fix'd before his Apollonius.

God always acts Geometrically.

H

OW great a Geometrician art thou, O Lord! For while this Science has no Bounds; while there is for ever room for the Discovery of New Theorems, even by Human Faculties; Thou art acquainted with them all at one View, without any Chain of Confequences, without any Fatigue of Demonftrations. In other Arts and Sciences our Understanding is able to do almost nothing; and, like the Imagination of Brutes, feems only to dream of fome uncertain Propofitions: Whence it is that in so many Men are almost fo many Minds. But in thefe Geometrical Theorems all Men are agreed: In thefe the Human Faculties appear to have some real Abilities, and thofe Great, Wonderful and Amazing. For thofe Faculties which feem of almost no force in other Matters, in this Science appear to be Efficaci ous, Powerful, and Successful, &c. Thee therefore do I take hence occafion to Love, Rejoice in, and Admire; and to long for that Day, with the Earneft Breathings of my Soul, when thou shalt be pleased, out of thy Bounty, out of thy Immenfe and Sacred Benignity, to allow me to behold, and that with

a 2

with a pure Mind, and clear Sight, not only thefe Truths, but thofe alfo which are more numerous, and more important; and all this without that continual and

painful Application of the Imagination, which we difcover thefe withal, &c.

Mathematical Notes or Abbreviations.

The Note for Equality. So ab fignifies that a and b are equal.

+ The Note for Addition. So ab fignifies the Sum of a and b together.

The note for Subtraction. So a-b fignifies the Difference between a and b.

x The Note for Multiplication. So axbora b fig. nifies a multiplied by b.

:: The Note for equality of Proportion. So A: B:: ab fignifies that A bears the fame Proportion to B, that a bears to b.

The Note of continued Proportion. So A, B, C fignifies that A bears the fame Proportion to B, that B bears to C.

q The note for a Square. So CBq fignifies the Square of the Line CB.

c The Note for a Cube. Cube of the Line C B.

So CBc fignifies the

THE

I.

The Elements of EUCLID.

A

BOOK I.

DEFINITIONS.

Point is a Mark in Magnitude, which is
[fuppofed to be] indivifible.

That is, which cannot be divided fo
much as in Thought. A Point is the be-
ginning, as it were, of all Magnitude,
as Unity is of Number.

2. A Line is a Magnitude which hath Length only, and wants all Breadth; forafmuch as it is understood to be produced from the flowing of a Point.

3. Points are the Terms of a Line.

4. A right Line, is that which lies evenly betwixt its Fig. 1. Terms.

Or as Archimedes: A right Line is the leaft of all those which have the fame Terms; or, is the fhortest of all thofe which can be drawn betwixt two Points.

Or as Plato hath it: A right Line is that whose Extremes hide all the reft; [that is, when the Eye is placed in a Continuation of the Line.]

The Senfe is the fame in all. The Inftrument whereby right Lines are described, is [called] a Rule; which whether it be ftrait or not you may know by this Tryal.

Describe a Line according to the Rule; then turning the Rule fo, that that which before was the Right-hand End may now become the Left-hand End, apply it again to the Line before described; if it doth now entirely

B

fall

Table 1.

Fig. 2, 4.

fall in with the Line, the Rule is ftrait; if not, the Rule is not ftrait. The Reafon hereof depends on Axiom 13. 5. A Surface is a Magnitude which hath only Lengt. and Breadth.

It hath two Dimenfions therefore: And is understood to be produc'd by the flowing of a Line.

6. Lines are the Extremes of a Surface.

7. A Plane, or a plain Surface, is that which lies evenly betwixt its extreme Lines.

Or as Hero, that, to all the Parts whereof a right Line may be accommodated.

For it is produc'd from the Motion of a right Line. Or, A plain Surface is that whofe Extremes any of them hide all the rest, [the Eye being placed in a Continuation of the Surface.]

Or, It is the leaft of all Surfaces which have the fame Terms. The Senfe is the fame in all.

Euclid hath not here defined a Body or Solid, because he was not yet about to treat concerning it. But left any one fhould want the Definition thereof, take it here thus: A Body is a Magnitude long, broad and deep. A Body therefore hath three Dimenfions, a Surface two, a Line one, a Point none.

8. A plain Angle is the mutual Inclination to each other of two Lines, which touch one another in a Plain, but fo as not to make one Line.

Therefore the two Lines AB, CA, touching one another in A, but fo as not to make one Line, conftitute an Angle.

9. The Sides or Legs of an Angle are the Lines which make the Angle.

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10. The Vertex or Top of an Angle is the Point (A) in which the Legs do meet and touch one another.

Note, that a fingle Angle is defigned by one Letter put at the Top: When there are more at one Point, they are defigned by three Letters, the middlemoft of which denotes the Top of the Angle; and many times alfo by one Letter interpos'd betwixt the Sides near the Top. So in Fig. 5. the Angle made by the Lines BA, CA, is defigned either by three Letters BAC, or by one only O. 11. Angles are called Equal, if when the Tops of them are laid upon one another, the Sides of one agree with the Sides of the other. But unto this it is not required that the Sides fhould be of an equal Length.

3

12. They

12. They are called Unequal when the Top and one Side agreeing, the other doth not agree; and that is called the Greater, whofe Side falls without. So the Angle B AE is greater than the Angle BAC.

An Angle is not diminifh'd or increas'd by the Diminution or Augmentation of the Sides that include it.

Fig. s

13. A right-lin❜d Angle is that which right Lines con- Fig. 2,4 Atitute; a curvi-linear, which crooked Lines; a mixt one, that which a right Line and a crooked one make.

14. When the right Line [CA] ftanding upon the Fig. 6. Right one [BF] leans unto neither Part, and therefore makes the Angles on both Sides equal, CAB CAF, both of the equal Angles are called Right ones: But the right Line CA which ftands upon the other, is called a perpendicular Line, or barely a Perpendicular, A right Angle may alfo be defined thus. Fig. 6. A right Angle is that (BAC) to which on the other Side an equal one arifeth (CAF) if you produce or draw forth a Side, as (BA).

Two Rules fo joined as to contain a right Angle, make an Inftrument, which is called a Square. Pythagoras was the Inventor of it, as Vitruvius affirmeth, c. 2. 1. 9. So So great is the Ufe and Force of a right Angle in Framing, Meafuring, and Strengthning all Things, that nothing almoft can be done without it. The Proof of a Square is made thus: Apply the Side of it, A E to the right Line A F, and defcribe the right Line CA along the other Side. Then turning the Square towards B, if on both Sides it agrees to the right Lines CA, AB, you may know that it is true and exact. The Reason hereof appears from the 14th Definition it felf.

15. The Angle BAC, which is greater than the right Fig. 7. one FAC, is called an obtufe Angle.

16. The Angle (LAC) which is lefs than the right Fig. 8 Angle (FAC) is called an Acute one.

17. A plain Figure is a plain Surface, bounded on every Side with one or more Lines.

18. A Circle is a plain Surface contained within the Fig. 9. Compafs of one Line called the Circumference; from which Line all the right Lines that can be drawn unto one certain Point, within the contained Space (A), are equal.

19. That Point is called the Center.