefore the termes given are 364 ∶ 756 ∷ 196 ·

Set one Arme of the Index at the Antecedent terme 364, and the other Arme at the consequent 196: and keeping the same distance, bring the Antecedent arme vnto 756; and the consequent arme will shew 407+: And therefore for so many Souldiers will the corne laid vp suffice for 364 dayes, or 13 moneths.

Example IIII. 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 finde that the perpendicular cutteth in the nearer Station 28 degrees 7 minutes almost: and in the further Station 21 degr. 58 min. almost: and be­tweene both the Stations, the distance was 76 feet.

The Rule of measuring heights by two Stations is con­tained in these Theoremes.

Theor. As the difference of the Tangents of the arches cut in either station, is to the distance be­tweene [Page 11] the stations; so is the Tangent of the lesser arch, to the nearer distance from the Tower. Againe

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 finde the Declination of the Sunne the 9^th day of May.

The place of the Sunne for every day, may be had nere inough out of this Table, by Adding vnto the place of the Sun in the beginning of that moneth so many degrees, as there are dayes past in that mo­neth: But if the number of degrees exceed 30, the ex­cesse is to be accounted in the Signe next following.

Wherefore the 9^th of May the place of the Sun is ♉ 20+9, that is ♉ 29: which is 59 degr. distant from the next Aequino­ctiall point.

[Page 12] These things being knowne, the Rule is delivered in this Theoreme.

Theor. As the Radius is to the sine of the sunnes distance from the next Aequinoctiall point; so is the sine of the sunnes 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 finde the Right ascension of the Sunne, the 9^th day of May.

Seeke the place of the Sunne for the day proposed in the former Table; and the Sunnes distance from the next Aequinoctial point, as in the former example.

These things being knowne, the Rule is by one of these two Theoremes.

Theor. As the Radius, is to the sine of the com­plement of the sunnes greatest declination; so is the the tangent of the sunnes distance from the next Aequinoctiall point, to the tangent of the distance of the right ascension of the sunne, from the same Aequinoctiall point. Or

Theor. As the tangent of the greatest declinati­on of the Sunne, is to the Radius; so is the tangent of the declination of the sunne for the time propo­sed, vnto the sine of the right ascension of the sunne 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 Continuall proportion, Or of Pro­gression Geometricall.

1 THE Ratio of a Progression is the quotient of the consequent terme divided by his ante­cedent. And therefore in the Instrument it is the distance taken betweene the termes 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 termes, as is shewed in Chap 1. Sect. 7.

As for example, if the Ratio 60, to 65 be proposed to be septupled.

Set the Armes of the Index at 60, and 65: and then with the same opening, bring the Antecedent arme which was at 60, vnto 1, and the consequent arme will cut 1⌊08384 in the fourth circle, and 03476 [...] in the fift cir­cle: this latter number being multiplied by 7 maketh 23444; vnto 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 termes of the Ratio out of the Canon, and to multiply them by the number given: As I have done in these examples.

Or if the Canon be wanting; you may come neerer the marke, if that first single opening of the Index being kept, and the Antecedent arme set at 1; you transferre the Antecedent Arme, unto that place which the consequent arme doth cutt; and the consequent arme will cut the same space duplicated. Then holding the consequent arme in that same place, open the Antecedent arme unto 1. and afterward with that duplicated opening, bring the Antecedent arme to the duplicated space, and the conse­quent arme will cut the space quadrupled. Thirdly, bring the Anticedent arme unto the quadrupled space and the consequent arme, keeping that duplicated opening, will cut the space sextupled. Lastly, having againe taken a single opening, bring the Antecedent arme unto the number, or space sextupled; and the consequent arme will shew the Ratio sought for septupled.

And this manner of working may bee observed for as many Multiplications as you please of any Ratio given.

[Page 15] 3. The Ratio, and first terme being given, to continue the same unto any number of termes.

Open the Armes 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 Arme vnto the first terme given, and the consequent arme will shewe the second terme: againe bring the Antecedent arme vnto the second terme found, and the consequent arme will shew the third. After that bring the Antecedent arme unto the third terme found, and the consequent arme wil shew the fourth. And in this manner you may proceed as farre 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 termes in either progression will be as followeth.

[Page 16] 4 Theor; The ratio of any former terme, in a row of concinuall proportionals, vnto any of the termes following, is aequall to the ratio of the first terme unto the second, multiplied into it selfe ac­cording to the distance of that latter terme from the former.

As for example, The Ratio 5⌊4 unto 6⌊ [...]024448 which is the third terme from it, is aequall to the ratio of 100 vn­to 108 triplicated, Or as the Cube of 100 unto the Cube of 108. Wherefore

5 The Ratio, and the first terme of the Progression being given, to finde out any other terme required.

First, multiply the ratio given into it selfe, accor­ding to the distance of the terme sought from the first terme, by the 2 sect: then say

As 1 is to that mul [...]ple found; so is the first terme, to the terme sought for.

Example, What will be the amount of 26^th, in 7 yeares by Interest upon Interest, after the rate of 20 pence in the pound?

Because 1 pound which is 20 shillings containeth 60 groates, and 20 pence conteine 5 groates, the rate of the Interest will bee 60 unto 65, Or 100 unto 108⌊ [...] × [...] But the first terme is given 26, unto which there are to be ac­quired 7 other termes in continuall proportion.

First, by the 4 Chap: and 2 Sect. Let the ratio given be septuplicated, that is multiplied sevenfold into it selfe, which will be 1⌊552. Then set the consequent arme of the Index vnto the septuplicate number 1⌊552, and open the Antecedent arme vnto 1; and keeping the same ope­ning, [Page 17] bring the Antecedent arme unto 26; and the con­sequent arme will shew 45⌊5312^li the amount sought for.

6 The Ratio, and any other terme of the Progression being given, to finde the first terme.

First multiply the Ratio given into it selfe according to the difference of the terme given from the first terme. Then say

As that multuple is unto 1; so is the terme given, unto the first terme.

Example, what summe in 7 yeares 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 arme of the Index to that septuple 1⌊7512, open the consequent arme unto 1: and keeping the same opening, bring the Arme unto 45⌊5312^li: and the consequent arme will shew 26, which was the stocke, or summe of money, from which that amount did arise.

7 The Ratio, the First terme, and any other terme of a Progression being given, to finde how many places the terme given is from the first terme.

First say, As the first terme is unto 1; so is the other terme given, unto the ratio multiplied into it selfe according to the distance of that terme from the first.

Wherefore according to 1 Chap. 7 Sect. and 3 Chap. 2 Sect. by helpe of the fift circle, see how often the ratio given, is contained in that multuple found.

Example, In how many yeares did 26^li, by Interest [Page 18] upon Interest after the rate of 100 unto 108⌊33+ + increase unto 45⌊5312^li?

First set the Antecedent arme of the Index at 26, and the consequent arme at 1: and keeping the same ope­ning bring the Antecedent arme vnto 45⌊5312, and the consequent arme will shew 1⌊7 [...]12: to which in the fift circle answereth 24333. Then because unto the conse­quent terme 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 yeares sought for.

8 The First terme, and any other terme of the Progression being given to finde the ratio of the Progression.

First say, As the first terme is unto 1: so is the other terme given, unto the ratio multiplied into it selfe according to the distance of that terme from the first.

Wherefore according to Chap. 1. Sect. 7, by helpe of the fift circle, Let the multuple found be divided by the distance of the terme from the first.

Example, 26^li by Interest upon Interest in 7 yeares amounted unto 45⌊5312: what was the ratio of the Interest compared unto 100?

First set the Antecedent arme of the Index at 26, and the consequent arme at 1: and then keeping the same opening, bring the Antecedent arme unto 45⌊5312: and the consequent arme will shew 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 terme of the Inte­rest sought for.

