only longitude and latitude.

VI. The extremes, or limits of a superficies are lines. VII. A Plain-superficies is that which lies equally be. twixt its lines.

VIII. A Plain-angle is the inclination of two lines the one to the other, the one touching the other in the same plain, yet not lying in the same strait-line.

IX. And if the lines which contain the angle, be right.

Jines, it is called a right-lined angle.

Plate I.
Fig. 1.

Fig. 2.

X. When a right-line CG, standing upon a right-line AB, makes the angles on either fide thereof, CGA, CGB, equal one to the other, then both those equal angles are right-angles; and the right-line CG, which standeth on the other, is termed a Perpendicular to that (AB) whereon it standeth.

Note, When several angles meet at the fame point (as at G) each particular angle is described by three letters; whereof the middle letter sheweth the angular point, and the towo other letters the lines that make that angle: As the angle which the right-lines CG, AG make at G, is called CGA, or AGC.

XI. An Obtufe-angle is that which is greater than a right-angle; as DGB.

XII. An Acute-angle is that which is less than a rightangle; as DGA.

XIII. A Limit, or Term, is the end of any thing.

XIV. A Figure is that which is contained under öne

or more terms.

XV. 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 the one to the other.

XVI. And that point is called the center of the circle. XVII. A Diameter of a circle is a right-line drawn thro' the center thereof, and ending at the circumference on either fide, dividing the circle into two equal parts.

XVIII. A Semicircle is a figure which is contained under the diameter and that part of the circumference which is cut off by the diameter.

In the circle EABCD, E is the center, AC the diameter,
ABC the femicircle.
XIX. Right-lined figures are such as are contained
under right-lines.

XX. Three-sided or trilateral figures are such as are contained under three right-lines.

XXI. Four-fided or quadrilateral figures are such as are contained under four right-lines.

XXII. Many-fided figures are such as are contained under more right-lines than four.

XXIII. Of trilateral figures, that is, an equilateral triangle, which hath three equal fides; as the triangle ABC. Fig. 3.

XXIV.

XXIV. An Isosceles, is a triangle which hath only two fides equal; as the triangle ABC. Plate I. Fig. 4.

XXV. A Scalenum, is a triangle whose three fides are all unequal; as ABC. Fig. 5.

XXVI. Of these trilateral figures, a right-angled triangle is that which hath one right-angle, as the triangle ABD. Fig. 5.

XXVII. An Amblygonium, or obtuse-angle triangle, that is which hath one angle obtufe ; as ABC. Fig. 5.

XXVIII An Oxgonium, or acute-angled triangle, is that which hath three acute angles; as ABC. Fig. 4.

An equiangular, or equal-angled figure is that whereof all the angles are equal. Two figures are equiangular, if the several angles of the one figure be equal to the several angles of the other. The same is to be understood of equilateral figures.

XXIX. Of Quadrilateral, or four-fided figures, a square is that whose fides are equal, and angles right; as ABCD. Fig. 6.

XXX. A Figure on the one part longer, or a long square, is that which hath right-angles, but not equal fides; as ABCD. Fig. 7.

XXXI. A Rhombus, or diamond-figure, is that which has four equal fides, but is not right-angled; as ABCD. Fig. 8.

XXXII. A Rhomboides, is that whose opposite sides, and oppofite angles are equal; but has neither equal nor right angles; as ABCD. Fig. 9.

XXXIII. All other quadrilateral figures besides these are called trapezia, or tables; as GNDH. Fig. 10.

XXXIV. Parallel, or equi-distant right-lines are such, which being in the same superficies, if infinitely produced, would never meet as AD and BC. Fig. 11.

XXXV. A Parallelogram is a quadrilateral figure, whose opposite sides are parallel, or equi-distant; as ABCD. Flate I Plate I. Fig. 7.

XXXVI. In a Parallelogram AGEL, Fig. 9. when a diameter AE, and two lines BK, CF, parallel to the fides, cutting the diameter in one and the same point D, are drawn, so that the Parallelogram be divided by them into four Parallelograms; those two LD, DG, through which the diameter does not pass, are called complements; and the other two CB, KF, through which the diameter passeth, the Parallelograms standing about the

diameter.

A Problem is, when something is proposed to be done or effected. A Theorem is, when something is proposed to be demonStrated.

A Corollary is a Confectary, or some consequent truth gained from a preceding demonstration.

A Lemma is the demonstration of some premise, whereby the proof of the thing in hand becomes the shorter.

1.

FR

Postulates or Petitions.

Rom any given point to any other given point, to draw a right-line.

2. To produce a finite right-line, strait forward continually.

3. Upon any center, and at any distance, to describe

a circle.

1.

Axioms.

Hings equal to the fame thing, are also equal one to the other.

Things

As A=B=C. Therefore A=C; or therefore all A,B,C, are equal the one to the other.

Note, When feveral quantities are joyned the one to the other continually with this mark, the first quantity is by virtue of this axiom equal to the last, and every one to every one: In which cafe we often abstain from citing the axiom, for brevity's fake; altho the force of the consequence depends

thereon.

2. If to equal things you add equal things, the wholes will be equal.

3. If from equal things you take away equal things, the things remaining will be equal.

4. If

1

4. If to unequal things you add equal things, the wholes will be unequal.

5. If from unequal things you take away equal things, the remainders will be unequal.

6. Things which are double to the same third, or to equal things, are equal one to the other. Understand the fame of triple, quadruple, &c.

7. Things which are half of one and the fame thing, or of things equal, are equal the one to the other. Conceive the fame of fubtriple, subquadruple, &c.

8. Things which agree together, are equal one to the other.

The converse of this axiom is true in right-lines and angles, but not in figures, unless they be like.

Moreover, magnitudes are faid to agree, when the parts of the one being apply'd to the parts of the other, they fill up an equal or the same place.

9. Every whole is greater than its part.

10. Two right-lines cannot have one and the fame

fegment (or part) common to them both.

11. Two right-lines meeting in the fame point, if they be both produced, they shall neceffarily cut one another in that point.

12. All right-angles are equal the one to the other. 13. If a right-line EF (Plate 1. Fig. 11.) falling on two right-lines, AD, BC, make the internal angles on the same side, GFE, FEG, less than two right-angles, those two right-lines produced shall meet on that fide where the angles are less than two right-angles.

14. Two right-lines do not contain a space. 15. If to equal things you add things unequal, the excess of the wholes shall be equal to the excess of the additions.

16. If to unequal things equal be added, the excess of the wholes shall be equal to the excess of those which were at first.

17. If from equal things unequal things be taken away, the excess of the remainders shall be equal to the excess of what was taken away.

18. If from things unequal things equal be taken away, the excess of the remainders shall be equal to the excess of the wholes.

19. Every whole is equal to all its parts taken to gether.

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