The mariners magazine, or, Sturmy's mathematical and practical arts containing the description and use of the scale of scales, it being a mathematical ruler, that resolves most mathematical conclusions, and likewise the making and use of the crostaff, quadrant, and the quadrat, nocturnals, and other most useful instruments for all artists and navigators : the art of navigation, resolved geometrically, instrumentally, and by calculation, and by that late excellent invention of logarithms, in the three principal kinds of sailing : with new tables of the longitude and latitude of the most eminent places ... : together with a discourse of the practick part of navigation ..., a new way of surveying land ..., the art of gauging all sorts of vessels ..., the art of dialling by a gnomical scale ... : whereunto is annexed, an abridgment of the penalties and forfeitures, by acts of parliaments appointed, relating to the customs and navigation : also a compendium of fortification, both geometrically and instrumentally / by Capt. Samuel Sturmy.

CHAP. II. Of what is needful first to be known in the Practick Part of the Compass, and how to divide the Circles and Parts.

THE principal Hand-maids that expert Sea-men are furnished with, that their Undertakings may be crowned with everlasting Credit, are these, viz. Arith∣metick, Astronomy, Geometry, and the Mathematiques. By the operation of these loving Sisters, and excellent Arts, as hath been said, Navigation is daily pra∣cticed by expert Sea-men: but much abused by hundreds of ignorant Asses, that know nothing what belongs to them, yet do undertake Voyages, to direct a Ship naviga∣ble upon the Terrestrial Globe, resting wholly upon favourable Fortune, which hath made some of them famous; but many times disasterous Periods have ended their Undertakings, with the loss of many mens Goods and Lives; which yet I must confess have and do happen to the best, but not so often as to them by great diffe∣rence. But to come to the Substance of what is here intended, I would have it to be understood, That he that intendeth the Art of Navigation, hath Arithmetick in readiness. If he want it, he may be instructed by divers Books now extant, as Re∣cord, Blundevill, and Mr. William Leybourn's Arithmeticks. As for the Mathematical and Astronomical Knowledge, so much as is useful for Sea-men, will be shewn in the Projection and use of divers Instruments, which will after follow in its due place. In this Treatise we will come to the Sea-Compass, that we may proceed in a regular form. The knowledge of it is the root of that famous Art we chiefly treat of, and presents himself as the first Principle framed by God in the Operation and Nature of the Magnet, which being in its quality beyond our capacities, yet it is the first thing to be learned and understood, it being the foundation to all the following Conclusions, and is first taught to our Youths and Boys which are intended for Navigators. They are taught first to know the Point on the Card, and by what Name it is called, and to say it perfectly backwards and forwards; and to know that to every Point of the Compass there is allowed for Time ¾ of an Hour, which is 11 Degrees 15 Minutes; and how to number the Hours from the North and South, either Eastward or West∣ward, readily to answer as soon as demanded: As also to know how the Ship Capes; that is, to know the Point of the Compass that looks straight forwards to the Head of the Ship: As likewise to know upon what point of the Compass the Wind blows over; that is, if the Wind be at North, it blows over the Flower de Luce toward the South; and so o•• the rest. So we teach them to know what Point the Sun is on; That in England a South-east Sun on the Aequator makes 9. 24 of the Clock; and when he is South, makes 12 of the Clock; and South-west, 2. 36 of the Clock. As also they learn to set the Moon in the same manner on the Full and Change-days, to know the Tides by, as shall be shewed.

The Compass we Steer our Ships by, is only a Circle of some 8 inches diameter; and is divided into 32 Points, which have several denominations, as you may see ex∣pressed in the Figure. The whole Circle is divided into 360 degrees, and 24 hours: The Compass contains also 16 distinct Rhombs or Courses; for each several Course hath two Points of the Compass, by which it is expressed. As for example, Where there is any place scituated South-east, in respect of another place, we say the Rhomb or Course that runneth betwixt them, is South-east and North-west: or if it bear South or North, so we call it: or if West, we say West and East. The Compass swings in the Boxes, the Wyers first well touched with a good Load-stone, and the Chard swimming well on the Pin perpendicular in the middle of the Box; it repre∣sents

the plain Superficies or Horizon as we call it, when looking round about us at Sea, and see the Heavens make intersection with the Waters, sheweth that you are in the Center, and that every place in the Horizon is equally distant from you: So that when you espy any Island, Rocks, Ships, or Cape-Lands, by looking straight up∣on the Compass, you shall know upon what Point of the Compass the Object beareth from you. But we will haste to shew the young Practitioners the Sea-Compass, with the 32 Points, expressed by the Letters upon each Line, and also how to make it, as followeth.

[illustration] diagram of a compass

The COMPASS.

How to divide the Circles of the Mariners Compass.

FIrst draw a Line at pleasure, and cross it in the midst with another Line at right Angles; Then in the crossing of these two Lines set one foot of your Compass, and open the other to what distance you please, and with that distance draw the Cir∣cle, which by the cross Lines of East and West, North and South, are divided into four Quadrants and equal parts, each of them containing 6 hours a piece; set VI at East and VI at West, XII at North, and XII at South, so have you the four first Divisions of your Figure: Then keeping your Compass at the same distance as you draw'd the Circle, set one foot in the crossing of the Line and the Circle at East 6, with the other make two marks, one of II, and X. Then set one Foot in the West at 6, on the other side mark out the hours of II and X, as before; keeping the Compasses still at the same distance, set one Foot at South XII, and with the other you shall mark out the Hours of VIII and IIII. Then set one Foot of your Compas∣ses at North at XII, and in the same manner mark out the Hours of VIII and IIII. Thus the Circle is divided into 12 equal parts, and each of them contains 2 hour

piece; so that it will be easie for you to divide each of these into two parts; which done, you have the 24 hours. Lastly, you may divide each hour into 4 equal parts, which will be quarters of an Houre, as you may see in the Figures.

To divide a Circle into 360 equal parts, is a thing very necessary; for in all Que∣stions in Astronomy, and in the Calculation of all Triangles, these parts are the mea∣sure of the Angles: so that in respect of this, every Arch is supposed to be divided into 360 equal parts or Degrees; and every Degree is supposed to be divided into 60 lesser parts, called Minutes. To divide a Circle after this manner, draw a Line at pleasure, and cross it at right Angles with another Line, and draw a Circle as before. Keep your Compasses at the same distance, and divide the Circle from the 4 Quar∣ters into 12 equal parts. Then closing your Compasses, divide each of these into 3; so you have in all 36 parts. Then you may easily divide with your Pen each of these parts into 10 little parts, as you may see in the middle Circle of the Figure, which are Degrees.

For the 32 Points of the Compass, draw the Line of North and South, and cross it at right Angles with the Line of East and West, and draw the Circle, as before; and with the same distance, set one Foot of your Compasses at East, and with the other draw a small Arch at A and B, and cross it from North to South with the same distance; the like do from the West Point to C and D. Then laying your Rule cross-ways to these Crosses, draw the Line BD and AC; so is your Circle divided in 8 equal parts. Then closing your Compasses, you may easily divide these 8 parts into 4; divide one, and that distance which will divide all the rest into equal parts, if you have followed Directions. And so you have the 32 Rhombs or Points of the Com∣pass; and so you may subdivide these Points into halves and quarters, as you may see in the Figure. So have you made the Mariner's Sea-Compass. The Use shall be shew'd in its place.

[illustration] diagram of a compass

The Figure of the Compass, and the Traverse Quadrat, joyned both together.

The Traverse Quadrat sheweth the making of the Traverse Table, in Chap. 3. Of Sayling by the plain Sea-Chart.

The Moons Motion, and the Ebbing and Flowing of the Sea.

THe Practitioner in Navigation, is next to learn to know the certain time of the Flowing and Ebbing of the Sea; In all Ports called by Sea-men the shifting of Tydes, which is governed by the Moon's Motion, as it is found by experience.

Wherefore I will first shew the use of a small Instrument, which is here framed, whereby the meanest Capacity (which is void of Arithmetick) shall be able to know the Age of the Moon, with what Flood and Ebb it maketh in all Channels, and in every Port and Creek at High-water; and also be able to know what a Clock it is at any time of the Night; and divers other Questions, only by moving the Instrument, according as shall be directed.

I shall also shew you, how you may do all these Questions of the Tyde by the Moon Arithmetically: But first by your Instrument it must be projected according to the following Figure, which you may make of three pieces of Board, well planed, and exactly divided, according as you see it formed in the Figure. The outward Circle, being the biggest Board, hath 32 Points of the Compass; The inward Cir∣cle on the same Board is divided into 24 Hours, being the thickest Board. The se∣cond Circle must be divided into 30 equal parts, representing the distance 30 times 24 Hours, or 30 natural days, attributed to the Sun. The uppermost Circle of the three, is attributed to the Moon, with an Index as that of the Sun, and to be turned and applied to either the 30 days, containing the Computation of the Moon betwixt Change and Change, or the 24 hours; as likewise to the Points of the Compass. And so may the Index of the Sun be applied either to Time, or the points of the Com∣pass, which shall be made plain by the following Questions; which will appear de∣lightful and easie; and the illiterate man will find in most useful; and he that hath better Knowledge, will sometime use the Instrument for variety sake. First, for the Figure of the Instrument.

A Ʋseful Variation-Compass.

UPon the two upper Circles of the Instrument I have set a most useful Variation-Compass, easie to be understood, and as exact as any Instrument whatsoever for that purpose. You shall have full direction how to use them in the following Discourse, when we come to treat of the Variation of the Compass. But this observe, The mid∣le Compass representeth the Compass you steer your Ship by, which is subject to Va∣riation; but the upper Compass and Circle representeth the true Compass, that never varieth; and by it you may very readily know the Variation of the Steering Compass, how much it varieth from the true Point. The inward Circle of the middle Com∣pass is divided into the 32 Points, with their halves and quarters: and likewise the outward Circle of the smaller or upper Compass. This is too hard for Practitioners at first to know how to use this Instrument, to rectifie the variation of the Compass; therefore I shall be no longer on this Discourse, and proceed to what was promised, and shew the farther use of it afterward.

PROPOSITION I. The Moon being 16 days old, I demand upon what Point of the Compass she will be at 8 of the Clock at night.

In all Questions of this nature you have the Hour and Time given, and the Moons Age, to find the Point of the Compass. To answer these Questions, place the middle Index of the Sun on 8 of the Clock at night; then bring the upper In∣dex of the Moon right over the 16th. day of her Age, in the middle Circle of the Sun, and the Index of the Moon or upper Circle points to E. b. S. half a point Southerly, the true Point of the Compass the Moon will be, when she is 16 days old, at 8 of the Clock at night.

[illustration] diagram of a compass

An Instrument Shewing y^e Changing of y^e Tides and y^e Variation of y^e Compas

Betweene fol: 6 & 7.

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that Point of the Compass, by turning the Index of the Moon, as before was shewed: So you may be sure to have the Hour always under the Index, on the Change-day, throughout all the Points of the Compass; and so we shall proceed to Examples.

PROPOSITION VII. The Moon being 16 days old, I demand, What a Clock it will be Full-Sea at Bristol, Start-point, Waterford, where an East-by-South Moon on the Change-day makes the Full-Sea?