9 Two numbers being given to finde as many Midle proportionals betweene them as you will.

[Page 19] Divide the distance of the greater number given from the lesser in the fourth circle iustly taken, according to Chap. 1. Sect. 7, by helpe of the fift circle into equall seg­ments, one more then are the number of Midle propor­tionals sought for. All these segments added orderly to the first terme, doe distinguish the termes of the Pro­gression which you seeke.

Example, Let there be foure Midle proportionals, sought out betweene 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 Vnite [...]ne, is indeede 12989, according to Chap. 1. Sect. 7, and so is the distance iustly taken. Then subducting 9039 from 12989, there will remaine 3950: which divided by 4 + 1, the quotient will give 790. wherfore 9039 + 790, scil. 9829 in the fift circle doth agree with 9⌊6 in the fourth circle, which is the first midle proportionall. And 9039 + twise 790, scil. 10619 in the fift circle doth agree with 11⌊52 in the fourth circle, which is the second midle pro­portionall. And in this manner 13⌊824, and 16⌊5888 will be found the third and fourth midle proportionals.

10 Theor. If from the Ratio given, being multi­plied in it selfe according to the number of termes, you subduct 1, and multiply the remainder, by the antecedent of the ratio. It will be

As the difference of the termes of the ratio, is unto the product; so is the first terme, 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 terme of the Progression α: and the whole summe of the termes Z it shall be, [Page 20] D · rat. mul^ta— 1 in R ∷ α · Z.

Which is the very Theoreme it selfe expressed in Sym­boles of words: that it may more easily be fixed in the phan­tasie. Which proportion also wee must consider, doth hold both alternly, and conuersly.

This Theoreme may otherwayes be expressed, by the equality which the product of the two midle termes hath to the product of the two extreames, thus Ratio multiplicata—1 in R in α = Z in D.

This manner of setting downe Theoremes, whether they be Proportions, or Equations, by Symboles or notes of words, is most excellent, artificiall, and doctrinall. Wherefore I earnestly exhort every one, that desireth though but to looke into these noble Sciences Mathema­ticall, to accustome themselues unto it: and indeede it is easie, being most agreeable to reason, yea even to sence. And out of this working may many singular consectaries be drawne: which without this would, it may be, for ever lye hid. As in this present proportion: because it is D · rat. mul^ta—1 in [...] ∷ α · Z · wherefore Rat. mul^ta—1 in R · D ∷ Z · α · And α · Z ∷ D · rat. mul^ta—1 in R

These are exceeding easie: but this following is more difficult, and requireth attention.

In the former equation it was Rat. mul^ta—1 in R in α = Z in D

Now because Rat. mul^ta—1 in R in α, and Rat. mul^ta [...] in R in α—R in α, and Rat. mul^ta in R—R in α, [Page 21] and Rat. mul^ta in α—α in R, are equall one to another, and also to Z in D, these aequa­tions shall also be consectarious. [...] and [...] and [...] and [...]

And besides these many more. The practise 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 terme, and the number of termes being given, to finde the summe of the whole progression.

For this operation the rule, or Theoreme last before serveth: for by it [...]

[Page 22] Example. If an Annuity of 5^li, be detained 7 yeares, what will be the amount thereof by interest upon interest after the rate of 100 unto 108?

Now because the Amount is the summe of the Progres­sion, whereof the first terme is the annuitie, Multiply the ra­tio into it selfe according to the number of yeares, by Chap. 4. Sect. 2, and it will be 1⌊71382: from which if you subduct an vnite, there remaines 0⌊71382, which multi­plied by 100 maketh 71⌊382: then set the Armes of the Index at 1 and 71⌊382; and bring the Antecedent arme which stood at 1, unto 5 the first terme: and the conse­quent arme will cut 356⌊915 (that is Rat. mul^ta—1 in R in α) and lastly this number being divided by 108—100, scil. by 8, the quotient will give 44⌊6 [...]40, for the summe of the whole progression: and so much is the amount sought for.

12. The Ratio, the Number of Termes, and the Summe of a Progression be­ing given, to find the first terme.

By the converse of the foregoing Theoreme, it is ma­nifest, that Rat ∶ mul^ta—1 in R · D ∷ Z · α ·

The declaration in words was in the 10 Sect.

Example. If an Annuitie detained 7 yeares by Inte­rest upon Interest, after the rate of 100 unto 108, did increase unto 44⌊6140^li. how much was that Annuitie?

Multiply the Ratio into it selfe according to the num­ber of yeares (per cap. 4. sect. 2.) and the product will be 1⌊7 [...]82: from which if you subduct an Vnite, there re­maines 0⌊7 [...]82: this being multipled by 100 doth make 71⌊ [...]82. Then say 71⌊382 · 8 ∷ 44⌊6140 · 5 · [Page 23] for the First terme: which was the Annuitie sought for.

13 The Ratio, the First terme, and the Summe of the Progression being gi­ven, to find the Number of termes.

By the Theoreme in the 10. Sect. it was α · Z ∷ D · rat. mul^ta—1 in R ·

Wherefore set the Antecedent arme of the Index unto the Antecedent terme of the ratio, and the consequent arme unto the Summe of the Progression: and with that same opening, bring the Antecedent arme unto the diffe­rence of the termes of the ratio; and the consequent arme wil shew a number (that is Rat. mul.—1 in R;) which if you divide by the Antecedent terme of the ratio, and unto the quotient adde an Vnite, you shall have the ratio mul­tiplied into it selfe according to the number of termes. Therefore taking the distance betweene the termes of the ratio, with the armes of the Index, measure by helpe 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 termes of the progression.

Example. If an Annuitie of 5^li detained by Interest up­on interest after the rate of 100 to 108, increased unto 44⌊6140^li. How many yeares was the Annuitie detained?

Set the Antecedent arme of the Index at 5, and the con­sequent arme at 44⌊6140, and with the same opening bring the Antecedent arme unto 8, and the consequent arme will shew 71⌊382 (that is Rat. mul^ta—1 in R:) this number being divided by 100, will be 0⌊71382: and if unto the quotient you adde 1, you shall have 1⌊71 [...]82 (the ra­tio multiplied into it selfe according to the number of yeares:) unto which in the fift circle answereth 2338: [Page 24] 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 yeares sought for.

Or such questions may be more easily performed by this Theoreme, which the industrious Reader may by himselfe practise. [...]

14 Theor. If the summe of the whole progressi­on be divided by the ratio multiplied into it selfe according to the number of termes, the quotient will be the first terme; and that summe given will be the last terme, of another progression, having the same ratio but one terme more.

15 And because a summe of mony the amount where­of in any number of yeares given, by Interest upon inte­rest, doth equall an Annuitie so long detained, is equiva­lent to the same Annuitie; and the amount of an Annuitie is the summe of a Progression continued from that Annui­tie. If therefore an Annuitie for any number of yeares be divided by the ratio multiplied into it selfe according to the number of yeares; the quotient will be the just price of an Annuitie to endure for so long. And because by the 10 and 11 Sect. it hath beene shewed that [...].

Therefore by the 14 Sect. [...] will be equall to the price of an Annuitie in ready money, which shall be the Rule for the operation following.

[Page 25] Wherefore also Rat. mul^ta—1 in R in α = Rat. mul^ta in D in Pret. which proportion is thus enuntiated in words.

Theor. If from the Ratio multiplied into it selfe according to the Number of yeares you subduct an Vnite, and the remainder be multiplied continu­ally by the Antecedent of the Ratio, and the An­nuitie it selfe: And againe, If the Ratio multiplied into it selfe, according to the number of yeares be multiplied continually by the difference of the termes of the Ratio, and by the Price: both those products will be equall.

Example. An Annuitie of 5^li, to endure for 7 yeares, is to be sold: what is it worth in ready money, after the Rate of 100 unto 108?

