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.

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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