You are to consider the Point of the Compass the Moon is upon in these Ports, when it is Full-Sea on the Change-day (as in all other Ports) which in these Ports is found by observation to be always East-by-South Moon (which is 6 hours 45 mi∣nutes) Then consider whether it be the Day or the Night-Tyde you would know the Time of Full-Sea; if it be the Morning-Tyde, bring the Index of the Moon to the West-by-North Point, staying it there; bring the 16th. day of her Age under the Index of the Moon, and the Index of the Sun will point you to 7 of the Clock and 33 Minutes, the time of Full-Sea in the Morning. If it be the Evening Tyde, bring the Index of Luna to the East-by-South, and stay it there, until you have brought 16 Days and half under the Index of Luna, and the Index of Sol will point directly upon 8 of the Clock at Night, the time of Full-Sea in the aforesaid Ports: Thus you see there is 27 Minutes difference in every Tyde in these Ports. So you may know in every other Port in the same manner, if you do as before-directed, and allowing half a day more for the Night-Tyde, by turning of it half a day further. And take this for a Rule, That the Moon betwixt Change and Full is ever to the Eastward of the Sun, and riseth by day, still separating it self from the Sun until she be at the Full: Then after the Full, in regard she hath gone more Degrees in her separation than is contained in a Semicircle, she is gotten to the Westward of the Sun (rising by night) and applieth towards the Sun again until the Change-day, which you may see plain∣ly demonstrated by the Instrument.

PROPOSITION VIII. The Moon being 16 days old, I desire to know at what hour it will be Full Sea at London, Tinmouth, Amsterdam, and Rotterdam, where a S. W. and N. E. Moon makes a Full-Sea upon the Change-day.

It is found by observation, That the S. W. and N. E. Moon makes Full-Sea in all the aforesaid Ports. You may know the Moon is to the Eastward of the Sun that is before him; Bring the Index of the Moon to the S. W. Point; then turn the 16 day of her Age under the Moons Index, and the Index of the Sun answereth the Que∣stion, That it is Full-Sea at all the aforesaid Ports at 3 of the clock and 48 minutes in the morning, the Moon being 16 days old.

PROPOSITION IX. At Yarmouth, Dover, and Harwich, where a S. S. E. Moon maketh Full-Sea on the Change-day, the Moon being 9 days old, I demand the time or hour of Full-Sea that day in the aforesaid Places.

Here it hath been found by experience, That a S. S. E. Moon makes Full-Sea on the Change-day, in the aforesaid places; therefore bring the Index of the Moon to the S. S. E. Point, keep it there fast directly on the Point, and bring the Moons Age to cut the edge of the Moons Index, and the Index of the Sun will shew you, That the time of Full-Sea in the aforesaid Ports, will be at 5 a Clock and 42 minutes in the morning.

PROPOSITION X. At St. Andrews, Dundee, Lisbone, and St. Lucas, where a South-West-and-by-South Moon makes High-Water or Full-Sea on the Change-day; The Moon being 28 days old, I demand the time of Full-Sea that day in these Places.

You have here given you the S. W. b. S. Point of the Compass; therefore bring the Index of the Moon, and stay it on that Point, and bring the 28 day of her Age un∣der the edge of the Index of the Moon, and the Index of the Sun will point you out the time of Full-Sea, which is at 39 minutes past 12 of the clock at noon, in the aforesaid places. And so are all Questions of this nature answered. And so I will conclude the Use of this Instrument, for finding the Ebbing and Flowing of the Tyde, and so will proceed to shew you Arithmetically how to find the Golden Num∣ber or Prime without a Table, the Epact, and Full, Change, and Qu••rters of the Moon, and how to know her Age for ever; and what Sign and Degree she is in the Zodiack, how long the Moon shineth, and what time of the day or night it is High-Water or Full-Sea in any Port; and also the Moon's Motion, as far as it is useful for Mariners.

[ I] How to find the Golden Number or Prime, according to the Julian, English, or Old Account.

YOu may observe this, That the Prime or Golden Number is the space of 19 years, in which the Moon performeth all her Motions with the Sun: At the expiration of which Term, she beginneth again in the same Sign and Degree of the Zodiack, that she was 19 years before; and always finisheth her Course with the Sun, and never exceedeth that Term. To find this useful Number, you must do thus; Al∣ways in what year you would know what is the Prime Number, add 1 to the date thereof, and then divide it by 19, and that which remaineth upon the Division, and cometh not into the Quotient, is the Number required. As for example,— In the year of our Lord 1665. I demand what is the Prime Number. Add to the year of our Lord always 1, which makes in this Question 1666. Then divide that sum by 19, the remain is the Prime or Golden Number, as you may see by the Work, which answereth the Prime or Golden Number for this present〈 math 〉〈 math 〉 year to be 13, it being left out of the Division that cometh not into the Quotient. Thus you see it is very easie to do it for any other year. Observe, That when you find nothing remaining upon the Division, that is the last year of the Moon's Revolution, and may conclude, that 19 is the Prime for that year. Note, The Prime beginneth always in January, and the Epact in March.

[ II] How to find the Epact, according to the Julian, English, or Old Account, and what it proceedeth from.

THe Epact is a Number that proceedeth from the difference which is made in the space of one whole year, between the Sun and the Moon. Note, The Solar year doth contain 365 days, 5 hours, 48 minutes; and the Lunar year, allowing 12 Moons, there being 29 days, 12 h. 44 min. between Change and Change, doth con∣tain but 354 days, 8 hours, 48 min. So that there is almost 11 days difference between the Revolution of the Sun and Moon, at every years end; which difference makes the Epact. Therefore to find the Epact for any year, first you must know the Prime Number for that year, which we found before for the year 1665. to be 13. Then you must multiply this Prime Number 13. by the difference 11. and it will make 143. which divide by 30. and there re∣maineth of the Division, that cometh not into the Quotient, 23. which is the Epact for the year required.〈 math 〉〈 math 〉 So I make no Question but that you understand how to find the Prime and the Epact for any year past, present, or to come. Therefore I hold this sufficient to express so facile a thing as this is. I have told you already, That the Epact always beginneth in March; but I shall make a small Table for those that are ignorant in Arithmetick, and cannot find these two Golden Numbers, as I may call them, for 45 years to

come, where any one may find the Prime and Epact most readily in any year you shall desire.

[ III] A Rule to find the Change, Full, and Quarters of the Moon.

ADd unto the Epact of the year proposed all the Months from March, including the Month of March, and substract that sum from 30. the remain sheweth the day of the Change: But if the Epact be above 26. there this Rule faileth a day at the least; but at other times it will be no great difference: Therefore it may serve for the following Conclusions.

As for Example, I desire to know the New-Moon in October, 1665. The Epact is 23. the Months from March are 8. which added makes 31. from it 30 substracted, remains 1. which taken from 30. one whole Moon, there remains 29. So that the 29th. day of October is the day of her Change, or New-Moon, which by exact Calcu∣lation it is at 58 min. past 4 in the morning.

Having thus found the time of the New-Moon, you may from thence reckon the Age of the Moon, and so find the Quarters, or Full-Moon.

Thus the Moon's Age is	Days	Hours	Min.
At the First Quarter	07	09	11
At the Full Moon	14	18	22
At the Last Quarter	22	03	33
At the Whole Moon	29	12	44

[ IV] How to finde the Age of the Moon at any time for ever.

ADd to the days of the Month you are in, the Epact, and as many days more as are Months from March, including March for one; and if these 3 Numbers added together exceed 30, take 30 from it as often as you can, and the remain is her Age: But if the Numbers added be under 30, that's her Age; As for Example, 1665. the Epact is 23. I demand, What Age the Moon is the 21th. day of September? From March to September is 7 Months, the Epact 23, and the day of the Month is 21. Added together, makes 51. From it substract 29, because the Month hath but 30 days in it, and the remain is 22, the Age of the Moon that day. Had it been the 22th. of August, and added them together, it would have made 51. Then to have taken 30 out, there had remained 21 for the Moon's Age the 22 day of August.

[ V] To finde what Sign the Moon is in, by which is gathered, what the Moon differeth from the Sun.

MUltiply the Age of the Moon by 4. divide the Augment or Sum by 10, the Quotient sheweth the Sign the Moon differeth from the Sun; the Remain multiplied by 4, giveth the Degrees to be added. As for Example,—The Moon 22 days old, I demand what she differeth from the Sun? 〈 math 〉〈 math 〉 Multiply 22 by 4, and the Product is 88. That divide by 10, and in the Quotient is 8, and 8 remaineth upon the Division: That multi∣plied by 4, is 32; from which take 30, the number of Degrees in a Sign, and add the 8 Signs in the Quotient, it makes 9 Signs. The odd 2, multiplied by 4, make 8 Degrees; to which add the Sun's Motion from his entrance into the Sign ♎ which was the 14 day, to the 21, make 7 days or Degrees to be added, to the 8 Degrees make 9 Signs 15; which counted after this manner, from ♎, saying, ♏ 1, ♐ 2, ♑ 3, ♒ 4, ♓ 5, ♈ 6, ♉ 7, ♊ 8, ♋ 9 Sign, and the odd 15 Degrees is 15 Degrees of Cancer. So the Question is answered, That the Moon is 9 Signs 15 Degrees from the Sun at 22 days old; which note, She differeth but 4 Degrees from her true Motion by the Tables, which is near enough for the Mariner to answer any Man.

How to find what Sign the Moon is in more exact; with the Moon's Motion for every day of her Age.

AStronomers divide the Compass of the Heavens into 12 Signs, which they set forth by these Names and Characters, which you must be a little acquainted with, and the place of the Sun in the Zodiack. Each of these Signes you have them as followeth.

A Table shewing the Moon's Motion according to her Age.

D. Age.	S.	D.	M.
1	00	13	11
2	00	26	21
3	01	09	32
4	01	22	42
5	02	05	53
6	02	19	04
7	03	02	04
8	03	15	26
9	03	28	35
10	04	11	46
11	04	24	56
12	05	08	07
13	05	21	18
14	06	04	28
15	06	17	39
16	07	00	49
17	07	14	00
18	07	27	11
19	08	10	21
20	08	23	32
21	09	06	42
22	09	19	53
23	10	03	03
24	10	16	14
25	10	29	25
26	11	12	35
27	11	25	46
28	12	08	56
29	12	22	07
30	13	05	17

First know, That the Sun entreth the first Sign ♈ the 11th of March, ♉ the 11th of April, ♊ the 12th of May, ♋ the 12th of June, ♌ the 14th of July, ♍ the 14th of August, ♎ the 14th of September, ♏ the 14th of October, ♐ the 13th of November, ♑ the 12th of December, ♒ the 11th of January, ♓ the 10th of Fe∣bruary. This known, the place of the Sun is well found, adding for every day past any of these, 1 Degree.

Thus you see, the Sun runs through these 12 Signs but once in a year; The Moon in less than a Month, viz. in 27 days, 7 hours, 43 minutes. Note, That every New-Moon, the Sun and Moon are in one Sign and Degree; but the Moon hath a Motion of about 13 Degrees every day, as is shewed in this Table. Therefore according to the Age of the Moon, add the Signs and Degrees of the Moon's Motion, to the place of the Sun at the New Moon, and so you shall have the Sign and Degree which the Moon is in at any time desired.

Thus for Example, A New-Moon 1665. the 26th. No∣vember, and the Sun and Moon are both in 14 Degrees of ♐. Now upon the 11th of December, the Moon being 14 days old, I would know what Sign the Moon is in. This Table shews, for the 14 days of the Moon's Motion, you must add 6 S, 4 D, 28 Min. to the said 14 Degrees of ♐.

Now counting those 6 Signs upon your Fingers, reckon∣ing the Names of the Signs in order from Sagittarius, ♑ 1, ♒ 2, ♓ 3, ♈ 4, ♉ 5, ♊ 6, it falls upon the Sign Gemini. Lastly, adding the odd 14 Degrees unto the 4 Deg. of the Moon's Motion together, shews the place of the Moon to be in 18 Degrees of Gemini.

There is much use made of the Moon's being in such and such Signs, in Physick and Husbandry, of which I shall say nothing; but give you one Conclusion which much depends hereon; that is,

To know the time of the Moon's Rising, Southing, and Setting.