The Ratio multiplied into it selfe according to the number of yeares (per Cap. 4 Sect. 2) is 1⌊7138: subduct 1, and there will remaine 0⌊71382: which multiplied by 100 maketh 71⌊382. Then set the Antecedent arme of the Index at 1, and the consequent arme 71⌊382: and keep­ing that same opening, bring the Antecedent arme unto 5; and the consequent arme will shew 356⌊915; keepe this (for it is [...]. After that set the Ante­cedent arme of the Index at 1, and the consequent arme unto the multuple ratio 1⌊7138; and with the same open­ing bring the Antecedent arme unto 8; and the conse­quent arme will cut 13⌊7106, keepe this number also (for it is Rat. mul. in α). Lastly, place the Antecedent arme of the Index at 13⌊7106, and the consequent arme at 1; and with the same opening bring the Antecedent arme vnto 356⌊015: and the consequent arme will shew 26⌊032^li. [Page 26] which is the iust Price of an Annuitie of 5^li in readie money.

16 And by the last praecedent Theoreme or Rule, also [...] which Theoreme may bee enuntiated in words as was there shewed.

Example. An Annuitie for 7 yeares is bought for 26⌊032^li. after the rate of 100 unto 108, by Interest upon interest: how much was that Annuitie?

The Ratio multiplied into it selfe for the number of 7 yeares (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) it selfe lesse by an Vnite, and multiplied by 100: and the quotient will be 5^li, the Annuitie 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 Annuitie be divided by it selfe, being diminished by the product of the dif­ference of the termes of the ratio multiplied by the Price: the quotient will bee equall to the ratio multiplied into if selfe according to the number of yeares. As

If the ratio be 100 unto 108; the Annuitie 5^li: and the price thereof 26⌊032^li. Set the Antecedent arme of [Page 27] the Index at 1, and the consequent arme at 8, the diffe­rence of the termes of the ratio: and with the same ope­ning bring the Antecedent arme unto 26⌊032—: and the consequent arme will shew 208⌊256: which subducted from 500, there will remaine 291⌊744, for the divisor. Set therefore the Antecedent arme of the Index at the divisor 291⌊744, and the consequent arme at 1: and with the same opening, bring the Antecedent arme unto the dividend 500: and the consequent arme will shew 1⌊71 [...]8, which is the ratio multiplied into it selfe according to the number of yeares. And by this number so found it will be easie (by helpe of the fift circle) the ratio of the Inte­rest being given, to finde the continuance of the Annuitie.

Example. An Annuitie of 5^il, was bought for 26⌊032^li—, after the rate of 100 unto 108: how many yeares is it to last?

First seeke out the ratio multiplied into it selfe accord­ing to the number of yeares, which will be 1⌊71 [...]8, accord­ing as was even now shewed 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 yeares sought for.

CHAP. V. Of the Quadrating, and Cubing of Numbers, Sides or Rootes: and of the Extraction of the Quadrate, and Cubic side, or roote, out of Numbers, or Powers given.

1 IF a number, side, or roote be multiplied into it selfe; the product will be a Quadrat. And if a quadrate bee multiplied into his owne side, or roote, 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 arme of the Index at 1, and the conse­quent arme at the number given: then with the some opening bring the Antecedent arme to the number given; and the con­sequent arme will shew the Quadrat thereof.

And the number of figures in a Quadrat of a single root (or which doth not exceede 9) is easily found out by those Rules, that have beene before delivered concerning Multiplication. But if a side or root consist of more fi­gures then 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 arme of the Index at 1, and the consequent arme at 7; and with that opening [Page 29] bring the Antecedent arme unto 7; and the consequent arme will shew 49 which is the Quadrat sought for.

Example 11. The Quadrat of the side 57 is required.

Set the Antecedent arme of the Index at 1; and the consequent arme at 57: then with that same opening bring the antecedent arme unto 57; and the consequent arme will shew 3249, which is the Quadrat sought for, consisting of foure 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 bee 324900, the quadrat sought for.

Example 1111. The quadrat of the side 574 is re­quired.

Set the Antecedent arme of the Index at 1, and the cor­sequent arme at 574; and with the same opening bring the Antecedent arme unto 574; and the consequent arme will shew 329476, the quadrat sought for consisting of sixe 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 arme of the Index at 1, and the conse­quent arme at the number given; and with that same opening, bring the Antecedent arme unto the number given; and the consequent arme will shew the Quadrat; then bring the An­tecedent arme unto the Quadrat, and the consequent arme with that same opening will shew the Cube of that side given.

The number figures in a Cube of a single side, or root which doth not exceede 9, is easily found by that which hath beene before delivered concerning Multiplication: [Page 30] But if the side, or root consist of more figures then 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 · againe 1 · 7 ∷ 49 · 343 ·

Set therefore the Antecedent arme of the Index at 1, and the consequent arme at 7; and with that opening bring the Antecedent arme unto 7, and the consequent arme will shew 49 the quadrat thereof: Then set the Antecedent arme at 49; and the consequent arme (with that first opening) will shew 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 arme of the Index at 1, and the consequent arme at 57; and with that same opening bring the Antecedent arme unto 57; and the consequent arme will shew the quadrat 3249. Then set the Antecedent arme at 3249; and the consequent arme keeping the for­mer opening will shew 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.

Hauing 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 needlesse to set downe.

[Page 31] 4 The Extraction of the Square, or Quadrat root, or side, is done by helpe 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 halfe: then set the In­dex at that halfe; and it will shew in the fourth circle, the side, or root sought for.

But you must know that if the number which is the Quadrat proposed, have onely 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 lesse then the true number if any thing remaine.

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; indeede 1. 6902 having the gradu­all number 1 praefixed, because 1 in the fourth circle sig­nifieth 10, one circuition thereof being finished: the halfe 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; indeede 3. 5118 prefixing the gradual number 3, because 1 in the fourth circle signifieth 1000, three circuitions thereof being finished: the halfe whereof is 1. 7559. Then set the Index at 7559, omit­ting the prefixed graduall number 1; and it will shew in the fourth circle 57 the side sought for, consisting of two figures, because 1 in the fourth circle signifieth 10.

[Page 32] 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; indeede 5. 5178 prefixing the graduall number 5, because 1 in the fourth circle signifieth 100000, five circuitions thereof being past over; the halfe whereof is 2. 7589; Then set the Index at 7589, omit­ting the graduall number 2 prefixed thereto; and it will shew in the fourth circle 574 the side, or root sought for, consisting of three figures, because 1 in the fourth circle doth signifie 100.

5 The Extraction of the Cubic roote, or side is done by helpe 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 shew in the fourth circle the side, or roote sought for.

And you must know that if the Cube proposed have onely three places of Integers, the side, or roote 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 lesse then the true number if any thing re­maine.

Example I. The side, or roote of the Cube 343 is re­quired.

Set the Index at 343 in the fourth circle, and it will cut in the fift circle 5353; indeede 2. 5353 prefixing the graduall number 2, because 1 in the fourth doth signifie 100, two circuitions thereof being past over: the third part whereof is 8451. Then set the Index at 8451, and it will shew in the fourth circle 7, the side sought for.

[Page 33] Example II. The side, or roote of the Cube 185193 is required.

Set the Index at 185193 in the fourth circle; and it will cut in the fift circle 2677; indeede 5. 2677 prefixing the graduall number 5, because 1 in the fourth circle doth signifie 100000, five circuitions thereof being past over: the third part whereof is 1. 7599. Then set the Index at 7599, omitting the prefixed graduall number 1; and in the fourth circle it will shew 57, the side, or root sought for, consisting of two figures, because 1 in the fourth circle doth signifie 10.

Examples of greater Cubes it will be needlesse to set downe.

CHAP. VI. Of Duplicated, and Triplicated pro­portion. And first of Duplicated proportion.

Theoreme. LIke Plaines 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, doe appertaine vnto 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 againe 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 plaines, the length of the greater, is 40 feete, 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 plaines 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 plaine.

Example II. How many Acres of woodland measured with a Perch, of 18 feete, are there in 73 Acres of cham­pane land, measured with a Perch of 16½ feete?

[Page 35] The measures given (18, 16½) being reduced into their lest termes, are as 12 unto 11: and the quadrats of these numbers, are, 144, and 121. And the Proportion is Re­ciprocall. Say therefore 144 (Q ∶ 12) · 121 (Q ∶ 11) ∷ 73 · and so many are the Acres of Wood land.

CHAP. VII. Concerning the Measuring of Circles, Cones, Cylinders, and Sphaeres.

1 ARchimedes in a peculiar Treatise found the proportion of the Diameter of a circle to the Circumference to bee a very smale deale greater then o [...] 7 unto 22: And of late Ludolph Ʋan Ceulen insisting in the same steps of Archimedes, hath more precisely found it to be of 1 vnto 3⌊141592653589793 but for our Instru­ment it will be sufficient to take the ratio of 1 vnto 3⌊1416, Or of 0⌊3183+ unto 1: leaving the diligent practizer, to more exactnesse, if he please to use his Pen.

And note that the Rules following are set downe in proportions, to be wrought as hath been taught in Chap. 2, Sect. 3. Wherein

• D, or Diam. signifieth the Diameter.
• Dq, or Q. Diam. the Quadrat of the Diameter.
• Dc, or C. Diam. the Cube of the Diameter.
• R, or Rad. the Radius, or Semidiameter.
• P, or Perif. the Periferie, or Circumference.
• Pq, the Quadrat of the Periferie.
• Long. the length.
• L the side, or latus
• Alt. the altitude.
• [...], sheweth that the two magnitudes betweene which it is set, are to be multiplied together.

CHAP. VIII. Concerning Plaine, and Solide Measures.

1 THe diuiding of the Carpenters ruler into Inches, and halfe, and quarters, and halfe quarters of Inches, that is of e­very Inch into eight parts, is most inartificiall, and vnfit for measuring, by reason of the manifold denominations, which must be brought into one, and is hard to bee done of them that are vnskilfull. I would wish therefore that euery Inch were diuided into 10 parts, or rather that the foot were diuided into 100 parts, which is best of all: for then there will neede no reduction. And all other di­visions must bee reduced vnto this, by these Rules following.

2. If the measures be taken vpon a Ruler diuided into Inches and tenth parts of an Inch, first take out all the whole feet, and then diuide the Inches remaining, with their decimall parts if there be any by 12.

Example. How many feet and decimall parts of a foot, are in Inches 17⌊3?

First take out the whole foot which is 12 Inches, and there will remaine Inches 5⌊3: which being diuided by 12, you shall haue 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 bee taken vpon a Ruler diuided into inches and halfe quarters, that is each inch into 8 parts, First you must reduce the eight parts into decimals of Inches, by diuiding the number of parts given by 8 the Denominator thereof: and afterward by the former [Page 43] Rule, reduce the Inches, and decimall parts, into deci­mall parts of a foot.

Example. How many feet, and decimall parts of a foot, are in 7 inches, and 5 eight parts?

First diuide the 5 eight parts by 8, and you shall haue 625 thousand parts: which being put to 7 inches, will make inches 7⌊625: Again diuide these by 12, as was shew­ed in the former rule: and the whole measure will be feete 0⌊635. And contrariwise feet 0⌊635, will be reduced in­to inches 7⌊625, being multiplied by 12.

4. I must aduise all those that haue occasion, to mea­sure Plaines, or Solids, to make themselues very perfect in this kind of Reduction (because most Rulers they shall ordinarily meet withall, are diuided into inches, and halfe quarters) which will be very easie to them, if they doe but remember, that In diuision the first terme of the proportion implyed, is the Diuisor it selfe: but in Multipli­cation, the first terme is euermore 1. as hath beene shewed in Chap: 2, Sect: 7. And therefore, presuming on the diligence of the Practiser herein, I shall not neede in this kind of measuring, to speake any more of inches, but of feet and decimall parts of feet, as if the Ruler were so diuided.

CHAP. IX. Concerning the Measuring, or Gauging of Vessells.

AWine, or Beere vessell, whether Pipe, Hogshead, Barrell, Kilderkin, or Pirkin, and such like, is in forme of a Sphaeroides, hauing the two ends equally cut off: and accordingly may be measured thus.

Measure the two diameters of the Vessell, in inches, or else in tenth parts of a foote, the one at the bung hole, the other at the head, and also the length within. And by the diameters found, finde out the circles; then adde together two third parts, of the greater circle, and one third part of the lesse: Lastly, multiply the aggregate by the length: so shall you haue the content of the Vessell, either in Cubic inches, or cubic tenth parts of a foot.

Example. Suppose a vessell, 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 euermore, 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 giueth 24839⌊56 cubic inches for the whole content of that vessell.

[Page 53] 2 Mr. Edm: Gunter in his second booke of the Crosse Staffe, Chap. 4. pretending to shew the manner of gau­ging Wine vessels, beginneth with these words.

‘The Vessels which are heere measured, are supposed to bee Cylin­ders, or reduced into Cylinders by taking the meane, betweene the Di­ameter at the head, and the Diameter at the boungue, after the usuall manner.’

And according to this supposition, teacheth to finde a Gauge point, For a gallon of wine, in that his imagined Cylindriacall vessell.

Because his words are cautelous, and subterfugious, wee must a little examine them; for if his way bee true, my Rule before set downe, though grounded vpon de­monstration, cannot stand.

Well then, that reduction of a wine Vessell into a Cy­linder, is either true, or false; if it bee true what neede those ambiguities of Vessels which are here measured: and are supposed to be, &c. and after the vsuall manner? if false why is it not noted, but deliuered as a Rule to confirme an er­ror. And what meaneth, the meane betweene the Diame­ter at the head, and the Diameter at the boungue? is it the meane Geometricall, or Arithmeticall, that is the meane pro­portionall, or that which equally differeth from both? such shifting is vnworthy an Artist.

First therefore, Let it be the meane in respect of difference, which is equall to halfe the summe of the two Diameters: I say that the Vessell cannot truely be reduced to a Cylin­der by such a meane Diameter. For seeing it is most ap­parent that such a Vessell, is greater in the middle then at the ends, the boords, or sides thereof, shall from the middle to the ends, goe either streight, and so the Vessel shall be as it were, two equall segments of Cones, set base to base: Or else arching, and so the Vessel shall (as be­fore I sayd, and is commonly taken for a truth) bee a Sphaeroides, hauing the two ends equally cut of.

[Page 54] If it be considered as two segments of Cones: the measure by that meane Diameter, or midle Section is quite false, as hath beene demonstrated in the former Chapter, Sect: 16, 17. and will be giuen lesse then the true con­tent, although the sides goe straight: much more then, if the sides goe arching; for that conuexnesse, must needs yeeld a greater capacity. And therefore in neither can that manner of Gauging be true.

Againe, if the meane Diameter be vnderstood to be the meane proportionall betweene the two Diameters, it is much more false, for betweene any two numbers, the meane Ge­ometricall is lesse then the meane Arithmeticall.

Thus much I haue thought good in this place ingenu­ously to signifie to the inexpert Learner, that hee might not beguile himselfe with a prejudged opinion.

3. I haue shewed the measuring of vessels by the cubic inch: but our vsuall reckoning is by the Gallon, and parts thereof. Wee must therefore doe the best we can, to in­quire the true quantitie of a Gallon in inch measure, which will bee difficult to doe exactly, both because the Stan­dards vsually are not streight sided, but a little arching, neither doe they agree perfectly one with another: but what partly by experience, both mine owne, and others, which hath come to my sight; and partly by reasoning shall seeme to me most probable, I will not refuse to set downe.