FOr her Rising (know this) having found the place, or what Sign she is in, seek out in the following Kalendar what time the Sun is in this Sign and De∣gree, and there you shall find the true time of the Sun-Setting, being in that place: This is half the continuance of the Sun above the Horizon in that Sign and Degree. Add this to the time of the Moon's coming to the South, it shews the time of her Setting; and substracted from it, shews the time of her Rising.

Thus upon the 11th of September, as before, the Moon being 14 days old, and in the 18 Degree of Gemini, I desire to know the time of the Moon's Rising and Setting.

	H.	M.
☽ So.	11	12
☉ Set.	8	10
☽ Ris.	3	02
Added	19	22
☽ Set.	7	22

First multiply 14, the Moon's Age, by 4. Divide the Product by 5. In the Quotient will be 11 a Clock, and the one Unite upon the Division is Min. 12, that the Moon will be South that night. Secondly, The Sun is in this Sign and Degree about the first day of June, and then sets at 8 a Clock 10 minutes past. This substracted, shews the Rising of the Moon to be at 3 of the Clock 2 minutes in the afternoon. The said 8 hours, 10 being added, makes 19 hours 22 min. which by casting away 12, the remain shews the Moon's Setting to be at 7 of the Clock, and 22 min. past in the morning, which answers the Question desired; which is as neer as can be for your use.

PROP. I. How to find when it is Full-Sea in any Port, Rode, Creek, or River.

I have shewed you already how to find the Prime, Epact, and Age of the Moon, at any time desired. Now we will proceed to shew you the finding of Full-Sea in any Place; as in manner following.— First, Carefully watch the time of High-Water, and what Point of the Compass the Moon is upon, on her Change-day, in that Port or Place where you would know the time of the Full-Sea, or find by the Table what Moon makes a Full-Sea in the said Port. Secondly, Consider the Age of the Moon; then by Arithmetick resolve it in this manner. Multiply the Moon's Age by 4, di∣vide the Product by 5, the Quotient shews the Moon's being South. If any thing remaineth upon the Division, for every Unite you must add 12 Min. If it was 4 re∣maining, it would be 48 Minutes to be added. Then add the hour that it Flows on the Change-day to it, and the Total is the hour of Full-Sea. If it exceed 12, sub∣stract 12 from it, the remain is the hour of the day or night of Full-Sea, in any Port, River, or Creek. Which I will make plain by some Examples, (viz.)

PROP. II. The Moon 16 days old, I demand, What a Clock it will be Full-Sea at Bristol, Start-point, and Waterford, where E. b. S. Moon maketh Full-Sea on Change-day?

Consider here an E. b. S. Moon maketh 6 hours 45 min. and the Age of the Moon is 16 days old: Therefore multiply the Age by 4, and it makes 64; divide that by 5, and it is 12, and 4 remaineth, which is 48 min. To it add 6 hours 45 min. E. b. S. it makes 19 ho. 33 min. Therefore substract 12 hours from it▪ there remaineth 7 a Clock 33 minutes, the time of Full-Sea in the morning at the aforesaid Ports; which you may compare with your Instrument, and find it very well agree. 〈 math 〉〈 math 〉

PROP. III. The Moon being 25 days old, I demand, What a Clock it will be Full-Sea at London, Tinmouth, Amsterdam, and Rot∣terdam, where it flows S. W?

Consider that at these Places on the Change-days a S. W. Moon maketh Full-Sea, which is 3 hours. Therefore multi∣ply 25, the Moon's Age, by 4, it makes 100. That divide by 5, in the Quotient will be 20, and nothing remain. To it add 3 ho. S. W. and it makes 23 hours. From it substract 12, and the Remainder shews you, That it will be Full-Sea at all the aforesaid Places, at 11 of the Clock in the mor∣ning. So you will find it agree with your Instrument. 〈 math 〉〈 math 〉

PROP. IV. The Moon being 9 days old, I desire to know the hour of Full-Sea at Quinborough, Southam. and Portsmouth.

Note, That a South-Moon on the Change-day, maketh Full-Sea at these Places. Therefore multiply the Moon's Age by 4, it makes 36. That divide by 5, and the Quotient is 7 of the Clock; and 1 remaineth, which is 12 minutes, the time of Full-Sea at the aforesaid Places, the Moon's Age being nine days. Note, If a North or South Moon makes Full-Sea on the Change-day, there is nothing to be added to the Quotient; but the Quotient is the hour of the day, and the Remainder is the min. as before directed. One Example more shall suffice. 〈 math 〉〈 math 〉

PROP. V. The Moon 5 days old, I demand the time of Full-Sea at Rochester, Malden, Blacktail, where S. b. W. Moon is Full-Sea.

Here you may note, That on the Change-day at these Places it flows S. b. W. which is but one Point from the South, being but ¾ of an hour, or 45 min. And it had been all one if it had been North-by-East. Multiply by 4, divide by 5, and the Quotient will be 4; to it add 45 min. S. b. W. shews you it will be Full-Sea at the aforesaid Places at 4 a Clok and 45 min. in the morning. But note, Had it been S. b. E. or N. b. W. it had been 11 ho. 15 min. By this time I hope I have made the Practitio∣ner able to know the time of Full-Sea in any Port, by Instrument and Arithmetick: Therefore I will leave him a small Table for his use.

A TIDE-TABLE.

	H.	M.
Rye, Winchelsey, Culshot, a S. b. E. Moon.	11	15
Rochester, Malden, Blacktaile, S. b. W.	0	45
Yarmouth, Dover, Harwich, S. S. E.	10	30
Gravesend, Downs, Blackness, Silly, S. S. W.	1	30
Needles, Orford, South and North Fore-land, S. E. b. S.	9	45
Dundee, St. Andrews, Lisbone, St. Lucas, S. W. b. S.	2	15
Poole, Isles of Man, Dunbar, Diepe, S. E.	9	00
London, Tinmouth, Amsterdam, Rotterdam, S. W.	3	00
Portland, Hartflew, Dublin, S. E. b. E.	8	15
Barwick, Flushing, Hamborough, S. W. b. W.	3	45
Milford, Bridgewater, Lands-end, E. S. E.	7	30
Baltimore, Corke, Severn, Calice, W. S. W.	4	30
Bristol, Start-point, Waterford, E. b. S.	6	45
Falmouth, Humber, Newcastle, W. b. S.	5	15
Plimouth, Hull, Lyn, St. Davids, W. & E.	6	00
Quinborough, Southam. Portsmouth, N. & S.	0	00

Add any two Numbers together of the foregoing Table, and they shall be 12 hours; Except the two last, N, S. and E. W. So that you may perceive, what hath been said from the South, either Eastward or Westward, the same answereth to the North, either Westward or Eastward. And so much for the Tydes. But we will know the Moon's Motion, and the Proportion between Tyde and Tyde.

PROP. VI. The Motion of the Moon, and the Proportion of Time betwixt Tyde and Tyde.

After all this, I will shew you in brief the Motion of the Moon, and the reason of the difference between Tyde and Tyde.

You must note, the Motion of the Moon is twofold. First, A violent Motion, which is from East to West, caused by the diurnal swiftness of the Primum Mobile. Secondly, A natural Motion from West to East, which is the reason the Moon requireth 27 days and 8 hours 8 min. to come into the same minute of the Zodiack from whence

she departed. But coming to the same Point and Degree where she was in Conjuncti∣on with the Sun last, she is short of the Sun, by reason the Sun's Motion every day is natural East, 1 Degree, or 60 Minutes, which maketh so much difference, that the Moon must go longer 2 days, 4 ho. 36 min. nearest, more than her natural Motion, before she can fetch up the Sun, to come into Conjunction with her: So that betwixt Change and Change is 29 days, 12 hours, 44 min. by my account. The Mariners always allow just 30 days between the Changes, by reason he will not be troubled with small Fractions of Time, in this Account of Tydes, which breedeth no great error: Experience therefore must needs shew me this, That I must allow the some Proportion to the Moon in every 24 hours, to depart from the Sun 12 De∣grees, which is 48 min. of time, untill her full East; but then having performed her Natural Motion above half the Globe, she is to the VVest, as we may know by Reason. Now if the Moon move in 24 hours, 48 min. then in 12 hours she must move 24 min. and in six hours, 12 min. By this proportion each hour she moveth 2 min. So the Tydes differeth as the time differeth.

I will add one old approved Experience for the Mariners use, though it is imper∣tinent in this place; that is, to cut Hair, the Moon in ♉, ♑, ♎: Cutting, shaving, clipping in the Wane, causeth baldness; but the best time in the Wane, is in ♋, ♏, or ♓. So I hope I have satisfied the Learner concerning the Moon.

THE Mariners Magazine; OR, STURMY's Mathematical and Practical ARTS. The Practick Part of Navigation, in working of a Ship in all Weathers at Sea.

WE have been shewing the Practitioner all this while, the Course and Motion of the Moon, and so by it to know how to shift the Tydes, or time of High-Water, in any Port, Road, Har∣bour, or Creek, Instrumentally and Arithmetically. The next thing to be observed by a Learner, is the Words of Command, with readiness to answer and obey, which is the most excel∣lent Ornament that can be in a Compleat Navigator, or Mari∣ner. And as Captains Exercise their Men on Shore, that their Souldiers may understand the Postures of War, and to execute it when the Word of Command is given by their Commander; In like manner are Seamen ond Mariners brought up in Practical Knowledge of Navigation at Sea, in working a Ship in all Weathers. Although the Rules here demonstrated, are but of little benefit to him, that hath been brought up all his Life-time at Sea; and less to those that be altogether ignorant in Marine Affairs: But that the Practick may be de∣livered in proper Sea-Phrases, according to each several Material that belongs to a Ship compleatly rigg'd, with the Use of the several Ropes in working and trimming of Sails at Sea on all Occasions, cannot be denied by those that know these things perfectly: Therefore it is impossible for any Man to be a Compleat Mariner or Navi∣gator, without he hath attained to the true Knowledge of Theorick and Practick, be∣ing both Sisters and inseparable Companions, that makes them perfect Navigators; Therefore I could not let this scape my Pen.

And to explain my self, that I may prevent the Censures of all such that will be curious, inquiring whether I am not lame, or incapable of that, and like themselves appear imperfect; I may speak it with trouble to my self, and shame to others, That there was never more lame and decrepit Fellows preferred by Favour and Fortune, as also by Kindred, and by Serving for Under-Wages (which a deserving Man might and would have) as is now adays. Let a man go aboard the best Ship at Sea, and it will be very rare to find Ignorance out of the Officers Cabins, and commonly able Mariners and more sufficient Men before the Mast, which are first to hawl a bowl∣ing, through the averseness of their Fates, which is great pity. I should be glad to live to see a more equaller Balance among Sea-men, and their Imployers, to further the industrious, and encourage the deserving Men; for if this partiality should con∣tinue long, it is to be feared, in some short time, the Compleat Mariner wil be hard∣ly found aboard any Ship, to the great disparagement of our English Nation, which hath from time to time so long deservingly had the Superiority over all other parts of the World, for breeding the most famous Navigators. The Hollander to his Loss knows it right well, that there are none like English for Courage at Sea; but that ma∣ny of them out-strip us in the Art of Navigation, which proceeds from the former

unequal Balance, which makes our expert Saylers to seek if Fortune will be favoura∣ble amongst them, They had not at this day been High and Mighty, and in such a flourishing Condition as now they are. Therefore I hope to see and hear, That the English Mariner will make better use of swift-stealing Time, that he may redeem what is lost, and attain to such perfection, as that he may parallel his Art with his V••lour and Courage; And that Imployers will use more Equity, in placing deserving men according to their merit. I shall not draw out my Digression to any longer Di∣scourse; for I know my plain Rhetorick will not rellish in some mens ears, though it may in others: Therefore I shall draw to a Conclusion, desiring that no man will censure me, before he knows what is in me, or is able to mend this. For some there are, being a little touched (as the common Saying is) that if they had me at Sea, they would put me to seek all my prescribed Rules; but I would have such to know, That when I at Sea, I shall work the Ship in all Assays as well as ever they did, and can as often as I shall be called thereunto, after this manner, (viz.)