4 Our English Gallon is vnderstood to bee 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 my selfe haue measured Bushels, and P [...]cks, which haue exactly beene fitted to the Standards, and haue still in my account found a Gallon to containe better then 270 [Page 55] cubic inches, indeed much about 272, or 273 as pre­cisely as I could measure in a vessell not truely regular. Also my worthy friend Master William Twine, who hath vndergone great paines and charge, in finding out the true content of our English measures, gaue vnto me two se­verall measures of an Ale-gallon, and those in due conside­ration but little differing. The one was found out by a brassen vessell made in manner of a Parallelepipedon, the base whereof was exactly sixe inches square, and the sides diuided into inches and twentieth parts: into which vessell he powring out a standard Gallon of Queene Eliza­beth, filled with water, found it therein to ari [...]e vnto 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 forme 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 bottome was inches 5⌊ [...]: and the height of it was in­ches 9⌊8: which being cast vp by Chap: 8, Sect: 15, will be found to containe 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 himselfe, but to giue me sa­tisfaction, shewed me the experience in the said Vessell and Standard: but the truth is, I obserued the Standard, be­sides the arching of the sides, to bee not exactly circular within, nor the brimme of an even height, nor the bot­tome plaine: and in taking the height of the water in the Vessell, our sight was not able to estimate the ascent thereof so precisely, that a spoonefull of water, more or lesse, could breed any sensible difference.

What therefore shall wee doe in this difficultie? in­deede looke to the first ground, and principle of our Eng­lish measuring, from Barley cornes. For the length of [Page 56] 3 Barley cornes taken out of the middle of the eare is an Inch, or Ʋncia, that is a twelfth part of a foot. 3 feete make a Yard: and 16 feet and an halfe, that is 5 yards and an halfe, a Perch, with which wee measure our land; for 40 perches is a Furlong, and 8 furlongs an English mile: and againe 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 feete 272¼, is as it were the beginning of all land measure in the superfi­ciall content.

Now therefore seeing in Vessels a gallon is as it were the beginning of Ʋessell measure (for a Pottell is but a dimi­natiue of it, and a quart, the quarter) it is not vnlikely that our wise Ancestors had such a consideration also in solide measures, that as a square Perch (the beginning of Superficiall land measure) did containe 372 square feete and a quarter; so a Gallon (the beginning of vessell mea­sure) should containe 272 Cubic inches and a quarter. And the rather seeing that the ancient Geographers, diuide a foote into 4 Palmes, a palme being 3 of our inches, as 3 feete are a yard. So that as the side of a square Perch consisteth of yards 5½, a Gallon also should consist, of a number of Cubic inches the square side whereof is palmes 5½.

Wherefore sauing the exact truth when it shall ap­peare, and in the meane time the more probable reasons of other men, I make bold to tender this my coniecture, to the censures of more diligent Inquirers, That the measure of an English Ale gallon should be a square Vessell of inch 16½, or Palmes 5½ every way, and 1 inch deepe: that is 272⌊2 [...] Cubic inches.

5. And this my opinion may peraduenture receiue some confirmation by the inquiry of an English wiue Gallon.

[Page 57] M. Henry Briggs that learned Geometrician, and my very louing friend, made an experiment, with a Cubicall vessell, which was 12 inches euery way, which hauing filled with water carefully measured, found it to containe 7 gallons and an halfe wanting a moment, as hee himselfe long since, being then of Gresham Colledge, signified to me. Now if it had contained exactly 7 gallons, and an halfe, a wine gallon should haue beene 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 containe 231 cubic inches.

Againe M. Gunter in the place before mentioned shew­eth, that the common opinion is that at London, a Cylin­driacall vessell, whose diameter is 38 inches, and length 66 inches, doth containe 324 gallons: wherefore by this account a gallon should bee 231 cubic inches almost ex­actly, which in both so neerely agreeing, wee may well conclude, That an English wine gallon doth containe 231 cu­bic inches.

It is also commonly receiued, that the reason of the greatnesse of an Ale gallon aboue the Wine gallon is, that be­cause of the frothing of the Ale or Beere, the quantity becommeth lesse, and therefore such liquors that did not so yeeld froth, as Wine, Oyle, and the like, should in rea­son haue a lesser measure. If then we compare these two gallons together, we shall finde that 272⌊25 · 231 ∷ 16⌊5 · 14 · which abatement might to our Ancestors, in apportio­ning those measures, seeme to be reasonable.

[Page 58] 6 To finde how many Cubic tenth parts of a foot are in a gallon, both of beere, and Wine: or also in any number of Cubic inches.

Because there be in a Cubic foote, 1000 Cubic tenth parts, and 1728 Cubic inches, say, 1728 (C ∶ 12) · 1000 (C ∶ 10) ∷ 272⌊25 · 157⌊ [...]521— cubic tenth parts of a Beere 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 vessell measured in Sect. 1.

7 And according to these measures so found, you may easily finde the content of a pint, or quart, or peck, or bushell, which two last are to be reckoned in Ale measure.

8 The Content of a Vessell, being giuen in cubic inches, or in cubic tenth parts of a foot, to finde how many gallons it con­taineth.

This is easily done if you diuide the content giuen in inches, by 272⌊ [...]5 for Ale measure: and by 231, for Wine measure. But if the content be giuen in decimall parts of a foote diuide it by 157⌊5521—for Ale measure; and by 133⌊6803—for Wine measure.

Example. How many wine gallons are in a vessell con­taining 24839⌊56 cubic inches, or 14374⌊746 cubic tenth parts of a foote. Diuide 24839⌊56 by 231, or diuide 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 foure pieces of Metalls, whereof the third is of the same kinde with the first, and the fourth of the same kind with the second, are proportionall, their grauities also, or weights, shall be proportionall.

2. If there bee foure pieces of metall, whereof the third is of the same kind with the first, and the fourth of the same kinde with the second: and the first and second be of equall greatnesse, and the third and fourth of equall weight; the weights of the first and second, shall be reci­procall to the magnitudes of the third and fourth.

3. Two Sphaeres of the same matter are in weight, as the cubes of their Diameters, are in magnitude. Et contra.

4. Pieces of Metall if they bee of equall magnitude, haue their weights in direct proportion, as is here set downe: but if they be of equall weight, they haue their magnitudes in proportion reciprocall: According to the experiments of Marinus Ghetaldi, in his tractate called Archimedes promotus.

5. To finde the Weight of a Sphaere of Tinne, hauing the Diameter 1 Inch, or else [...] tenth part of a foot.

Take a piece of Tinne, and turne it exactly in a Lathe, into a Cylinder, hauing both the Diameters of its base, and also the length equall, to the Diameter of the Sphere giuen. Then weight that Cylinder, that you may haue [Page 60] the weight thereof in graines: And lastly, take two third parts of the whole number of graines: for the weight of the Sphaere.

After this manner Marinus Ghetaldi found a Cylin­der of 1 inch, or twelfth part of a foot thick, and long to weigh 1824 graines: whereof [...] is 1216, the weight of a Sphaere of that thicknesse.

Againe if you say, 1000 (C ∶ 10) · 1728 (C ∶ 12) ∷ 1216 · 2101⌊248

You shall haue the weight of a Sphaere whose Diame­ter is one tenth part of a foot.

Wherefore also a Cubed inch of Tinne weigheth 2322⌊4—, and a Cubed tenth part of a foote weigheth 4013⌊1—

And note that Mar: Ghetaldi vseth the ancient Ro­man foot, which by the measure set downe in his booke seemeth to be very little lesse, then our-vsuall English foot, if not exactly the same.

Note also that he diuideth one pound, into 12 ounces, and euery ounce into 24 Scruples, and euery Scruple into 24 graines: So that an ounce with him weigheth 576 graines: and a pound 6912. Whereas our English pound of Iroy weight by Assize, or Gold smiths weight, is but 5760 graines, and our ounce 480. But whether the Raguzean graine, be the same with our English, I leaue to be tryed by the diligent Practizer.

6 To finde the Weight of a Sphaere of Tinne, at any other Diameter assigned.

Multiply the Cube of the Diameter giuen by 1216, if it be in inch measure; or by 2101⌊248, if the measure be by decimall parts of a foot: and the product will be the weight of that Sphaere.

And contrariwise to find the Diameter of a Sphaere of Tinne, by the weight giuen in graines. Diuide the weight giuen in graines by 1216 if you would haue inch [Page 61] measure: or by 2101⌊248 if you would measure by deci­mall parts of a foot, and the quotient shall be the Cube of the Diameter.

7. To finde the Weight of a Sphaere of any Mettall, at any Diameter giuen, either in Inch measure, or in decimall parts of a foot.

First by Sect. 6, seeke the weight of a Sphaere of Tinne, at that Diameter: then by Sect. 4, say, As the proportionall number of Tinne, is to the proportionall number of that other Metall: so is the weight of the Sphaere of Tinne now found, to the weight of the Sphaere proposed.

Example. Suppose a Sphaere of Iron, whose Diameter is 3 inches; what shall be the weight thereof?

First, the weight of a Sphaere of Tinne, of 3 inches Dia­meter, will be found to be 32832 graines. Then say, 1554 · 1680 ∷ 32832 · 35494⌊054 graines the weight of the Sphaere proposed.

8 To finde the Diameter of a Sphaere of any Me­tall in inch measure, or decimall parts of a foot, the weight thereof being giuen.

First by the contrary of Sect. 6, seeke the Cube of the Diameter of a Sphere of Tinne of that weight. Then by Sect. 4, say reciprocally. As the proportionall number of that other Metall, is to the proportionall number of Tinne: so is the Cube of the Diameter now found, to the Cube of the Diame­ter of the Sphaere proposed.

Example. A Sphaere of Iron weigheth 35494⌊0 [...]4 grains: how many inches is the Diameter thereof?

First, the Cube of the Diameter of a Sphere of Tinne of 35494⌊054 graines weight, will be 29⌊18910695, then say, reciprocally, 1680 · 1554 ∷ 29⌊18910691 · 27—

The Cubic root whereof is 3, the Diameter of a Sphaere of Iron of that weight proposed.

CHAP. XI. Concerning the Ordering of Soldiers, in any kinde of rectangular forme of battaile.

1 BAttailes are considered either in respect of the number of men, or in respect of the forme of ground. As asquare battaile of men is that which hath an equall num­ber of men, both in Rank and File, though the ground on which they stand, bee longer on the File, then on the Ranke. And a square battaile of ground is that which hath the Ranke as long as the F [...]le, though the men in Ranke be more then in File.

2. In respect of the number of men, it is called either asquare battaile, or a double battaile, or a battaile of the grand front, which is quadruple, or a battaile of any proportion, of the number in Ranke, to the number in File.

3. If it bee asquare battaile of men: Extract the qua­drat root out of the whole number of men, and the same shall be the number of Souldiers, to be set in a Ranke.

Example. 576 Souldiers are to bee martialled in a square battaile, that so many may be in Ranke, as in File.

Take the quadrat root of 576, which is 24: the same shall be the number to be placed in a Ranke.

4. If it be a double battaile of man: Extract the qua­drat root out of halfe the number of men, and the same doubled shall bee the number of Souldiers to bee set in a Ranke.

Example. 1458 Souldiers, are to be placed in a double battaile; so that twise so many may be in Rank as in File.

[Page 63] Take halfe the given number 1458, which is 729, the quadrat root whereof is 27: double it and you shall haue 54 men to be placed in a ranke.

5. If it be a quadruple battaile, 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 Souldiers to be set in a ranke.

Example. 1024 Souldiers are to bee martialled into a battaile of the grand front, so that fower times so many be in ranke 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 haue 64 men to be placed in a ranke.

6. If a battaile bee required of any other forme, that is, if a Ratio be given, according to which the number of men in Ranke, shall be to the number in file. Multiply the two termes of the Ratio given: Then say As the pro­duct is to the quadrat of the terme which is for the ranke, or As the terme which is for the file, is to the terme which is for the ranke: so is the whole number of Souldiers, to the quadrat of the number of men to be placed in a ranke.

Example. 1944 Souldiers, are to be martialled so, that the number of the ranke, be to the number of the file, as 8 vnto 3, that is for 8 men in ranke, 3 are to be set in file.

First multiply the two termes of the Ratio 8 and 3, the product whereof is 24, also quadrat 8, the terme of the ranke, which will be 64. Then say, 3 · 8 ∷ 1944 · 5184 · out of which extract the quadrat root 72, and it will giue you the true number of the ranke.

7. In respect of the forme of ground, the battaile is [Page 64] either a square of ground, or longer one way then the other. For the distance, or order of Souldiers martialled in array, is distinguished either into Open order, or Order.

Open order is when the very centers of their places, are distant 7 feet asunder, both in ranke and file.

Order is when the centers of their places, are distant 3 feet and a halfe in ranke, and so much in file. Or else 3 feet and a halfe in ranke, and 7 feet in file: which last order, and whatsoeuer order else there is, in which the distance of the rankes one from another is greater, then 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 ranke.

8 If it be a square battaile of ground, the centers of the distances being feet 3½ in ranke, and 7 feet in file. Because 3½ is halfe of 7, the ratio of the distances, is as 1 unto 2. And seeing the number in ranke, to the number in file, is reciprocall to the distances, the ratio of the number of men in ranke, 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 terme of the file, is to the terme of the ranke; so is the whole number of Souldiers, to the true number of the ranke.

Example, 1352 Souldiers, are to bee set in a square of ground, that their distances may be feet 3½ in ranke, and 7 feet in file.

The Ratio of the ranke to the file, shall reciprocally be, as 7 to 3½, that is as 2 to 1. Say therefore 1 · 2 ∷ 1352 · 2704, the quadrat roote whereof 52 is the number of men to be set in a ranke.

9 If a battaile wherein the distance in ranke is vnequall to that in file, be longer one way then the other, accor­ding [Page 65] to any Ratio giuen: there is to be considered a dou­ble ratio, one reciprocall in respect of the distances, the other according to the forme of the ground. Wherefore to finde the Ratio of men in rank, to the men in file, Mul­tiply the two termes of the ranke, for the ranke, and the two termes of the file, for the file. And then the Rule shall bee the same with that in Sect. 6, namely, As the terme of the file, is to the terme of the ranke: so is the number of Souldiers, to the quadrat of the true number of the ranke.

Example. 10290 souldiers, are to be set in a battaile, so that they may stand onely 3 feet asunder in ranke, and 7 feet in file, and the length of the ground for the ranke, to the length of the ground for the file, shall haue the ra­tio of 5 vnto 2.

First in respect of the distances, the Ratio of Rank to file, reciprocally is as 7 vnto 3. Secondly, in respect of the ground, the ratio of rank, to file, is as 5 to 2. Where­fore by multiplication of like termes, 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 ranke.

10 If 1000 Souldiers, may be lodged in a square, of 300 feete, how many feete must the side of a square be, which will serue to lodge 5000? Say, 1000 · 5000 ∷ 300 × 300 · 450000 · the quadrat roote 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 necessarie Astronomicall operations,

1 BEfore wee deliuer the Rules of such operations, it will not be inconuenient, to set downe certaine Reductions, wher­of we may haue frequent vse.

To reduce sexagesime parts into decimals. Diuide the sexagesimes giuen 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 Againe 60 · 1 ∷ 34⌊′ · 0⌊57° Wherefore 34′, 12″ are equall to 0⌊57 of a degree.

And contrariwise to reduce decimall parts of degrees in sexagesimes. Multiply the decimall part giuen by 60.

Example. How many sexagesime parts are 0⌊57°? 1 · 60 ∷ 0⌊57° · 34⌊ [...]′ Againe 1 · 60 ∷ 0⌊2 · 12″

To reduce houres into degrees. Multiply the houres with their decimall parts by 15.