PROP. I. The Wind is fair.

The Wind is fair, though but little; he comes well, as if he would stand; there∣fore up a hand and loose fore Top-sail in the Top, that the Ships may see we will Sail; Bring Cable to the Capston, have up your Anchor, loose your Fore sail in the Brailes; put abroad our Colours, loose the Misne in the Brailes. Is all our men on board? Those that be on Shore may have a Towe, and be blest with a Ruther; ••f ••r we will stay for no man. Come my Hearts, have up your Anchor, that we may ha•••• a good Prize. Come, Who say Amen? One and all. Oh brave Hearts, the Anc••or 〈◊〉〈◊〉 a Peck; have out fore Top-sail, have out main Top-sail, hawl home the Top-sa•••• Sheets. The Anchor is away, let fall your Fore-sail, hoise up your fore Top-sail, hoise up your main Top-sail; up and loose the Main-sail, and set him; loose Sprit-sail, and Sprit-sail Top-sail. A brave Gale. Bring the fore-Tack to the Cat-head, and trim our Sails quartering; hoise up our small Sails; have out the Misne Top-sail and set him. Now we are clear, and the Wind like to stand; hoise in our Boats before it is too much Sea; aboard Main-tack, aboard Fore-tack, a Lee the Helmne handsomly, and bring her to easily, that she may not stay. Breace the Fore-sail and Fore-top-sail to the Mast, and hawl up the Lee-Bowlings, that the Ship may not stay; pass Ropes for the Boot on the Lee-side, and be ready to clap on your Tackle, and hoise them in; stow them fast. Let go the Lee-Bowling of Fore-sail, and Weather-Braces. Right your Helmnes, hale aft the Fore-sheet, trim the Sails quartering as before: Let go the Sprit sail Breales, and haft of the Sheets; and hoise up the Sprit-sail Top-sail, and other small Sails. Set the main Stay-sail, and fore Top-sail, Stay-sail and Misne Stay-sail, and main Top-sail, Stay-sail and leese in your Boonets, that we may make most of our way. To our Station, and clear your Ropes. Come, get up our steering Sails. The Lee steering Sails of Main-sail, and Main-top-sail, Fore-sail and Fore-top-sail only; for they will set fairest, and draw most away. I have on purpose omitted several Words, by reason I would not trouble the Reader with such indiffe∣rent things as is conceived by all Mariners to be done; as Cooning the Ship, Breasing, Vereing, and haleing aft, and hoising, looring, and the like: but it is to be suppo∣sed all to be done at the same time. Thus have you a brave Ship under all her Sails and Canvas, in her swiftest way of Sailing upon the Sea. Now let us have her right before the Wind.

Right afore the Wind, and a fresh Gale.

The Wind is vered right aft, take in your Fore and Fore-top-sail, Steering-sail, and Fore-top-sail, and Main and Main-top-sail, Stay-sails; for they are becalmed by the after Sails, and will only beat or rub out. The Wind blows a fresh Gale, round aft the Main-sheets, and Fore-sheets, brasse. Square your Yards, take in your Main and Main-top-sail, Steering-sails. Unlease your Bonnets. Take in your Main and Fore-top-gallant-sails, In the Sprit-sail, and Misne Top-sail, let go the Sheets, hale from

the Cholyens, cast off Top-gallant Bowlings. Thus you have all the small Sails in, and furled, when it blows too hard a Wind to bear them.

The wind vereth forward, and scanteth.

The Wind scanteth, vere out some of your Fore and Main-sheets, and Sprisle-sheets, and let go your Weather Braces; tope your Sprit-sail Yard. The Wind still vereth forward; Get aboard the Fore and Main-Tack; cast off your Weather-sheets and Braces: The Sails are in the Wind, hawle off Main and Fore-sheets; the Wind is sharp, hawl forward the main Bowline, and hawl up the Main top-sail, and Fore top-sail Bowline, and set in your Lee-braces, and keep her as near as she will lie. Thus have you all the Sails trimm'd sharp, full, and by a Wind.

The wind blows Frisking.

The Wind blows hard; settle our fore and main Top-sails two thirds of the Mast down. It is more Wind, come, hawl down both Top-sails close. Come, stand by, take in our Top-sails; Let go the Top-sail Bowlines, and Lee-Braces; let go the Lee-Sheets, set in your Weather Braces, spill the Sails, hawl home the Top-sail Clue-lines, square the Yeard. Now the Sail is furled, and you have the Ship in all her low Sails, or Courses at such a time.

It bloweth a Storm.

It is like to over-blow; Take in your Sprit-sail, stand by to hand the Fore-sail. Cast off the Top-sail Sheets, Clue-garnets, Leech-lines, Bunt-lines; stand by the Sheet, and brace; loure the Yeard, and furl the Sail (here is like to be very much Wind) See that your main Hallyards be clear, and all the rest of your Geer clear and cast off. (It is all clear.) Loure the main Yard. All down upon your doone hall; now the Yard is down, hawl up the Clue-garnets, Lifts, Leach-lines, and Bunt-lines, and furle the Sail fast, and fasten the Yards, that they may not travers and gall. Thus have you the Ship a trije under a Mizen.

A very hollow grown Sea.

We make foul weather, look the Guns be all fast, come hand the Mizen. The Ship lies very broad off; it is better spooning before the Sea, than trying or hulling; go reefe the Fore-sail, and set him; hawl aft the Fore-sheet; The Helmne is hard a weather, mind at Helmne what is said to you carefully. The Ship wears bravely stu∣dy, she is before it, and the Sheets are afle and braces; belay the fore doon hall, that the Yard may not turn up; it is done. The Sail is split; go hawl down the Yeard, and get the Sail into the Ship, and unbind all things clear of it, and bring too the Fore-bonnet; make all clear, and hoise up the Fore-yard; hawl aft the Sheets, get aft on the Quarter-Deck, therefore Braces.

Starb••ard; hard up, right your Helmne Port. Port hard, more hands, he cannot put up the Helmne. A very fierce Storm. The Sea breaks strange and dangerous; stand by to hawl off above the Lennerd of the Whipstaff, and help the man at Helmne, and mind what is said to you. Shall we get down our Top-masts? No, let all stand; yet we may have occasion to spoon before the Sea with our Powles. As we mast, get down the Fore-yeard— She scuds before the Sea very well; the Top-mast being aloft the Ship is the holsomest, and maketh better way through the Sea, seeing we have Sea-Room. I would advise none in our condition to strike their Top-masts, before the Sea or under. Thus you see the Ship handled in fair weather and foul, by and learge. Now let us see how we can turn to windward.

The Storm is over, let us turn to Windward.

The Storm is over, set Fore-sail and Main sail; bring our Ship too; set the Misne, and Main Top-sail, and Fore Top-sail. Our Course is E. S. E. the Wind is at South: Get the Starboard-Tacks aboard, cast off our Weather Braces and Lifts; Set in the Lee-Braces, and hawl forward by the Weather Bowlines, and hawl them laught and belaye them, and hawl over the Mizen Tack to Winerd; keep her full, and by as near as she will lye. How Wind you? East. A bad quade Wind. (No near) hard, no near. The Wind vereth to the Eastward still. How Wind you? N. E. hard, no near. The Wind is right in our teeth; no near still. How wind you? N. W. b. N. The Wind will be Northerly, make ready to go about; we shall lay our Course another way, no near, give the Ship way, that she may stay: ready, ready a Lee the Helmne. Vare out there Fore-sheet, cast off your Lee-Braces, brace in upon your Weather Bra∣ces. The Fore-sails is a back stays, hawl Main-sail, hawl, about, let rise the main Tack, cast off your Larboard-Braces, let go main Bowline, and main Top-sail Bow∣line; brace about the Yard, hawl forward by the Larboard-Bowlines; get the main Tack close down, in the Cheese-tree: hawl up the Weather Bowline, and set the Lee Brace of Main and Main Top-sail Yards, and the Sheets is close aft; hawl, of all; hawl; get to fore-Tack, let go fore-Bowline, and fore-Top-sail Bowline; hawl afle the fore-Sheet, hawl taught, the main Bowline, and main Top-sail Bowline; shift the Mizen, tack, hawl bout fore Bowline, and fore Top-sail Bowline; set in the Lee-Braces taught, fore and aft, keep her as near as she will lie.— No near, How Wind you? N. N. E. thus werr no more; no near, keep her full. The Wind is at N. N. E. thus werr no more. (How Wind you?) E. N. E. The Wind is at N. keep her away her Course E. S. E. Cast off the Lee-Braces, and Weather-Bowlines, and set in your Weather-Braces. Vere out the main Sheet, and fore Sheet, loose the Sprit-sail, and Spit-sail, Top-sail, and Mizen Top-sail, and Top-gallant-sails; hoise them up, the Wind vears afle still; let rise the fore-Tack: the Wind's quartering, hawl aft the fore-Sheet, bring him down to the Cat-head with a pass-a-ree; studdy in your Weather Braces; the Wind stands, here. Thus you have the Ship as at first, steering under all her Canvas, quarter Wind, as she did at first, setting Sail. She hath been wrought with all manner of Weather, and all sorts of Winds. Therefore we will draw to the last with a Man of War in Chase and taking of her Prize, and so leave this Practick Part to your Censure.

The Man of War in her Station.

Now we are in our Station, and a good Latitude, hand your Top-sails, and furle your Main-sail and Fore-sail, and brail up the Mizen, and let her lie at Hull, until Fortune appear within our Horizon. Up alaft to the Top-mast-head, and look abroad, young-men; look well to the Westward, if you can see any Ships that have been nipt with the last Easterly Winds. A Sail, a Sail. Where? Fair by us. How stands she? To the Eastward, and is two Points upon her Weather Bow, and hath her Larboard-Tacks aboard. O then she lies close by a wind; we see her upon the Decks plainly. A good man to Helmne. Up young-men, and loose the Fore-sail, Main-sail, and Mizen. Get the Larboard Tacks aboard; have out the Main top-sail, and Fore-top-sail, and loose the Sprit-sail, aloofe. Keep her as neer as she will lie; hawl aft the Sheets, and hawl up your Bowlines laught. Do you see your Chase? How Wind you? E. N. E. Then the Wind is at N. Hoise up your Top-sails as high as you can; have out Sprit-sail, Top-sail, and Mizen Top-sail; hawl home the Sheets, and hoise them up: A young-man and loose the Main-top-gallant-sail, and Fore-top-gallant-sail; hawl home the Sheets, and hoise them up; hoise up Main Stay-sail, and Mizen Stay-sail, and loose the Main-top-sail, and Fore-top-sail; Stay-sail, and set them. It blows a brave chaseing Gale of Wind; The Ship makes brave way through the Sea; we rise her apace; if she keep her Course, we shall be up with her in three Glasses. No near, keep the Chase open with little of the Fore-sail. So, thus, keep her thus. Come ofle all hands, the Ship will Stear the better when you sit all

quiet; by, by her small Sails, for she is too much by the Head. The Chase is a lusty brave Ship. So much the better, she hath the more Goods in her Hold. The Ship hath a great many Guns (no force) it may be she is a Private _____ _____ Port, the Chase is about, come fetch her wack, and we will be about after her. We sayl far better than she; we have her Wack; a Lee the Helmn, about Ship, vere out Fore-sheet. Every man stand handsomly to his business, and mind the Bowlines and Bra∣ces, Tacks and Sheets; hawl Main-sail, hawl about. Let go Main Bowline, Top Bowline, Top-gallant Bowline; Hawl off all, hawl Fore-sail, about, shifts the-Helmn; bring her too, Hawl the Main-sheets close afle, and fore-sheet. Set in the Lee-Braces, hawl too the Bowlines. Thus the Chase keeps close to the Wind; keep her open under our Lee. Gunner, see that you have all things in readiness, and that the Guns be clear; and that nothing pester our decks.—Down with all Hammocks and Cabins that may hinder and hurt us. Gunner, is all our Geer ready? Is the store of Cartrages ready fill'd, all manner of Shot at the Main-mast? Is there Rammers, Sponges, Ladles, Priming-Irons, and Horns, Lyntstocks, Wads, and Water at their several quarters sufficient for them? Be sure that none of our Guns be cloy'd; and when we are in Fight, be sure to load our Guns with Cross-bar and Langrel. Always observe to give Fire when the Word is given. See that there be Half-Pikes and Jave∣lings in a readiness, and all our Small-shot well furnished, and all their Bandaliers fill'd with Powder, and Shot in their Pooches.