[Page 67] 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 Houres 8, 34′, 12″ doe containe 128⌊55 degrees.

And contrariwise to reduce degrees into houres. Diuide the degrees with their decimall parts by 15.

Example. How many houres are in degrees 128⌊55? 15 · 1 ∷ 128⌊55 · 8⌊57

2 It is to be vnderstood, that if foure numbers are proportionall, their Order may be so transposed, that each of those termes, may bee 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.
• IIII. As the fourth is to the third; so is the second to the first.

Wherfore euery proportion doth implicitly containe foure Orders, two descending, and two ascending, as may be seene by their combinations: By one of which orders, if of foure proportionall numbers, any three be giuen, that other which is vnknowne, may be found out.

[Page 68] Example. To finde out any of these,

If the first, second, and third termes be giuen, the fourth shall be found out by the I order.

If the first, third, and fourth termes be giuen, the se­cond shall be found out by the II order.

If the first, second, and fourth termes bee giuen, the third shall be found out by the III order.

If the second, third, and fourth termes be giuen, the first shall be found out by the IIII order.

3. To finde out any one of these.

[Page 69] 4. To finde out any one of these.

5. To finde out any one of these.

6. To finde out any one of these.

[Page 70] 7. To finde out any one of these.

8. To finde out any one of these.

9. To find out any one of these.

[Page 71] Or also of these.

10. To finde out any one of these.

11. To finde out any one of these.

[Page 72] 12. To finde out any one of these.

13. To finde out any one of these.

14. To finde out any one of these.

[Page 37] 15. To find out any one of these.

16. To find out any one of these.

17 The hower of the Sunnes Rising, and setting is found out by the Ascensionall difference. For if you reduce the degrees of the Ascensionall difference, into howers, it will shew you how much the Sunne riseth, or setteth be­fore, or after 6 a Clock.

18. The Oblique ascension also of the Sunne is found out by the Ascensionall diff [...]rence. For if you subduct the Sunnes Ascensionall difference, out of the right ascension of the Sunne, from the beginning of Aries, for the sixe Northerne signes which are ♈, ♉, ♊, ♋, ♌, ♍, or if you adde it thereto, for the sixe Southerne Signes, which are ♎, ♏, ♐, ♑, ♒, ♓, you shall have the Sunnes oblique as­cension.

19. The declination of the Sunne, and his Al­titude aboue the Horizon at any time, together with the height of the Pole being giuen, to find the hower of the day. Say,

[Page 74] As the Radius, is to the Sine of the complement of the Sunnes declination.

So is the Sine of the compl. of the Poles height, to a fourth number. Keepe it,

Then out of the Sunnes distance from the North pole, subduct the complement of the Pole; and of that re­maines, and the complement of the Sunnes altitude, take both the Summe, and also the difference. And say againe,

As the fourth before kept, is to the Sine of halfe that summe: so is the Sine of halfe the difference, vnto a number which being multiplied by the Radius, is equall to the quadrat, of the Sine of halfe the Angle of the Sunnes horary distance from the Meridian.

20. The declination of the Sunne, and his alti­tude aboue the Horizon at any time together with the height of the Pole being given, to find the Sunnes Azumith. Say,

As the Radius is to the Sine of the compl. of the Suns altitude. So the Sine of the compl. of the Poles height, is to a fourth, Keepe it,

Then out of the complement of the Sunnes alti­tude, subduct the complement of the Pole; and of that re­maines, and the distance of the Sunne from the North Pole, take both the Sunne, and also the difference: and say againe,

[Page 75] As the fourth before kept, is to the Sine of halfe the summe: So is the Sine of halfe the difference, vnto a number which being multiplied by the Radius, is equall to the quadrat of the Sine of halfe the angle of the Sunnes horizontall distance from the Meridian.

21. To find the length of the Crepuscu­lum, or Twilight.

Betweene the light of the day, and the darkenesse of the night, the Twilight is set by the wise Creator; that wee here vpon the earth might not in an instant, passe from one extreame into another, but by successiue de­grees. The Twilight is nothing else but the refraction of the Sunnes beames, in the densitie of the aire. And Pet-Nonnius to find the length of the Twilight, watched the time after Sunne 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 neere the Horizon: then by one of the knowen fixed Stars, having taken the true hower of the night, found by many obseruations, that at the time of shutting in the Twilight, the Sunne was vnder the Horizon 18 degrees, and vntill the Sunne was gone so low, the Twilight continued. Say therefore

As the Radius is to the Sine of the compl. of the Sunnes declination:

So the Sine of the compl. of the heigth of the Pole, is to a fourth, Keepe it.

Then out of the Sunnes distance from the South Pole, subduct the complement of the Pole; and of that re­maines, [Page 76] and degrees 62, take both the Summe and also the difference; and say againe,

As the fourth kept, is to the Sine of halfe the summe: so is the Sine of halfe the difference, to a number which being multiplied by the Radius is equall to the quadrat of the Sine of halfe the angle of the Sunnes distance at the ending of the Twi­light, from the high Noone next to it.

Wherefore if out of the whole angle conuerted into howers, you subduct halfe the diurnall arch, or the hower of the Sunnes setting, you shall haue the true length of the Crepusculum, or Twilight.

22. To find the length of the least Cre­pusculum in the yeare.

The Sunne being in the winter Tropic maketh the lon­gest Crepusculum, of the whole winter halfe yeare, and from thence, as the dayes increase, the Crepuscula doe decrease vntill they come to bee shortest, which is in a certaine Parallel, betweene that Tropic, and the Aequi­noctiall: the declination whereof is thus found out.

As the tangent of the complement of the Pole, is to the Sine thereof: So is the tangent of 9 degrees, to the Sine of the declination of the Parallell, in which the Sunne maketh the shortest Cre­pusculum of the whole yeare.

[Page 77] 23 But before the Crepusculum come to bee shortest, there is another Parallel, in which the Crepusculum is equall to that in the Aequinoctiall: the declination wher­of 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 Sunne maketh the Twilight equall to that in the Aequinoctiall.

24. If an Arch of the Ecliptic, be equall to his Right ascension, one end thereof beeing knowne, to find out the other end. Say,

As the Sine of the Compl. of the declination of the arch giuen. 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 poynt of one quadrant of the Ecliptic, wherein the difference of longi­tudes cease to be greater, then the differences of the right ascensions.

Multiply the Sine of the complement of the greatest declination, by the Radius, and out of the product ex­tract the quadrat root: the same shall bee the Sine of the complement of the declination sought for.

[Page 78] 26. To find the quantitie of the angles, which the circles of the 12 Howses make with the Meridian. Say

As the Radius is to the Tang. of 60 degr. for the 11^th, 9^th, 5^th, and third howers, or to the Tang. of 30 deg. for the 12^th, 8^th, 6^th, and second howses; so is the Sine of the complement of the Pole, to the tang. of the compl. of any howse with the Meridian.

And note that on the Easterne part of the vpper hemi­sphaere, there are three circles of Howses, the Horoscope, which is also the Horizon, and next to that is the circle of the 12^th Howse, then the circle of the 11^th Howse. On the Westerne part also, are three circles of Howses, the circle of the 7^th Howse, which also is the Horizon, and next thereto the circle of the 8^th Howse, then the cir­cle of the 9^th Howse. But the circle of the 10^th Howse, is the very vpper Meridian it selfe. Contrary Howses are 1 and 7, 2, and 8: 3 and 9: 4 and 10: 5 and 11: 6 and 12.

27. Resolue the whole time from the Noone last past into degrees (by multiplying the howers with their decimall parts by 15, according to Sect: 1) which adde vnto the right Ascension of the Sunne: and you shall haue the right ascension of the point of the Aequator in the vpper Meridian, which is called the Right ascension of Medium coeli.

28. Adde 99 degrees to the Right ascension of Med. Caeli: and it shall be the degree of the Aequator then rising vpon the East Horizon.