See that our Murtherers and Stockfowlers have their Chambers fill'd with good Powder, and Bags of small Shot to load them, that if we should be laid aboard, we might clear our Decks.—Starboard, the Chase pays away more room, Starboard hard; Vere out some of the Main-sheet and Fore-sheet; Cast of the Larboard-Bra∣ces, (Steady) Keep her thus: Well Steer'd; the Chase goes away room, her Sheets are both aft, she is right before the Wind: Stereboard hard; Let rise main Tack, let rise fore-Tack; Hawl afle Main-sheet, hawl afle fore-Sheet. We have a stearn-Chase, but we shall be up with her presently, for we fetch upon her hand-going. The Chase hawls up his Main-sail and furles it; she puts aboard her Waste-clothes; she will fight us. Come up from alow young men, and furle our Main-sail; sling our Main Yard, with the Chains in the Main-top; sling our Fore-yard, put aboard our Waste-Cloaths; he will fight us before the Winde I see; She is full of Men; It is a hot Ship, but deep and foul. Come chearly my Hearts, It is a Prize worth fighting for; The Chase takes in her small Sails; Up aloft and take in our Top-gallant-sails, Sprit-sails, Top-sails, Mizen Top-sail, and furle the Sprit-sail, and get the Yard alongst under the Bowsprit. She puts abroad her Colours, It is Red, White, and Blew; they are Dutch Colours; no force, the worst of Enemies. Boy, up and put abroad St. George's Colours in our Main-top; step oft a hand, and put abroad our bloody Ancient; Call all hands aloft; Come up aloft all hands. They are all up Captain.

Gentlemen, We are here employed and maintained by his Majesty King CHARLES and our Country, to do our Endeavours to keep this Coast from Pyracy and Robbers, and his Majesties Enemies; whereof it is our Fortune to meet this Ship at this time: Therefore I desire you in his Majesties Name, and for the Sake of our Country, and the Honour of our English Nation, and our selves, for every man to behave himself coura∣geous like Englishmen; and not to have the least shew of a Coward: but to observe the Words of Command, and do his utmost endeavour against this barbarous and inhumane Enemy the Dutch, who have treacherously and inhumanely murthered so many of our English Nation, in the East-Indies and other Parts, whose Blood cries for vengeance. Therefore our Quarrel is just, and into Gods hands we commit our Cause, and our selves. So every man to his Quarters, and shew his Courage, and God be with you.

She settles her Top-sails, we are within shot; let all our Guns be loose, In the Tackles and the Ports, all knockt open, that we may be ready to run out our Guns when the Word is given. Up noise of Trumpets, and hail our Prize; she answer∣eth again with her Trumpet: Hold fast Gunner, do not fire till we hail them with our Voices. (Haye, Hooe) Amain for King CHARLES. (Port) edge towards him, he fires his Broad-side upon us. (What chear my Hearts?) Is all well betwixt

Decks? Yea, Yea, only he rackt us through and through. No force, it is his turn next; but give not Fire until we are within Pistol-shot. Port, edge towards him. He plies his small Shot; hold fast Gunner. Port, right your Helmne. We will run up his Side. Starboard a little; Give fire, Gunner. (That was well done.) This Broad-side hath made their Deck thin, but the small Shot at first did gaul us. Clap in some Case-shot in the Guns you are now a loading; Brace too the Fore-top-sail, that we may not shoot a head; He lies broad off to the Southward, to bring his other Broad-side to bear upon us. (Starboard hard.) Get to Larboard Fore-tack; trim your Top-sails, run out your Larboard Guns. He fires his Sterboard Broad-side upon us; He pours in his small Shot. Starboard give not fire until he fall off, that the Prize may re∣ceive our full Broad-side. Steady: Port a little; give fire Gunner: His Fore-mast is by the board. This last Broad-side hath done great Execution. Cheerly my Mates, the day will be ours; He is shot a Head; He bears up before the Wind to stop his Leaks: Keep her thus; Well Steered. Port, Port hard; Bear up before the Wind, that we may give him our Starboard Broad-side. Gunner, Is there great store of Case-shot and Langrel in our Guns? Yea, yea. Port, make ready to board him; Have your Lashers clear, and able men with them. Edge towards him Guns when you give fire; Bring your Guns to bear amongst his Men with the Case-shot. Well steered, we are close on boord. Give fire Starboard; Well done Gunner; They lie Heads and Points aboard the Chase. Come, Aboard him bravely; Enter, Enter. Are you lached fast? Yea, Yea, We will have him before we go here-hence. Cut up the Decks; Ply your Hand-Granadoes and Stink-Pots. He cries out Quarter; Quarter for our Lives, and we will yield up Ship and Goods. Good Quarter is granted, Provided you will lay down all your Arms, open the Hatches, hawl down all your Sails and furle them; loose the Lachings, we will sheer off our Ship, and hoise out our Shallop. If you of∣fer to make any Sail, expect no Quarter for your Lives. Go with the Shallop, and send aboard the Captain, Lieutenant, and Master and Mates, with as many more as the Shallop will carry. So we will leave the Man of War to put his Prisoners down into the Hold, and secure. And so likewise I have shewn you thus much of the Pra∣ctick part of Navigation, in which you may perceive that I have wrought the Ship in all Essays, in Words and proper Sea-Phrases; and if I was at Sea, I should perform it both in Word and Deed: therefore I leave it all to your Judicious Censures. And let not Ignorance, the Arch-enemy of Arts deceive you, and cause you to think that I have writ what I cannot do; but that I can as easily turn him in the Theorick, which way I list, as I can the Ship in the Practick. And so I will conclude with Ovid, when he sailed in the Straight Ionian,

Nothing but Waves we view in Sea where Ships do float, And Dangers lie, huge Whales, and all Fish play: Above our Heads, Heaven's Star-embroidered Coat, Whose Vault contains two Eyes, for Night and Day; Far from the Main, or any Marine Coast, 'Twixt Borean Blasts, and Billows, we are tost.

If Ovid in that straight Ionian Deep Was tost so hard, much more are we on Seas Of larger Bounds, where Staff and Compass keep Their strict observance: Yet in this unease Of Tackling Boards, we so the way make short, That still our Course draws nearer to our Port.

Between the Stream and silver-spangled Skie We rolling climbe, then hurling fall beneath; Our way is Serpent-like, in Meads which lie, That bows the Grass, but never makes a Path: But fitter, like young Maids and Youths together, Run here and there, all where, and none know whether.

Our way we know, and yet unknown to other; And whiles misknown to us, before we dive: The Hand and Compass that governs the Ruther Do often erre, although the Pilots strive With Card and Compass, yet our Reck'nings fall Too narrow, short, too high, too wide, too small.

To discon this, remark when we set Land, Some this, some that do guess, this Hill, that Cape. For some howers our Skill in suspence stand, Terming this Shore, that Head-land Points the Map; Which when mistook, this forg'd Excuse goes clear, O such and such a Land it first did 'pear.

In all which strife, stress'd Sailers have the pain, By drudging, pulling, hawling, standing to it In Cold and Rain, both dry and wet, they strain Themselves, and toyl; none else but they must do it. Both Prow and Poop do answer to the Helm; The Stearsman sings, no Grief his Joy can whelm.

By Night our Watch we set, by day our Sight, And furle our Sails: If Pirates do appear, We rest resolv'd; 'Tis Force makes Cowards fight, Though none more dare, than they that have most fear. It's Courage makes us rash, and Wisdom cold; Yet Wisemen stout, and stung, grow Lion-bold.

Sapientiam Sapiens dirigit, Artes Coartifex, &c.

The Wise-man knows his Wisdom how to use, Th' Artificer, what Art 'tis best to chuse. 'Tis a true Saying, and approved long, The Wise-man is more worthy than the Strong. The Fields he tills, the City he can guide, And for the Ships in Tempests well provide.

THE GEOMETRICAL DEFINITIONS.

THe ARTS, saith Arnobius, are not together with our Minds sent out of the Heavenly Places; but all are found out on Earth, and are in process of time sought and fairly forged by a conti∣nual Meditation. Our poor and needy Lives perceiving some casual Things to happen preposterously, while it doth imitate, attempt, and try, while it doth slip, reform, and change, hath out of these, some Fiduous Apprehension made by small Sci∣ences of Art, the which afterwards by Study are brought to some perfection.

Yet the Practice of Art is not manifest but by Speculative Illustration; because by Speculation we know that we may the better know. And for this cause I chose a Spe∣culative Part; And first of Geometry, that you may the better know the Practice.— To begin then.

I. A Point is that which hath no Parts.

A Point is supposed to be a Thing indivisible, or that cannot be divided into parts; yet it is the first of all Dimensions. It is the Philosopher's Atome. Such a Nothing, as that it is the very Energie of All Things. In God it carrieth its Extremes from Eternity to Eternity; which proceeds from the least imaginable thing, as the Point or Prick noted with the Letter A; and is but only the Terms or Ends of Quan∣tity.

[illustration] geometrical diagram

II. A Line is a supposed Length, with∣out Breadth or Thickness.

A Lines Extremes or Bounds are two Points, as you may see the Line a; b is made by moving of a Point from a to b. A Line is either straight or crooked; and in Geometry of three kinds of Magnitudes, viz. Length, Breadth, and Thickness. A Line is capable of Division in Length only, and may be di∣vided equally in the Point C, or unequally in D, and the like.

[illustration] geometrical diagram

III. The Ends or Bounds of a Line are Points.

You are to understand, the Ends or Bounds of a finite Line is A, B, as before: but in a Circular Line it is otherwise; for there the Point in its Motion returneth again to the Place where it first began, and so maketh the Line infinite.

[illustration] geometrical diagram

IV. A Right Line is the shortest of all Lines, drawn from any two of the said Points,

As you may see the Right Line AB straight, and equal between the Points A and B, with∣out bowing, which are the Bounds thereof.

[illustration] geometrical diagram

V. A Superficies is a Longitude, having only Latitude.

Superficies is That which hath only length and breadth, whose Terms and Limits are two Lines. In the first kind of Magnitude the Motion of a Point pro∣duceth a Line: So in the second kind of Magnitude, the Motion of a Line produ∣ceth a Superficies. This is also capable of two dimensions, as the length AB and CD, and the breadth AC and BD; and may be divided into any kind or number of Parts,

[illustration] geometrical diagram

VI. The Extremes of a Superficies are Lines.

As the Ends of a Line are Points, so the Bounds or Extremes of a Superficies are Lines; as before, you may see the Ends of the Lines AB, and BD, and DC, and CA.

VII. A Plain Superficies lieth equally between his Lines.

So the Superficies ABCD is that which lieth direct and equally between his Lines. And whatsoever is said of a Right Line, the same is meant of a Plain Superficies.

VIII. An Angle is when two Lines are extended upon the same Superficies, so so that they touch one another in a Point, but not directly.

As you may see the two Lines AB and BC incline one towards the other, and touch one the other, in the Point B. In which Point, by reason of the bowing in∣clination of the said Lines, is made the Angle ABC. And here note, That an Angle is most commonly signed by three Letters, the middlemost whereof sheweth the Angular Point, as in this Figure, when we say Angle ABC, you are to understand the very Point at B.

[illustration] geometrical diagram

IX. A Right Angle is that which is produced of a Right Line, falling upon a Right Line, and making two equal Angles on each side the Point where they touch each other.

As upon the Right Line CD sup∣pose there doth stand another Right Line AB, in such sort that it ma∣keth the Angles on either side there∣of; namely, the Angle ABD on the one side: equal to the Angle ABC on the other side; then are either of the two Angles Right An∣gles; and the Right Line AB, which standeth erected upon the Right Line CD, without bowing or inclining to either part thereof, is a Perpendicular to the Line CD.

[illustration] geometrical diagram

X. An Obtuse Angle is that which is greater than a Right Angle.

So the Angle CBE is an Obtuse An∣gle, because it is greater than the Angle ABC, which is a Right Angle; For it doth not only contain that Right An∣gle, but the Angle ABE also, and therefore is Obtuse.

[illustration] geometrical diagram

XI. An Acute Angle is less than a Right Angle.

Therefore you may see the Angle EBD is an Acute Angle; for it is less than the Right Angle ABD, in which it is contained by the other Acute Angle ABE.

XII. A Limit or Term is the End of every Thing.

As a Point is the Limit or Term of a Line, because it is the End thereof; so a Line likewise is the Limit and Term of a Superficies, and a Superficies is the Limit and Term of a Body.

XIII. A Figure is that which is contained under one Limit or Term, or many.

As the Figure A is contained under one Limit or Term, which is the round Line; also the Figures B and C are contained under four Right Lines: likewise the Figure E is contained un∣der three Right Lines, which are the Limits or Terms thereof; and the Figure F under five Right Lines: And so of all other Figures.

[illustration] geometrical diagram

And here note, We call any plain Superficies, whose Sides are unequal (as the Figure F) a Plot, as of a Field, Wood, Park, Forest, and the like.

XIV. A Circle is a plain Figure contained under one Line, which is called a Circumference; unto which all Lines drawn from one Point within the Figure, and falling upon the Circumference thereof, are equal one to the other.

As the Figure AECF is a Circle contained under the Crooked Line AECD, which Line is called the Circumference. In the middle of this Figure is the Center or Point B, from which Point all Lines drawn from the Circumference are equal, as the Lines BA, BE, BD, BC; and this Point B is called the Center of the Cir∣cle.

[illustration] geometrical diagram

XV. A Diameter of a Circle is a Right Line drawn by the Center thereof, and ending at the Circumference, on either side dividing the Circle into two equal Parts.

So the Line ABC in the former Figure, is the Diameter thereof, because it passeth from the Point A on the one side, and passeth also by the Point B, which is the Center of the Circle; and moreover, it divideth the Circle into two equal parts, namely, AEC being on one side of the Diameter, equal to AFC on the other. And this Observation was first made by Thales Milesius; For, saith he, if a Line drawn by the Center of any Circle do not divide it equally, all the Lines drawn from the Center of that Circle, from the Circumference, cannot be equal.

XVI. A Semicircle is a Figure contained under the Diameter, and that part of the Circumference cut off by the Diameter.

As in the former Circle, the Figure AFC is a Semicircle, because it is contained of the Right Line ABC which is the Diameter, and of the crooked Line AFC, being that part of the Circumference which is cut off by the Diameter: Also the part AEC is a Semicircle.

XVII. A Section or Portion of a Circle, is a Figure contained under a Right Line, and a part of the Circumference, greater or less than a Semicircle.

So the Figure ABC, which consisteth of the part of the Circumference ABC, and the Right Line AC, is a Section or Portion of a Circle, greater than a Semi∣circle.

[illustration] geometrical diagram

Also the other Figure ACD, which is contained under the Right Line AC, and the parts of the Cir∣cumference ADC, is a Section of a Circle less than a Semicircle. And here note, That by a Section, Seg∣ment, Portion, or part of a Circle, is meant the same thing, and signifieth such part as is greater or lesser than a Semicircle: So that a Semicircle cannot properly be called a Section, Segment, or part of a Circle.

XVIII. Right-lined Figures are such as are contained under Right Lines.

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XIX. Three-sided Figures are such as are contained under three Right Lines.

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XX. Four-sided Figures are such as are contained under four Right Lines.

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XXI. Many-sided Figures are such a•• have more Sides than four.

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XXII. All Three-sided Figures are called Triangles.

And such are the Triangles ABC.

[illustration] geometrical diagram

XXIII. Of Four-sided Figures, A Quadrat or Square is that whose Sides are equal, and his An∣gles right, as the Figure A.

[illustration] geometrical diagram

XXIV. A Long Square is that which hath right An∣gles, but unequal Sides, as the Figure B.

[illustration] geometrical diagram

XXV. A Rhombus is a Figure Quadrangular, having equal Sides, but not equal or right Angles, as the Figure C.

[illustration] geometrical diagram

XXVI. A Rhomboides is a Figure whose opposite Sides are equal, and whose opposite Angles are also equal: but it hath neither equal Sides, nor equal Angles, as the Figure D.

[illustration] geometrical diagram

XXVII. All other Figures of Four Sides are called Trapezia's.

[illustration] geometrical diagram

XXVIII. Such are all of Four Sides, in which is ob∣served no equality of Sides or Angles, as the Figures L and M, which have neither equal Sides nor Angles, but are described by all Adventures, without the observa∣tion of any Order.

[illustration] geometrical diagram

XXIX. Parallel or Aequi-distant Right Lines, are such which being in one and the same Superficies, and produced infinitely on both sides, do never in any part concur; as you may see by the two Lines AB, CD.

[illustration] geometrical diagram

XXX. A Solid Body is that which hath Length, and Breadth, and Thickness, as a Cube or Die; and the Limits and Extremes of it are Superficies, as the Figure I.

[illustration] geometrical diagram

XXXI. Axis is the Diameter about which the Sphere or Globe is turned.

[illustration] geometrical diagram

XXXII. The Poles of a Sphere are the Extremes or Ends of the Diame∣ter, and are terminated in the Superficies of the Sphere.

[illustration] geometrical diagram

XXXIII. A Sphere is defined by Euclid to be made, when the Diameter of a Semi-circle remaining fixed, the Semicircle is turned about, till it be returned to the Place whence it began to move at first.

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Geometrical Theoremes.

I. ANY two Right Lines crossing one another, make the contrary or vertical Angles equal. Euclid. 15. 1.

II. If any Right Lines fall upon two parallel Right Lines, it maketh the outward Angles of the one, equal to the inward Angles of the other; and the two inward opposite Angles, on contrary sides of the falling Line, also equal. Euclid 29. 1.

III. If any Side of a Triangle be produced, the outward Angle is equal to the in∣ward opposite Angles, and all the three Angles of any Triangle are equal to two Right Angles. Euclid 32. 1.

IV. In Aequi-angled Triangles all their Sides are proportioned, as well such as con∣tain the equal Angles, as also the subtendent Sides.

V. If any four Quantities be proportional, the first multiplied in the fourth, produceth a Quantity equal to that which is made by multiplication of the second in the third.

VI. In all Right-angled Triangles, the Square of the Side subtending the Right Angle, is equal to both the Squares of the containing sides. Euclid 47.1.

VII. All Parallellograms are double to the Triangles that are described upon their Basis, their Altitudes being equal. Euclid 41.1.

VIII. All Triangles that have one and the same Base, and lie between two Parallel Lines, are equal one to the other. Euclid 37.1.

Geometrical Problems.

PROBLEM I. Ʋpon a Right Line given, how to erect another Right Line which shall be perpendicular to the Right Line given.

THE Right Line given is AB, upon which from the Point E it is required to erect the Perpendicular EH. Opening your Com∣pass at any convenient distance, place one Foot in the assigned Point E, and with the other make the two Marks C and D, equal on each side the Point E; then opening your Compasses again to any other convenient distance wider than the former, place one Foot in C, and with the other describe the Arch GG; also (the Compasses remaining at the same distance) place one Foot in the Point D, and with the other describe the Arch FF. Then from the Point where those two Arches in∣tersect or cut each other (which is at H) draw the Right Line HE, which shall be Perpendicular to the given Right Line AB, which was the thing required to be done.

[illustration] geometrical diagram

PROBL. II. How to erect a Perpendicular on the end of a Right Line given.

LEt AB, be a Line given, and let it be required to erect the Perpendicular AD. First upon the Line AB, with your Compasses opened to any small distance, make five small Divisions, beginning at A, noted with 1, 2, 3, 4, 5. Then take with your Compasses the distance from A to 4, and place one Foot in A, and with the other describe the Arch e e: Then take the distance from A to 5, and placing one Foot of the Compasses in 3, with the other Foot describe the Arch h h, cutting the former Arch in the Point D: Lastly, from D draw the Line DA, which shall be perpendicular to the given Line AB.

[illustration] geometrical diagram

This operation is grounded upon this Conclusion, viz. These three Numbers 3, 4, and 5, make a Right-angled Tri∣angle, which is very necessary in many Mechanical Operations, and easie to be re∣membred.

PROBL. III. How to let fall a Perpendicular upon any Point assigned, upon a Right Line given.

THe Point given is C, from which Point it is required how to draw a Right Line which shall be perpendicular to the gi∣ven Right Line AB.

[illustration] geometrical diagram

First from the given Point C, to the Line AB, draw a Line by chance, as CE, which divide into two equal parts in the Point D. Then placing one Foot of the Compasses in the Point D, with that distance DC, describe the Semicircle CFE, cutting the given Line AB in the Point F. Lastly, If from the Point C you draw the Right Line CF, it shall be a Perpendicular to the given Line AB, which was required.

PROBL. IV. How to make an Angle equal to an Angle given.

LEt the Angle given be ACB, and let it be required to make another Angle equal thereunto.

First draw the Line EF at plea∣sure; then upon the given Angle at C (the Compasses opened to any di∣stance) describe the Arch AB; and also upon the Point F, the Compasses unaltered, describe the Arch DE; Then take the distance AB, and set the same from E to D; Lastly draw the Line DF: So shall the Angle DFE be equal to the given Angle ACB.

[illustration] geometrical diagram

PROBL. V. A Right Line being given, how to draw another Right Line which shall be parallel to the former, at any distance required.

THe Line given is AB, unto which it is required to draw another Right Line pa∣rallel thereunto, at the distance AC or DB. First open your Compasses to the distance AC or BD; then placing one Foot in A, with the other de∣scribe the Arch C; also (at that distance place one Foot in B, and with the other describe the Arch D. Lastly, draw the Line CD, that it may only touch the Arches C and D: So shall the Line CD be parallel to the Line AB, and at the distance required.

[illustration] geometrical diagram

PROBL. VI. To divide a Right Line into any number of equal Parts.

LEt AB be a Right Line given, and let it be required to divide the same into five equal Parts.

[illustration] geometrical diagram

First, From the given Line A, draw the Line AC, making any Angle from the end of the given Line which is at the Point B. Then draw the Line BD equal to the Angle CAB. Then from the Points A and B, set off upon these two Lines any Number of equal parts, being less by one than the Parts into which the Line AB is to be divided, which in this Example must be 4. Then draw small Lines from 1 to 4, from 2 to 3 twice, and from 1 to 4, &c. which Lines crossing the given Line AB, shall divide it into five equal Parts, as was required.

PROBL. VII. A Right Line being given, how to draw another Right Line parallel thereunto, which shall also pass through a Point assigned.