[Page 79] 29. If the first quadrant of the Aequator doe 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 qua [...]rant of the Aequator doe 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 Eclip­tick with the Meridian, on the East side is obtuse, and on the West side acute: and the 90^th degree of the Ecliptick, commonly called non agesimus gradus, is on the East part.

30. If the third quadrant of the Aequator doe 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 doe 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 Ecliptick with the Meridian, on the East side is acute, and on the West side obtuse: and on the 90^th degree is on the West part.

31. The point of the Ecliptick culminant in the Me­ridian, which is called Medium coeli, or Cor coeli, and is the cuspis of the 10^th house, may be found by Sect. 5.

32. The declination of the said culminant point, may be found by Sect. 3. VVherefore also by adding or sub­ducting that declination, to, or from the eleuation of the [Page 80] Aequator, (which is the complement of the Pole) the Altitude of Med. coeli may be had.

33. The Angle of the Ecliptick with the Maridian, may be found by Sect. 7.

34. To finde the Altitude of the 90 degr. Or the Angle of the Ecliptick 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 Ecliptick with the Meridian, to the Sine of the compl. of the angle sought for.

35. To finde the Azumith of 90 degr. which is also the Amplitude ortiue 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 Ecliptick with the Meridian, to the tang. of the compl. of the distance of that Azumith from the Meridian.

36. To finde the Horoscopus, or As­cendent degree of the Ecliptick, Or the Cuspis of the first house.

The Distance of the Azumith of 90 degrees from the Meridian, is equall to the Amplitude ortiue of the Ascendent degree. Wherefore the Ascendent degree [Page 80] or the Ecliptic, may thence bee found, by Sect: 8, or 9. Or else thus.

As the Radius is to the Sine of the complement of the angle of the Ecliptic with the Meridian: So is the tang. of the complement 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 Howses. 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 Howse 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 howse, 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 Howse sought for.

[Page 82] 39. To find the Altitude of the Pole aboue any of the circles of the Howses.

First find out the Angle which the circle of the Howse 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 Howse with the Meridian: So is the Sine of the height of the Pole a­boue the Horizon of the place, to the Sine of the height of the Pole aboue that circle of position.

40. The longitude, and latitude of any fixed Starre being given, to find out the Right ascension, and Declination there­of.

The angle which the Circle of the Sunnes 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 Sunnes Right ascension, maketh with the Meridian at the Pole of the world, I call the Angle of right Ascension. The conditi­on and quantitie of which two angles, is thus found out.

CHAP. XIII. Of Trigonometria, or the manner of calcu­lating both Plaine, and Sphaericall tri­angles. And first concerning certaine ge­nerall notions, and rules necessary thereto.

IN euery triangle both Plaine, and Sphaerical, the greater side subtendeth the greater angle. And the greater angle hath the greater side opposite vnto it. Also the greater angle ly­eth to the lesser side, and the greater side hath the lesser angle lying vnto it.

In euery plaine triangle, any two angles being giuen, the third is also giuen: and one of the angles being giuen, the summe of both the other two is giuen. For all the three angles together, are equall to two right angles, that is to 180 degrees.

In a plaine rectangle triangle, one of the acure angles is the complement of the other. Where note that when the complement 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 excesse is meant of the arch, which is aboue a quadrant. But when it is said the complement to a semicircle, it is vnderstood of so many degrees as will make vp 180.

But in a Sphaericall rectangle triangle, one of the oblique angles is alwayes greater then the compl: of the other.

If two arches together make vp a Semicircle, the ex­cesse of the greater arch, is equall to the complement of the lesser.

[Page 95] The same Right Sine, and the same Tangent, and Se­cant, doth belong both to the arch it selfe, and also to the complement of it to a Semicircle. But their versed Sines differ: For the versed Sine of an arch lesse then a qua­drant, is equall to the difference of the Radius, and the Sine of the complement of that arch: and the versed Sine of an arch greater then a quadrant, is equall to the summe of them. And the versed Sine is thus found out, As the Radius is to the Sine of halfe the arch; so is the Sine of that halfe arch, to halfe the versed Sine of the whole arch.

In a right angled triangle both Plaine, and Sphaericall, 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 euery rectangled triangle, is most fitly noted with the letters ABC; so that BA 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 euery oblique-angled triangle with the letters BCD; so that out of the angle C a perpendicular CA, being let downe, it may in the base BD distinguish the two cases BA and DA, which are the bases of the two particular triangles into which it is cut. And in noting the triangles with letters, obserue diligently, that if any angle be giuen 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 shal fall within the triangle: And BD = BA + DA that is BD is equall to the summe of BA 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 − BA, that is BD is equall to the excesse of DA aboue BA or if the angle B. be obtuse, the perpen­dicular CA: shall fall without the triangle beyond the obtuse [Page 96] angle D: And BD = BA − DA, that is BD is equall to the excesse of BA aboue DA. The lesser case being still taken from the greater angle.

And note that this signe +, or pl (that is plus) sheweth that the magnitude before which it is set, is affirmed and positiue in nature; and therefore to bee added. And that this signe—or mi (that is minus) sheweth the magni­tude before which it is placed, is denied, and priuatiue in nature; and therefore to be substracted, as you may see in those former examples.

Againe, some magnitudes are taken seuerally and apart; as s BA, that is the Sine of the Base; s co BC, that is the Sine of the complement 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 complement of the angle at the Cathetus: So also VqZ, that is the quadrat side of the plaine Z. And some magnitudes are taken vniuersally, and then they are included in pricks: as [...]: that is the Sine of halfe the arch, which is composed of the summe 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 plaines Z and X put together: also Vq ∶ Q in R ∶ Or Vq ∶ Q × R ∶ that is the quadrat side of a rectangular plaine, the two sides whereof are the lines Q and R, or some fourth proportionall already found, and the Radius, or Semidiameter, which is the totall Sine. For by the signe in, or ×, I vse to expresse multiplication.

When any triangle is giuen to be resolued by Trigono­metrie, note the parts thereof (either-sides or angles) which are giuen and knowne, with a little line drawne crosse each such part: and note the vnknowne part which is sought for with a little circle.

[Page 97] And if a triangle Sphaericall (bee it right angled or ob­lique-angled) proposed hath two sides each of them se­uerally greater then a quadrant: you shall in resoluing thereof, keepe the least side with the least angle opposed to it: and for the two other both sides, and angles, take the complements of them to a semicircle.

Lastly, if a triangle with all the three angles giuen, be required to be conuerted into a triangle hauing the three sides giuen. You shall for the greatest angle of the tri­angle proposed, & for the greatest side subtending it, take the complements to a semicircle; keeping the other two lesser angles, with their subtendent sides as they are.

CHAP. XIIII. Of the Nocturnall Dials.

THere are in the Instrument, two seuerall Nocturnall Dyals. The innermost of them is fitted to the starre in the rump of the great Beare, commonly called Aliot. The other is composed of 12 seuerall starres: whose names you shall finde written within, neere to the center.

The outermost circle of the Nocturnall Dyall is diui­ded into twise 12 houres: each houre being subdiuided into quarters, and are noted with figures belonging to the houres, as may be seene in the Instrument.

The middlemost circle of the Nocturnall is diuided in­to 12 moneths, hauing their names written: each moneth being distinguished into tenth dayes with longer lines; and into fift dayes with shorter lines. And if the Instru­ment be large enough, each day of the monethes through­out the yeare, may be noted with pricks.

In the innermost circle are the diuisions and names of 12 fixed starres: which are these.

• The bright star in the head of ♈.
• the Bulls eye
• the latter shoulder of Orion
• the little dogge
• the heart of the Lyon
• the tayle of the Lyon
• Spica Virginis
• the North ballance
• the head of Ophiuchus
• the heart of the Vultur
• the mouth of Pegasus
• the tip of the wing of Pegasus.