LEt AB be a Line given, and let it be required to draw another Line paralle thereunto, which shall pass through the given Point C. First, Take with your Com∣passes the distance from A to C, and place∣ing one Foot thereof at B, with the other describe the Arch DE; then take in your Compasses the whole Line AB, and place one Foot in C, and with the other describe the Arch FG, crossing the former Arch in the Point H: Then if you draw the Line CH, it shall be parallel to AB, the thing required.

[illustration] geometrical diagram

PROBL. VIII. Having any three Points given which are not scituated in a Right Line, How to find the Center of a Circle which shall pass directly through the three Points given.

THe three Points given are A, B, and C; now it is required to finde the Center of a Circle whose Circumference shall pass through the three Points given.

[illustration] geometrical diagram

First open your Compasses to any distance greater than half the distance between B and C; then place one Foot in the Point B, and with the other describe the Arch FG; then the Compasses remaining at the same distance, place one Foot in the Point C, and with the other turn'd about make the marks F and G in the former Arch, and draw the Line FOG at length, if need be.

In like manner open your Compass at a di∣stance greater than half AB; Place one Foot in the Point A, with the other describe the Arch HK: Then the Compasses remaining at the same distance, place one Foot in the Point

C, and turning the other about, make the marks HK in the former Arch. Lastly, draw the Right Line HK, cutting the Line FG in O, so shall O be the Center, upon which you may describe a Circle at the distance of OA, and it shall pass directly through the three given Points ABC, which was required.

PROBL. IX. How to describe a Circle in a Triangle, that shall only touch the three Sides; and to find the Centre.

LAy down the Triangle ABC, the three Sides equal; then divide the Sides of the Triangle AB in two equal parts, as at E, and draw the Line CE; and likewise di∣vide BC, and draw the Line AD; and where they cross one the other, as at O, that is the Center: Therefore put one Point of the Compasses in the Center O, and extend the other to either side, and describe the Cir∣cle GF, which will only touch the Sides A BC of the Triangle.

[illustration] geometrical diagram

PROBL. X. How to lay down a Triangle in a Circle, and to find the Center of the Circle in the Triangle.

DRaw the three Sides of a Triangle AB C, it is no matter if they be equal or not; then put one Foot of your Compasses in the Point B, open the other to more than half the length of the greatest side, as to C; and with that distance describe the Arch FHDG; and so removing the Compasses to C, cross the former Arch at F and D, and draw the Line DF. Again, the Compasses at the same distance, put one Foot in A, and describing a small Arch, cross the for∣mer Arch at H and G; and laying a Ruler over the Intersections of these two Arches at H and G, draw the Line GH; and where these two Lines cross one the other, as at K, that is the Center of the Triangular Points. From it extend the Compasses to either of the Points, and describe the Circle ABC, and the Triangle will be within the Circle.

[illustration] geometrical diagram

PROBL. XI. Any three Right Lines being given, so that the two shortest together be longer than the third, To make thereof a Triangle.

LEt it be required to make a Triangle of these three Lines A, B, and C, the two shortest whereof, viz. A and B together, are longer than the third C.

[illustration] geometrical diagram

First draw the Line DE equal to the given Line B; then take with your Com∣passes the Line C, and setting one Foot in E, with the other describe the Arch FF: also take with your Compasses the given Line A, and placing one Foot in D, with the other describe the Arch GG, cutting the former Arch in the Point K. Lastly, from the Point K, if you draw the Lines KE and KD, you shall constitute the Triangle KDE, whose Sides shall be equal to the three given Sides ABC.

PROBL. XII. Having a Right Line given, How to make a Geometrical Square, whose Sides shall be equal to the Right Line given.

THe Line given is RI, and it is required to make a Geometrical Square whose Sides shall be equal to the Line RI, First draw the given Line RI, then (by the first and se∣cond Problem) upon the Point B raise the Per∣pendicular BC, making the Line BC equal to the given Line RI also: Then taking the said RI in your Compasses, place one Foot in C, with the other describe an Arch at D; The Compass at the same distance, set one Foot in A, and cross the former Arch at D; then draw the Lines D, C and DA, which shall conclude the Geometrical Square ABCD, which was required.

[illustration] geometrical diagram

PROBL. XIII. Two Right Lines being given, How to find a third which shall be in pro∣portion unto them.

LEt the given Lines be A and B; and it is required to find a third Line which shall be in proportion unto them.

[illustration] geometrical diagram

First draw two Right Lines, making any Angle at pleasure, as the Lines LM and MN, making the Angle LMN: Then take the Line A in your Compasses, and set the length thereof from M to E; also take the Line B, and set the Length

thereof from M to F, and also from M to H: Then draw the Right Line EH, and then from the Point F draw the Line FG parallel to EH. So shall MG be the third Proportional required: For Arithmetically say,

As ME to MH: So is MF to MG 18.

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PROBL. XIV. Three Right Lines being given, To find a fourth in proportion to them.

THe three Lines given are ABC, unto which it is required to find a fourth Pro∣portional Line. This is to perform the Rule of Three. As in the last Problem, you must draw two Right Lines, making any Angle at pleasure, as the Angle EFG; then take the Line A in your Compass, and set it from F to I; then take the Line B in your Compas∣ses, and set that from F to K; then take the third given Line in your Com∣passes, and set that from F to H, and from that Point H draw the Line H L, parallel to IK; So shall the Line FL be the third Proportional required.

[illustration] geometrical diagram

Note, That these Lines are taken off a Scale, that is divided into 20 parts to an Inch: To do it Arithmetically say,

As FI is to FK: So is FH to FI.

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Here note, That in performance of the last Problem, That the first and third Terms, namely the Lines A and C, must be set upon one and the same Line, as here upon the Line FE, and the second Line B must be set upon the other Line FG, upon which Line also the fourth Proportion will be found.

PROBL. XV. How to work the Rule of Proportion by a Scale of equal Parts, and such other Conclusions as are usually wrought in Lines and Numbers, as in Mr. Gunter's 10 Prob. 2 Chap.

The Scale of Inches is a Scale of equal Parts, and will perform (by protraction upon a Flat or Paper) such Conclusions as are usually wrought in Lines and Numbers, as in Mr. Gunter's 10 Prob. 2 Chap. Sector, may be seen, and in others that have writ in the same kind. This way Mr. Samuel Foster hath directed in the I Chap. of his Posthumus Fosteri.

An Example in Numbers like his Tenth Probl.

As 16 to 7: So is 8 to what?

Here because the second Term is less than the first, upon the Line AB, I set AC the first Term 16, and the second Term AD 7, both taken out of the Scale of equal parts: thence also the third Number 8 being taken, with it upon the Center C, I de∣scribe the Arke EF, and from A draw the Line AE, which may only touch the same Arke; then from D, I take DG, the least distance from the Line AE, and the same measured in the same Scale of equal parts, gives 3½, the fourth Term required.

[illustration] geometrical diagram

But if the second Term shall be greater than the first, then the form of working must be changed, as in the following Example.

EXAMPLE.

As 7 to 16: So 21 to what? — 48.

Upon the Line AB, I set the second Term 16, which is here supposed to be AD; then with the first Term 7 upon the Center D, I describe the Arke GH, and draw AG that may just touch it: Again, having taken 21 out of the same Scale, I set one Foot of that Extent upon the Line AB, removing it until it fall into such a place, as that the other Foot being turned about, will justly touch the Line AG before-drawn; and where (upon such Conditions) it resteth, I make the Point C; then measuring AC upon your Scale, you shall find it to reach 48 Parts, which is the fourth Num∣ber required.

The form of Works (although not so Geometrical) is here given, because it is here more expedite than the other by drawing Parallel Lines; but in some Practice the other may be used. I have been the more large upon this, because in the following

Treatise I shall quote some more remarkable Places in Posthuma Fosteri: and the So∣lution of Proportions must be referred thither, the form of their Operations being the same with this. In them therefore shall only be intimated what must be done in ge∣neral, the particular way of working being here directed.

PROBL. XVI. To divide a Right Line given, into two parts, which shall have such pro∣portion one to the other as two given Right Lines.

THe Line given is AE, and it is required to divide the same into two parts, which shall have such proportion one to the other, as the Line C hath to the Line D.

First, From the Point A draw a Right Line at pleasure, making the Angle BAE; then take in your Compasses the Line C, and set it from A to F; and also take the Line D, and set it from F to B, and draw the Line BE: Then from the Point F draw the Line FG, parallel to BE, cutting the given Line AE in the Point G: So is the Line A B divided into two parts in the Point G, in proportion to the other, as the Line C is to the Line D.

[illustration] geometrical diagram

Arithmetically, let the Line AE contain 40 Perches or Foot, and let the Line C be 20, and the Line D 30 Perches; and let it be required to divide the Line AE into two parts, being in proportion one to the other, as the Line C is to the Line D.

First, Add the Lines C and D together, their Sum is 50: Then say by the Rule of Proportion, If 50 (which is the Sum of the two given Terms) give 40, the whole Line AE; What shall 30 the greater given Term give? Multiply and divide, and you shall have in the Quotient 24 for the greater part of the Line AE; which being taken from 40, there remains 16 for the other part AG: For

As AB is to AE: So is BF to EG.

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PROBL. XVII. How to divide a Triangle into two parts, according to any proportion as∣signed, by a Line drawn from any Angle thereof; and to lay the les∣ser part unto any Side assigned.

LEt ABC be a Triangle given, and let it be required to divide the same by a Line drawn from the Angle A, into two parts, the one bearing proportion to the other, As the Line F to the Line G; And that the lesser part may be towards the Side AB.

[illustration] geometrical diagram

By the last Problem divide the Base of the Triangle BC in the Point D, in propor∣tion as the Line F is to the Line G (the lesser part being set from B to D.) Lastly, draw the Line AD, which shall divide the Triangle ABC in proportion as F to G.

As the Line F, is to the Line G:

So is the Triangle ADC, to the Triangle ABD.

PROBL. XVIII. The Base of the Triangle being known, To perform the foregoing Problem Arithmetically.

SUppose the Base of the Triangle BC be 45, and let the Proportion into which the Triangle ABC is to be divided, be as 2 to 4. First add the two proportional Terms together, 2 and 4, which makes 6; then say by the Rule of Proportion, If 6, the Sum of the Proportional Term, give 45 (the whole Base BC) What shall 4 the greater Term given? Multiply and divide, and the Quotient will give you 30, for the greater Segment of the Base DC, which being deducted from the whole Base 45, there will remain 15 for the lesser Segment BD.

As 2/4 is to 45: So is 4 DC 30.

〈 math 〉〈 math 〉

PROBL. XIX. How to divide a Triangle (whose Area or Content is known) into two Parts, by a Line drawn from an Angle assigned, according to any Proportion required.

LEt the Triangle ABC contain 9 Acres, and let it be required to divide the same into two Parts, by a Line drawn from the Angle A, the one to contain 5 Acres, and the other 4 Acres. First, measure the whole length of the Base, which suppose 45; Then say, If 9 Acres the quantity of the whole Triangle, give 45 the whole Base, What parts of the Base shall 4 Acres give? Multiply and divide, the Quotient

will be 20 for the lesser Segment of the Base BD; which being deducted from 45 the whole Base DC, then draw the Line AD, which shall divide the Triangle ABC according to the proportion required.

If 9 Acres give 45, What shall 4 Acres give?

〈 math 〉〈 math 〉

PROBL. XX. How to divide a Triangle given into two parts, according to any Propor∣tion assigned, by a Line drawn from a Point limited in any of the Sides thereof; and to lay the greater or lesser part towards any Angle assigned.

THe Triangle given is ABC, and it is required from the Point M to draw a Line that shall divide the Triangle into two parts, being in proportion one to the other, as the Line N is to the Line O; and to lay the lesser part to∣wards B.

[illustration] geometrical diagram

First, from the limited Point M draw a Line to the opposite Angle at A; then divide the Base BC in proportion as O to N, which Point of Division will be at E; then draw ED parallel to AM: Lastly, from D draw the Line DM, which will divide the Triangle into two parts, being in Proportion one to the other, as the Line O is to the Line N.

PROBL. XXI. To perform the foregoing Problem Arithmetically.

IT is required to divide the Triangle ABC, from the Point M, into two parts in proportion as 5 to 2.

First divide the Base BC according to the given Proportion; then because the lesser Part is to be laid towards B, measure the distance from M to B, which suppose 32: Then say by the Rule of Proportion, If MB 32, give EB 16, what shall AR 28 (Perpendicular of the Triangle) give? Multiply and divide, the Quotient will be 14, at which distance draw a Parallel Line to BC, namely D; then from D draw the Line DM, which shall divide the Triangle according to the required Proportion.

PROBL. XXII. How to divide a Triangle (whose Area or Content is known) into two Parts, by a Line drawn from a Point limited, into any Side thereof, according to any number of Acres, Roods, and Perches.

IN the foregoing Triangle ABC, whose Area or Content is 5 Acres 1 Rood, let the limited Point be M in the Base thereof; and let it be required from the Point M to draw a Line which shall divide the Triangle into two parts between Johnson and Powell, so as Johnson may have 3 Acres 3 Roods thereof, and Powell may have 1 Acre and 2 Roods thereof.

First, Reduce the quantity of Powell's, being the lesser, into Perches (Observe, 160 square Poles contains 1 Acre, half an Acre contains 80 Perch, a quarter or one Rood 40 Perch.) which makes 240. Then considering on which side of the limited Point M this part is to be laid, as towards B, measuring the part of the Base from M to B 32 Perch, whereof take the half, which is 16, and thereby divide 240, the Parts be∣longing to Powell, the Quotient will be 15, the length of the Perpendicular DH, at which Parallel-distance from the Base BC, cut the Side AB in D, from whence draw the Line DM, which shall cut off the Triangle DBM, containing 1 Acre 2 Roods, the part belonging to Powell: Then the Trapezia ADMC (which is the part be∣longing to Johnson) contains the residue, namely, 3 Acres 3 Roods.

〈 math 〉〈 math 〉

PROBL. XXIII. How to divide a Triangle according to any Proportion given, by a Line drawn parallel to one of the Sides given.

The following Triangle ABC is given, and it is required to divide the same into two Parts, by a Line drawn parallel to the Side AC, which shall be in pro∣portion one to the other, as the Line I is to the Line K.

First (by the 16th Problem) divide the Line BC in E, in proportion as I to K; then (by the 27th Problem following) find a mean Proportional between BE and BC, which let be BF, from which Point F draw the Line FH, parallel to AC, which Line shall divide the Triangle into two parts, viz. the Trapezia AHFC, and the Triangle HFB, which are in proportion one to the other, as the Line I is to the Line K.

PROBL. XXIV. To perform the foregoing Problem Arithmetically.

LEt the Triangle be ABC, and let it be required to divide the same into two parts, which shall be in proportion one to the other, as 4 to 5, by a Line drawn Parallel to one of the Sides.

First let the Base BC, containing 54, be divided according to the propor∣tion given; so shall the lesser Segment BE con∣tain 24, and the greater EC 30; Then find out a mean Proportional be∣tween BE 24, and the whole Base BC 54, by multiplying 54 by 24, whose Product will be 1296; the Square Root thereof is 36, the mean Proportional sought, w^ch is BF. Now if BF 36 give BE 24, what shall AD 36? The Answer is HG 24, at which distance draw a Parallel Line to the Base, to cut the Side AB in H, from whence draw the Line HF, Parallel to AC, which shall divide the Triangle as was required.

[illustration] geometrical diagram

〈 math 〉〈 math 〉

PROBL. XXV. To divide a Triangle of any known Quantity into two Parts, by a Line Parallel to one of the Sides, according to any Number of Acres, Roods, and Perches.

THe Triangle given is ABC, whose Quantity is 8 Acres, 0 Roods, and 16 Per∣ches; and it is desired to divide the same (by a Line drawn up parallel to the Side AC) into two Parts, viz. 4 Acres, 2 Roods, 0 Perches; and 3 Acres, 2 Roods, and 16 Perches.

First, Reduce both Quantities into Perches (as it is hereafter taught) and they will be 720, and 576; then reduce both these Numbers by abbreviation into the least proportional Term, viz. 5 and 4; and according to that proportion, divide the Base BC 54 of the given Triangle in E: then seek the mean Proportion between BE and BC, which Proportion is BF 36, of which 36 take the half, and thereby divide 576, the lesser Quantity of Perches, the Quotient will be HG 32, at which Pa∣rallel-distance from the Base, cut off the Line AB in H, from whence draw the Line HF parallel to the Side AC, which shall divide the Triangle given, according as it was required.

〈 math 〉〈 math 〉

PROBL. XXVI. From a Line given, To cut off any Parts required.

THe Line given is AB, from which it is re∣quired to cut off 3/7 Parts.

[illustration] geometrical diagram

First, draw the Line A C, making any Angle, as CAB; then from A set off any 7 equal Parts, as 1, 2, 3, 4, 5, 6, 7; and from 7 draw the Line 7 B. Now because is to be cut off from the Line B, there∣fore from the Point 3, draw the Line 3 D, parallel to 7 B, cutting the Line AB in D; So shall AD be the 3/7 of the Line AB, and DB shall be 4/7 of the same Line.

As 7 is to AB: So is A 3 to AD.

PROBL. XXVII. To find, a Mean Proportional between two Lines given.

IN the following Figure, let the two Lines given be A and B, between which it is re∣quired to find a Mean Proportional. Let the two Lines A and B be joyned toge∣ther in the Point E, making one Right Line as CD, which divided into two equal Parts in the Point G; upon which Point G, with the distance GC or GD, describe the Semicircle CFD: Then from the Point E, where the two Lines are joyned toge∣ther, raise the Perpendicular EF: So shall the Line EF be a Mean Proportional between the two given Lines A and B. For,

As ED	is to EF:	So EF	to CF.
9	12	12	16

PROBL. XXVIII. How to finde two Lines, which together shall be equal in Power to any Line given; And in Power the one to the other, according to any Propor∣tion assigned.

IN this Figure let CD be a Line given, to be divided in Power, as the Line A is to the Line B.

[illustration] geometrical diagram

First, divide the Line CD in the Point E, in proportion as A to B (by the 16th Probl.) Then divide the Line CD into two equal Parts in the Point G, and on G, at the distance GD or GC, describe the

Semicircle CFD, and upon the Point E raise the Perpendicular EF, cutting the Se∣micircle in F. Lastly, draw the Line CF and DF, which together in Power will be equal to the Power of the given Line CD; and yet in Power one to the other, as A to B.

PROBL. XXIX. How to divide a Line in Power according to any Proportion given.

FIrst, Divide the Line C D in the Point E, in pro∣portion as A to B: Then di∣vide the Line CD in two e∣qual Parts in the Point G, and upon G as a Center, at the di∣stance CD, describe the Se∣micircle CFD, and on E raise the Perpendicular of EF, cutting the Semicircle in F: Then draw the Line CF and DF, and produce the Line CF to H, till FH be equal to FD, and draw the Line D. H. Lastly, draw the Line FK, parallel to DH: Then shall the Line CD be divi∣ded in K; so that the Square of CK shall be to the Square of KD, as CE to ED, or as B to A.

[illustration] geometrical diagram

PROBL. XXX. How to enlarge a Line in Power, according to any Proportion assigned.

IN the Diagram of the 28th Problem, let CE be a Line given, to be enlarged in Power as the Line B to the Line C.

First (by the 16th Problem) find a Line in proportion to the given Line CE, as B is to C, which will be CD; upon which Line describe the Semicircle CFD, and on the Point E erect the Perpendicular EF: Then draw the Line CF, which shall be in power to CE, as C to B.

PROBL. XXXI. To enlarge or diminish a Plot given, according to any Proportion required.

LEt ABCDE be a Plot given, to be diminished in Power as L to K.

Divide one of the Sides, as AB in Power as L to K, in such sort that the Power of AF may be to the Power of AB, as L to K; then from the Angle A draw Lines to the Point C and D. That done, by F draw a Parallel to BC, to cut AC in G, as FG: again, from G draw a Parallel to DC, to cut AD in H. Lastly, from H draw a Parallel to DE, to cut AE in I: So shall the Plot AFGHI be like ABCDE, and in proportion to it, as the Line L to the Line K, which was required.

Also if the lesser Plot was given, and it was required to make it in proportion to it as K to L; then from the Point A draw the Lines AC and AD at length; also increase AF and AI: That done, enlarge AF in Power as K to L, which set from A to B; then by B draw a Parallel to FG, to cut AC in C, as B C: likewise from C draw a Parallel to GH, to cut AD in D: Lastly, a Parallel from D to HI, as DE, to cut AI being increased in E; so shall you include the Plot ABC DE, like AFGHI, and in proportion thereunto, as the Line K is to the Line L, which was required.

[illustration] geometrical diagram

PROBL. XXXII. How to make a Triangle which shall contain any Number of Acres, Roods, and Perches, and whose Base shall be equal to any (possible) Num∣ber given.

LEt it be required to make a Triangle which shall contain 6 Acres, 2 Roods, 25 Perches, whose Base shall contain 50 Perches. You must first reduce your 6 Acres 2 Roods, and 25 Perches, all into Perches, after this manner.

First, Because 4 Roods makes 1 Acre, multiply your 6 Acres by 4. makes 24; to which add the 2 odd Roods, so have you 26 Roods in 6 Acres 2 Roods; then because 40 Perches makes 1 Rood, multiply your 26 by 40, which makes 1040, to which add the 25 Perches, and you shall have 1065, and so many Perches are contained in 6 Acres, 2 Roods, and 25 Perches.—Now to make a Triangle that shall contain 1065 Perches, and whose Base shall be 50 Perches, do thus; double the number of Per∣ches given, namely 1065, and they make 2130; then be∣cause the Base of the Trian∣gle must contain 50 Perches, divide 2130 by 50, the Quo∣tient will be 42 ⅗ which will be the length of the Perpen∣dicular of the Triangle. This done, from any Scale of equal Parts, lay down the Line BC equal to 50 Perches; then upon C raise the Perpendicular CE, equal to 42 ⅗ Perches, and draw the Line AE, pa∣rallel to BC; then from any Point in the Line AE, as from G, draw the Line BG, and GC, including the Triangle BGC, which shall contain 6 Acres, 2 Roods, 25 Perches, which was required.

[illustration] geometrical diagram

PROBL. XXXIII. How to reduce a Trapezia into a Triangle, by a Line drawn from any Angle thereof.

THe Trapezia given is ABCD, and it is required to reduce the same into a Triangle.

[illustration] geometrical diagram

First, extend the Line D C, and draw the Diagonal BC; then from the Point A draw the Line AF, pa∣rallel to CB, extending it till it cut the Side DC in the Point F. Lastly, from the Point B draw the Line BF, constituting the Triangle FBD, which shall be equal to the Trapezia ABDC.

And so I have concluded what I did intend of Geometrical Problems: Neither had I gone so far as I have, in regard Mr. William Leybourn hath ingeniously and very fully demonstrated them in his First Book of his Compleat Surveyor. But no Book (as I remember) now extant of Navigation, hath the foregoing Problems so large. Be∣sides, I shall direct (in the following Treatise) the Mariner to Survey any Plantation or Parcel of Land very exactly and easily, by his Sea-Compass.

Notes

• Here the Variation Compass is to be placed.
• Phocilides.